Basin boundary metamorphoses: Changes in accessible boundary orbits
Deskripsi Singkat
Nuclear Physics B (Proe. Suppl.) 2 (1987) 281300 NorthHolland, Amsterdam
281
BASIN BOUNDARY METAMORPHOSES: CHANGES IN ACCESSIBLE BOUNDARY ORBITS* Celso G R E B O G I , a E d w a r d O T T a,b and James A. Y O R K E c
University of Marylan~ CollegePark, MD 20742, USA Received 5 November 1985 Revised manuscript received 28 April 1986
Basin boundaries sometimes undergo sudden metamorphoses. These metamorphoses can lead to the conversion of a smooth basin boundary to one which is fractal, or else can cause a fractal basin boundary to suddenly jump in size and change its character (although remaining fractal). For an invertible map in the plane, there may be an infinite number of saddle periodic orbits in a basin boundary that is fractal. Nonetheless, we have found that typically only one of them can be reached or "accessed" directly from a given basin. The other periodic orbits are buried beneath infinitely many layers of the fractal structure of the boundary. The boundary metamorphoses which we investigate are characterized by a sudden replacement of the basin boundary's accessible orbit.
I. Introduction 1.1. Preliminary metamorphoses
discussion o f basin boundary
Basin b o u n d a r i e s * for dynamical systems with multiple attractors can be either smooth or fractal; they can a p p e a r to be very convoluted; or they can be simple in appearance. As an example illustrating this, fig. 1 shows some computergenerated pictures of basins of attraction for the d a m p e d driven p e n d u l u m , 0 + v~ + to2 sin 0 = f c o s t. Given the variety of basin b o u n d a r y structure observed in fig. 1, it is natural to ask how basins of attraction c h a n g e as a system parameter varies continuously. In particular, are there qualitative changes o f basin boundaries as the parameter passes
t h r o u g h certain critical values? This paper is devoted to a study of such changes, which we call basin boundary metamorphoses. In particular, basin b o u n d a r y m e t a m o r p h o s e s can occur as a result of a homoclinic tangency of the stable and unstable manifolds o f a saddle orbit on the basin boundary. Perhaps the simplest nontrivial system exhibiting the p h e n o m e n a of interest here is the H t n o n map, and so our numerical investigations are devoted to this case. More generally we believe that the type o f m e t a m o r p h o s e s we investigate are the typical large scale metamorphoses of maps in two dimensions and in m a n y ordinary differential e q u a t i o n systems (e.g., the forced d a m p e d p e n d u lum). T h e H t n o n m a p is Xn+ 1 =
A  x ,2 _ j y . ,
Y,,+l = x,,, ~Laboratory for Plasma and Fusion Energy Studies. band Departments of Electrical Engineering and of Physics and Astronomy. c Institute for Physical Science and Technology and Department of Mathematics. • *The basin of attraction of an attractor is the set of initial conditions which approach the attractor as time approaches positive infinity. The boundary of the closure of this region is invariant under the dynamics and is called the basin boundary. *Reprinted from Physica 24D (1987) 243
09205632/87/$03.50 ©Elsevier Science Publishers B.V. (NorthHolland Physics Publishing Division)
(la) (lb)
where the p a r a m e t e r J is the determinant of the J a c o b i a n matrix of the map. (Hence J is the net c o n t r a c t i o n ratio for any finite area in the x  y plane u n d e r the action of the map.) F o r eqs. (1), as A increases past the value A x =  ( J + 1)2/4, a s a d d l e  n o d e bifurcation occurs in which one at
C. Grebogi et al. / Basin boundary metamorphoses
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(a)
(b) r
3
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3
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(d)
3
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.
....
". . . . . . .
x
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Fig. 1. Basins of attraction for the damped driven pendulum, 0"+ v0 + 2 sin 8 = j cos t. To generate each one of these pictures, we choose over 1 000000 initial conditions in a grid. The differential equation is then integrated for each initial condition until the orbit is close to one of the possible attractors. For a given picture, if an orbit goes to the attractor indicated below, a black dot is plotted corresponding to that initial condition. Thus, the black region in each picture is essentially the basin of attraction for that attractor. (a) o~= 1, v = 0.1, f = 1.2; attractor in the black basin: O = 2.2055, 0'= 0.3729. (b) w = 1, v = 0.1, f = 2.0: attractor in the black basin: 0 = 0.8058; # = 0.9375. (c) oa = 1, u = 0.2, f = 1.8; attractor in the black basin: 0 = 2.9320; 0 = 0.5200. (d) ~ = 1, ~, = 0.3, f = 1.85; attractor in the black basin: 0 = 0.4186, 0 = 0.5744.
t r a c t i n g f i x e d p o i n t a n d o n e s a d d l e fixed p o i n t a r e
A~ t h e f i x e d p o i n t a t t r a c t o r m o v e s a w a y f r o m t h e
c r e a t e d (cf. t a b l e I). F o r A j u s t s l i g h t l y l a r g e r t h a n
s a d d l e f i x e d p o i n t . T h e s a d d l e fixed p o i n t lies o n
A 1 there are two attractors, one at
(Ixl, lYl)= ~ ,
the boundary
of the basin of attraction of the
a n d t h e o t h e r t h e a t t r a c t i n g fixed p o i n t ( f o r A < A~,
f i x e d p o i n t a t t r a c t o r . I n fact, t h e s t a b l e m a n i f o l d
i n f i n i t y is t h e o n l y a t t r a c t o r ) . A s A i n c r e a s e s p a s t
o f t h e s a d d l e is t h e b a s i n b o u n d a r y (cf, fig. 2a). A s
C Grebogi et al. / Basin boundary metamorphoses Table I Notation for special values of the Hdnon map parameter A
Notation Event signified A,,
Asf A rf
A,*
A,,. ,, ÷ ~
saddlenode bifurcation of a periodn orbit. smoothfractal metamorphosis fractalfractal metamorphosis chaotic attractor has a crisis by colliding with an accessible periodn saddle on the basin boundary tangency of the stable and unstable manifolds of an accessible periodn saddle on the basin boundary tangency of the unstable manifold of a periodn saddle with the stable manifold of a periodn + 1 saddle
Where notation first appears section 1.1 section 1.1 section 1.1 section 1.3
section 2.1
section 3.3
A increases further, the fixed point attractor undergoes various bifurcations, most prominently a perioddoubling cascade. We shall be interested in following the evolution of the basin boundary that is created at A = A 1 as the parameter A increases. Figs. 2 show the basin of attraction for the point at oo in black for three parameter values: (A, J ) = (1.150,0.3) (fig. 2a); (A, J ) = (1.395,0.3) (fig. 2b); and (A, J)=(1.405,0.3) (fig. 2c). This figure was obtained by starting with a grid of 640 × 640 initial conditions. Initial conditions in the basin of attraction of infinity were determined by seeing which ones yield orbits with large x and y values after a large number of iterations. Those which did are plotted as dots. These dots are so dense that they fill up the black region in the figure. For the case of fig. 2a the boundary of the basin of infinity (i.e., the black region) is apparently a smooth curve (the stable manifold of the period1 saddle). All initial conditions in the white region generate orbits which remain bounded and are generally asymptotic to the fixed point attractor labeled in the figure. For the case of fig. 2b the basin boundary shows considerably more structure, and, in fact, magnifications of it reveal that it is fractal (cf. refs. 14 for discussion of fractal basin boundaries). Initial conditions in the white region are generally asymptotic to a period
283
two attractor (labeled by two dots in the figure) which results from a period doubling bifurcation of the fig. 2a fixed point attractor*. One of our purposes in this paper will be to describe how a smooth basin boundary, as in fig. 2a, can become a fractal basin boundary, as in fig. 2b, as a system parameter is varied continuously (e.g., as A is varied from 11150 to 1.395 with J fixed at 0.3). It is found that the boundary becomes fractal following a homoclinic tangency of the period one saddle. Furthermore, we find that this is preceded by a sequence of heteroclinic intersections of stable and unstable manifolds of an infinite number of unstable periodic saddles (section 3). It will be shown that this sequence of events, preceding the actual metamorphosis, determines to a large extent the boundary structure following the metamorphosis to fractal. Henceforth, we shall denote by A u the value of A for which the boundary becomes fractal as A increases through Asf, and we shall call the accompanying conversion a smoothfractal basin boundary metamorphosis. We conjecture that the dimension of the boundary is greater than one following Asr. We also find another type of basin boundary metamorphosis in which the extent of a fractal basin can grow discontinuously by suddenly sending a Cantor set of thin fingers into the territory of another basin. This is illustrated by a comparison of figs. 2b and 2c which have slightly different values of the parameter A and the same J value: A1.395 in fig. 2b and A=1.405 in fig. 2c. Comparing these two figures we see that the basin *The "basin boundary" under study is usually the boundary of the basin of the attractor at infinity. Note that saddlenode bifurcations in the white region do occur and can lead to the presence of more than one attractor there. The white region would be the union of the basins of the attractors not at infinity. Most commonly, however, additional attractors created by saddlenode bifurcations are only seen over a comparatively small range in A before they are destroyed by a. crisis [1] (for example, see section 3.1). Thus it is common for the white region to be the basin of a single attractor. We have not observed any saddlenode bifurcations occurring in the black region.
284
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4
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O' ~).
2
{
0
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2
2
f
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2
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0
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2
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Fig. 2. Basin of infinity in black for the Hdnon map with J = 0.3 and (a) A = 1.150, (b) A = 1.395 (the numerals 1,2, 3, 4 denote the accessible period4 saddle in the sequence in which the points are visited), and (c) ,4 = 1.405 (1,2, 3 denote the accessible period3 saddle).
of attraction of infinity (the black region) has enlarged by the addition of a set of thin filaments, some of which appear well within the interior of the white region of fig. 2b (in particular, note the region  1.0 7z x ~  0.3, 2.0 7zy 7z 5.0 in fig. 2c). We call the change exemplified by the transition from fig. 2b to fig. 2c a fractalfractal basin boundary metamorphosis, and we denote the value of A at Which it occurs Art. As A is decreased toward A ff, the filaments become even thinner,
their area in the frame of the figure going to zero, but they remain in position, not contracting to the position of the basin boundary shown in fig. 2b. Hence, the position of the boundary changes discontinuously although the area of the wlaite and black regions apparently changes continuously. (This type of discontinuous jump in the boundary also occurs at A = A~f.) To conclude this subsection we note that Moon and Li [4], in work done at the same time as ours,
C Grebogi et al. / Basin boundary metamorphoses have also noted that basin boundaries can become fractal as a result of a homoclinic crossing of stable and unstable manifolds, and they have demonstrated how Melnikov's criterion can be used to find the smoothfractal metamorphosis condition for a periodically forced Dufling equation. Our analysis, on the other hand, reveals several basic underlying aspects not discussed by Moon and Li [4]: the essential role of accessible orbits, the fractalfractal metamorphosis, jumps in the location of the boundary at metamorphoses, crisis transfers, and the phenomena discussed in section 3.
1.2. Preliminary discussion of accessible boundary
points Both types of metamorphoses lead to a change in the accessible saddle orbits on the basin boundary. The concept of accessible boundary points and its importance for understanding the structure of fractal basin boundaries will be discussed in section 2.
285
saddle and its stable manifold*. (No other points are accessible. The stable manifold is dense in the boundary but most boundary points are not accessible.) In particular, the transition from (i) to (ii) is considered in section 3 and leads us to a study of the macroscopic collision processes that take place as a system parameter is varied. Considering this transition, as well as the transition from (ii) to (iii), we find that a nonattracting saddle set lying in the closure of the basin grows via a complex chain of crossings of stable and unstable manifolds. (This saddle set is a saddle plus the closure of all the crossing points of the stable and unstable manifolds of that saddle.) Eventually this inflated saddle set collides with the boundary, resulting in the basin boundary metamorphosis described above. Alternatively, the saddle set may collide with the attractor (an interior crisis). These virtually invisible saddle sets play a key role in the observed metamorphoses. In the appendix we describe our numerical techniques for finding accessible boundary saddle orbits.
Definition. A boundary point p is accessible from a region if there is a curve of finite length connecting p to a point in the interior of the region such that no point of the curve lies in the boundary except for p. As we will describe later (sections 2 and 3), we find that i) For A 1 < A < Asf, the set of boundary points accessible from the white region is the saddle fixed point and its stable manifold (and this is the entire boundary). ii) For Asf < ,4 < Aff, the set of boundary points accessible from the white region is a period4 saddle and its stable manifold. (No other points are accessible. The Stable manifold is dense in the boundary but most boundary points are not accessible.) iii) For A > A n, the set of boundary points accessible from the white region is a period3
1.3. Crises transfers One reason for studying basin boundary metamorphoses is that, as a system parameter is varied, a chaotic attractor can be destroyed by colliding with an unstable orbit on its basin boundary (a boundary crisis [1]). Basin boundary metamorphoses affect boundary crises. We find that basin boundary metamorphoses can lead to changes in the type of crisis which ultimately destroys the attractor, and we call this phenomenon a crisis transfer. To limit the scope of the problem of describing these metamorphoses we examine a particularly simple scenario. A typical saddlenode bifurcation *The filamentsmentionedabove are the stable manifoldof the period3 saddle. The period3 saddle came into existence much earlier (namelyat A ' 1.16) but, beforeArt, it was in the interior of the whiteregion.
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creates a new attractor (namely the node), and the saddle is on the basin boundary. Often the attractor takes a perioddoubling route to becoming chaotic. The chaotic attractor then grows as a parameter is varied, collides with the boundary (a boundary crisis), and the attractor and its basin are suddenly destroyed. We wish to describe the major basin boundary metamorphoses that are seen in this scenario, from saddlenode bifurcation to the final boundary crisis. If one consideres the onedimensional quadratic map (i.e., eqs. (1) with J = 0 ) , then it is well known that the attractor in txl < ~ no longer exists for A > 2. What happens is that, as A increases, the chaotic attractor widens, until, at A = 2, it touches the unstable period one point on the basin boundary (a boundary crisis [1]). For J, a nonzero fixed but small value, the J = 0 boundary crisis which occurs as A is increased is qualitatively unchanged. But for J larger a qualitative change does occur. This is illustrated in figs. 3a and 3b. These figures are obtained at fixed values of J ( J = 0.05 in fig. 3a and J = 0.30 in fig. 3b) by plotting a large number of consecutive x coordinates for an orbit generated by the map, eqs. (1), as the parameter A is raised in small increments (vertical axis). For figs. 3 the map is iterated 103 times for each value of A. Fig. 3a shows two clear crisis events: one, an interior crisis, at A = A~1.78, in which the chaotic attractor suddenly widens due to a collision with a period3 saddle orbit which is inside the attractor's basin; and another, a boundary crisis at A = A~ = 1.8874, in which the chaotic attractor is destroyed due to a collision with a saddle fixed point on the basin boundary. The crisis at A = A~ is essentially the same event as occurs for J = 0 and A = 2 in the 1 D quadratic map. As J is raised above the value 0.05 (which applies to fig. 3a), the values A~ and A]~ move closer together and eventually a crisis transfer occurs. The result is that the final boundary crisis killing the chaotic attractor becomes a period3 crisis, in which the attractor collides with the unstable period three orbit, which is now on the basin boundary. Figs. 4a and 4b show the
(a) PERK BOUN CRISI
PEF INT CRI
2
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0
I
2
x
(b) i
I.O
0.5
J
0.0
I
0.5
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Fig. 3. Bifurcation diagrams (A versus x) for the H6non map with (a) J = 0.05 and (b) J = 0.3. For a given value of A, an initial condition (x, v) is chosen and the x coordinate of its orbit is plotted after the first several thousand iterates are discarded.
chaotic attractor, its basin boundary, and the boundary saddle with which it collides (labeled by arrows) for a case ( J = 0.05, A = A~) in which the collision is with a period1 saddle (fig. 4a) and for a case ( J = 0.3, A = A3c) in which the collision is with a period3 saddle (fig. 4b). As the above discussion implies, for J = 0.05, the period3 sad
C. Grebogi et al. / Basin boundary metamorphoses
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(b)
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Fig. 4. (a) The chaotic attractor and its basin at the crisis, A = 1.8874, for J = 0.05. The period1 saddle on the basin boundary is indicated by an arrow. (b) The chaotic attractor and its basin at the crisis, A = 2.12467, for J = 0.3. The elements of the period3 saddle on the basin boundary are indicated with arrows.
die is not on the basin boundary, but, for J  0.3, it is.
2. Accessible points 2.1. Accessible points for the Hbnon map As mentioned earlier, a point p on the basin boundary is accessible from the white (black) region if one can draw a finite length curve from a point in the interior of the white (black) region to p in such a way that the curve touches the boundary only at that one accessible boundary point. As an illustration, consider the stable and unstable manifolds of the period1 saddle after they have crossed, i.e., A > A~' where A~' denotes the value of A at which these manifolds become tangent. We say l u x ls holds when the period1 unstable manifold crosses the period1 stable manifold. (The test will make it clear which saddle is under discussion and also
which branch of an unstable manifold is under investigation.) A schematic illustration is given in fig. 5. This figure shows that, as a result of the homoclinic intersection, lu x ls, a series of progressively longer and thinner tongues of the basin of the attractor at infinity (black regions in fig. 2b) accumulate on the outer portion of the stable manifold through the period1 saddle. [The (n + 1)st tongue is the preimage of the nth tongue.] A finite length curve connecting a point in the interior of the white region and the period1 saddle would have to circumvent all the tongues accumulating on the stable manifold of the period1 saddle. This, however, is not possible because the length of the nth tongue approaches infinity as n ~ oo. Thus the period1 saddle is not accessible from the white region. By contrast the period1 saddle is always accessible from the black region. This behavior is evident from fig. 2b. In fig. 2b we have also labeled a period4 saddle orbit. Evidently this saddle lies on the basin boundary and is accessible from the white region. We also note that there are tongues accumulating on the stable manifold of the period4 saddle but they do so on the side away from the finite attractor (cf. fig. 6a).
C Grebogi et al. / Basin boundary metamorphoses
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TONGUE
ATTRACTOR,,~
PERtOD I
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~'TONGUE 2 ~TONGUEI
SADDLE
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After the metamorphosis, however, the boundary includes the period4 saddle. In this sense we can say that as A increases past A~ the basin boundary suddenly jumps inward into the white region. Note, however, that the area of the basin of infinity in any finite regien of the plane, numerically, at least, appears to change smoothly as A increases through A~'. As soon as 1u x 1s holds, the basin boundary is fractal, i.e. A~ = A s f .
TONGUE 4
Fig. 5. Schematic illustration of tongues accumulating on the stable manifold of the period1 saddle.
Hence, a white region is left open for a curve to join the period4 saddle to the interior of the white region. There are also infinitely many other saddle periodic orbits lying in the basin boundary (cf. section 2.2 and section 3). These, however, are not accessible either from infinity or from the finite attractor because the tongues of the two basins accumulate on both sides of those orbits. This is illustrated in fig. 6b where we show successive blowups of the basin structure in fig. 2b in a small region around one element of the period5 saddle. Thus as A increases past A~' the boundary saddle which is accessible from the inner region suddenly changes from being the period1 saddle to being the period4 saddle. Just before the metamorphosis the period4 saddle is in the interior of the white region. As A approaches A~' from below, the distance between the basin boundary and the period4 saddle does not approach zero.
Indeed, the fractal dimension of the boundary for the case shown in fig. 2b has been numerically determined to be d = 1.530 + 0.006. We have done this by using the numerical technique of "uncertainty exponent measurement" introduced in ref. 3. Next consider the transition from fig. 2b to fig. 2c. We find that the essential difference between these two figures is that, for the parameters of fig. 2b, the accessible saddle is the period4, while for fig. 2c the period4 saddle is no longer accessible from the finite attractor, but a period3 attractor is, as indicated in fig. 2c. Furthermore, the transition between the two cases occurs instantaneously at A = A~" when the stable and unstable manifolds of the period4 saddle are tangent*. Thus A~ = A~. Fig. 7 shows a schematic illustration of the tangency of the numericaly determined period4 stable and unstable manifolds at A = A~ = 1.396.
*One branch of the unstable manifold of the period4 saddles slices through the basin boundary at a Cantor set of points. Here, we are interested in the other branch of the unstable manifold (i.e., the branch which is entirely in the white region in fig. 2b). When it becomes tangent and then crosses the stable manifold of the period4 saddle, the period4 saddle will no longer be accessible.
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Fig. 6. (a) Blowup of the basin structure in a small region around one element of the accessible period4 saddle orbit for J = 0.3 and A = 1.395 (magnification  105). (b) Three blowups around one element of the period5 saddle orbit for the same parameter values. The sequential blowups illustrate how an inaccessible orbit has tongues of both basins accumulating on it from both sides.
As expected, this value is between those applying for figs. 2b and 2c. Fig. 8a shows a blowup around an element of the period3 saddle illustrating that it is accessible. Fig. 8b shows blowups of the basin structure in a small region around one element of the period4 saddle for a case where A > A ~. The period4 saddle, which was accessible for A < A~' (fig. 6a), no longer is when A > A~' (fig. 8b).
2.2.
A simple example
In this subsection we describe a model onedimensional map, x.~: 1 = F(x.), which illustrates the concept of accessible boundary points in a
particularly simple way. The specific map* which *This m a p is similar to the onedimensional version of a twodimensional m a p treated in ref. 2 (i.e., set J = 0 in ref.. 2), a n d is similar to 1D maps treated by Mira [4], and Takesue and K a n e k o [4].
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Fig. 7. Schematic diagram of the stable and unstable manifolds of the period4 saddle orbit at tangency A = A,~ = 1.396. This diagram shows how the unstable manifold of one element of the period4 saddle orbit accumulates on the unstable manifold of the next element of the same orbit, which in turn accumulates on the next, and so on cyclically.
we investigate is piecewise linear and is illustrated in fig. 9a. This map has only two attractors, namely, x = + oo and x =  o o . In addition, as we shall show, it also has the following additional properties: i) There is a C a n t o r set basin b o u n d a r y separating the basins of the two attractors. ii) The only periodic orbit on the basin b o u n d a r y which is accessible from the attractor basin of x = + ~ ( x =  o o ) is the unstable fixed point at x= +1 (x=l). iii) Each b o u n d a r y point which is accessible f r o m the x = + m (x =  m ) basin is mapped to x = + 1 (x =  1) in a finite n u m b e r of iterates. iv) The C a n t o r set basin b o u n d a r y contains an infinite n u m b e r of unstable periodic orbits, and, except for the points x = _+1, all of these are inaccessible f r o m either basin. T o d e m o n s t r a t e (i)(iv) we first note that, for any point x , > 1, the map may be expressed ( x , + ~  1) = 5 ( x , 
1).
(2)
T h u s any initial condition in x > 1 generates an orbit which tends to + ~ , and x > 1 is part of the basin for the x = + ~ attractor. By symmetry,
x + 1). Hence, 0.2 < x < 0.6 is in the  ~ basin, and  0 . 6 < x <  0 . 2 is in the + ~ basin. This is illustrated in fig. 9b [the symbol 1 + (1  ) signifies that the designated interval maps to x > +1 (x <  1) in one iterate of the map]. Continuing the construction, one finds three open intervals which map to 1 + interval in one iterate and hence to x > 1 in two iterates (these intervals are labeled 2 + in fig. 9c), and three more open intervals which map to x <  1 in two iterates (labeled 2  ) . The 2 + ( 2  ) intervals are in the basin of + ~ (  ~ ) . We see that the basin b o u n d a r y must lie in [  1, + 1] and does not contain any of the eight intervals labeled 1 + , 1  , 2 + , or 2  . This is clearly the first two stages in a standard C a n t o r set construction; this demonstrates (i). [The fractal dimension of this C a n t o r set basin b o u n d a r y is ( l n 3 ) / ( l n 5 ) = 0.6826 . . . . ] Further. we see that the b o u n d a r y points accessible from 1 + (1  ) are x =  0 . 6 and x =  0 . 2 (x = 0.6 and x = 0.2), and these map in one iterate to x = +1 ( x =  1 ) . Similarly, the b o u n d a r y points accessible from the 2 + ( 2  ) intervals are seen to all map to x = + 1 (x =  1) in two iterates. Continuation of the Cantor set c o n s t r u c t i o n implies point (iii) and hence point (ii). Since the map takes three subintervals of [  1 , + 1 ] and linearly stretches them into the whole interval (the subintervals are [  1 ,  0 . 6 ] , [  0 . 2 , +0.2], and [0.6,1], there are an infinite n u m b e r of unstable periodic orbits* in [  1 , + 1]. These points cannot lie in either basin and hence must lie on their boundary. However, by (iii) they are also not accessible. Hence (iv) follows.
*To see this, note that each of the three subintervals themselves have three subintervals (cf. fig. 9c) which are mapped onto [  1, + 1] in two iterates, and so on. Thus, the n times iterated map has 3" subintervals which are linearly stretched and mapped to [ 1, + 1]. Each of these thus contains a fixed point of the n times iterated m a p . Hence, there are 3"/n periodic orbits of period n.
C. Grebogi et al. / Basin boundary metamorphoses
(o)
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Fig. 8. (a) Same as fig. 6a b u t showing an element of the period3 saddle orbit and A = 1.5. (b) Same as fig. 6b b u t showing an element of the period4 saddle orbit and .4 = 1.5.
As an example of an inaccessible boundary point, consider the fixed point x = 0. From the construction of the Cantor set, we see that this point certainly is on the boundary but that any e neighborhood of it contains an infinity of alternating segments of the two basins. [Note that the accessible points are those points which are end points o f one of the intervals deleted as one con
structs the Cantor set. In addition, they are precisely those points that map exactly onto one or the other fixed points at x =_+1. Such points could be considered the (generalized) stable manifold of the fixed points.] The example just given is analogous to the fractal basin boundary in fig. 2b in the following ways. First, a straight line cutting across the basin
C. Grebogi et al. / Basin boundary metamorphoses
292
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'
(c)
Fig. 9. (a) The map x,+ 1 = F(x,). (b) The interval 1 + (1  ) maps on one iterate to x > 1 (x <  1). (c) The intervals 2 + (2  ) map on two iterates to x > 1 (x <  1).
boundary of fig. 2b intersects it in a Cantor set along that line. Second, there are two accessible periodic orbits in the boundary in both cases: the period one saddle and the period four saddle, for fig. 2b; and the two unstable fixed points x = + 1, for the onedimensional map model. Third, all points accessible from the white (black) region of fig. 2b are on the stable manifold of the period four (one saddle); while all accessible points of the + oo(  oo) attractor for the onedimensional map model m a p to x = + l ( x   1) in a finite number of iterates. Fourth, both boundaries contain an infinite n u m b e r of inaccessible unstable periodic orbits, which, in the case of fig. 2b, are saddles. Section 3 will, in fact, show how an infinite number of saddles get buried in the boundary as a result of the basin boundary metamorphosis.
Now we consider how the one dimensional map with a fractal basin boundary, given in fig. 9, might come about with variation of a system parameter. That is, we consider a basin boundary metamorphosis for a onedimensional noninvertible map. As shown in fig. 10, we imagine that as some parameter p is raised from Pl to P2 the lower minimum of the map moves from F ( x ) >  1 at P = P l [fig 10a] to F ( x ) =  1 at P=Psf [fig. 10b] to F ( x ) <  1 at P=P2 [fig. 10c]. For P = P l the basin boundary is simply the single point x = 1, but there is also an invariant Cantor set in 1 > x >  1 (i.e., entirely in the + oo basin) generated due to the fact that the positive maxim u m of F(x) is greater than one. For fig. 10c there is a fractal basin boundary (as in fig. 9), and the conversion to fractal occurs at p = Psf. When this occurs the basin boundary dimension changes discontinuously. In particular, it is zero for p < psf (the boundary is the single point x = 1), but as p approaches Par from above it is the dimension of the invariant Cantor set in the limit as p approaches p~f from below. Furthermore, as p increases through p~f the basin boundary jumps far into the interior of the + oo basin. Indeed, for p just slightly bigger than p~f, there are thin interval pieces of the  o o basin near all the elements of the Cantor set which, for p < Pat, existed entirely in the + oo basin. Just as the thin filaments which j u m p into the white basin in fig. 2c approach zero width as A~Aef from above, so too do the abovementioned interval pieces of the  oo basin approach zero as p , p~f from above. F r o m the point of view of directly observable consequences, perhaps the most significant aspect of the basin boundary structure we have described here in section 2 is its implications for the final crisis in which the chaotic attractor is destroyed as it collides with a saddle on the basin boundary. Although there are an infinite number of saddles embedded in the basin boundary, the attractor can collide only with that one which is accessible to it. Thus, the type of crisis is, in a sense, determined by the accessible boundary saddle at the crisis (cf. figs. 4). Further, a crisis transfer may be thought
C Grebogiet al. / Basin boundarymetamorphoses
293
of as a change of the accessible boundary saddle at the crisis.
Xn+I =F(Xn)
/I ////// II
'i/
/ ////
/
3. HOW saddles get buried in the basin boundary and how the boundary jumps inward
,/ /
I
V
:t
Xn
/'11 (a)p:p I Xn+I : F(Xn)
// ////
1/1 i [ [1
/// // t
J
Xn
(b) p= Psf Xn+I = F(Xn)/,
1I ......  ~ .
!
/
.
,
.
.
.
.
:;I
//// 1 // I
///
/"
.
/ii
"
:~n
i
(c)p:p 2 Fig. 10. Metamorphoses. in onedimensional noninvertible maps. (a) F(x)> 1 at p=p]; (b) F(x)= 1 at P =Psf; (C) F(x)< 1 at P=P2.
It is evident from the discussion of figs. 2 in section 2.1 that the two events occurring at A = Asf and at A = A u are closely analogous. Both involve homoclinic crossings of accessible boundary saddle orbits: l u x ls at Asr and 4u × 4s at Aff. (At those values the relevant stable and unstable manifolds are tangent.) Also, both transitions are accompanied by a jump of the basin boundary inward into the white region. For the case of the smoothfractal basin boundary metamorphosis, the inward jump occurs because the period4 saddle is a finite distance from the basin boundary w h e n A < Asf , while for A > Ast there are parts of the basin of the attractor at infinity which are arbitrarily close to the period4 saddle. A similar statement applies for the period3 saddle at the fractalfractal basin boundary metamorphosis. With this in mind, we study the events leading up to the metamorphosis at A u (since Art is similar). In particular, we are interested in determining why the basin boundary suddenly jumps inward to the period4 saddle at the onset of l u x ls, and in determining how an infinite number ofother inaccessible saddles get buried in the basin boundary. We do not have a complete understanding of the answers to these questions. For example, we shall show how a particular class of saddles get buried in the basin boundary. We call saddles in this class simple Newhousesaddles*. While there are an infinite number of simple Newhouse saddles they are a comparatively small subset of all the buried saddles. Nevertheless, our study will be.sufficient for understanding why the basin boundary jumps inward at Asp *Newhouse [5] was one of the first to give an extensive picture of how saddlenodebifurcationsoccur near homoclinic tangencies, and he developedan extensivetheory of them. See also references in [6].
294
C. Grebogi et aL / Basin boundary metamorphoses (a)
3.1. Simple Newhouse orbits We begin by describing in more detail the sequence of events for J = 0.3 as A is increased. As A increases a saddlenode bifurcation occurs at A t =  ( 1 + J ) 2 / 4 =  0 . 4 2 2 5 in which a period1 attractor and a period1 saddle are created. In addition, an infinite number of other saddlenode bifurcations of other higher periods also occur for larger A. F o r these higher period orbits the stable orbit created is generally observed to proceed rapidly (with increase of A) through a perioddoubling cascade to a final crisis that destroys the attractor and its basin. The remnant of such a sequence following a periodN saddlenode bifurcation (i.e., what is left following the final crisis) is a horseshoe (or at least it rapidly becomes a horseshoe) containing an infinite n u m b e r of saddle orbits. In particular, one of these unstable orbits will be the periodN saddle created at the original saddlenode bifurcation. We call these saddles original saddles. These saddles have the property that their associated eigenvalues of the linearized m a p are b o t h positive (since J > 0). Furthermore, we shall concentrate our discussion on a special class of these original saddles. In order to delineate this class, consider fig. l l a which schematically illustrates the stable and unstable manifolds of the original period1 saddle for a value of A below that at which the manifolds are tangent. As A increases, a homoclinic tangency, as shown in fig. l lb, occurs at some value of A which we d e n o t e d A~'. (This value is also the value after which the basin b o u n d a r y of infinity becomes fractal, A~' = Asf. ) For now, however, we wish to discuss the occurrence of saddlenode bifurcations (and hence the creation of original saddles) for A < A~' as A ~ A~'. In particular, we observe an infinite sequence of saddlenode bifurcations of period 3, 4, 5, 6, 7 . . . . . which occur at parameter values A 3, A 4, A 5. . . . . where A3 < A4 < As < A 6 < A 7 < A s " " " < A ~'. The particular bifurcations which we refer to above are those which create original saddles that proceed once around a circuit following the unstable maplf,~!d, as sche
(b)
Fig. 11. Schematic illustration of the stable and unstable manifolds of the period1 saddle for (a) A < A~' (A~' is the value of A at homoclinic tangency) and (b) A = A?. matically illustrated in fig. 12 for a period5 case. W e call these orbits simple Newhouse orbits*. It can be shown [5] that for large n A~'  A , ,  X
n,
*Actually [5, 6] there is a much larger class of saddlenode bifurcations than the ones we concentrate on here. For example, in general, a high period orbit may proceed several times around the loop before returning to its original value.
C. Grebogi et at / Basin boundary metamorphoses
295
(a)
.
.
.
.
.
.
.
.
.
Fig. 12. Schematic illustration of a simple Newhouse orbit of period5. Dashed arrows indicate the sequence in which points on the orbit occur.
and, furthermore, the width zan of the window is
(b)
[6] I0 8
where ~, > 1 is the unstable eigenvalue at the original period1 saddle for A =A~', and A denotes the difference between A n and the value of A at the final crisis of the attractor created at A n. Thus we see that, for large n, the range of A over which the attractor exists is small compared to the range of A between the initial periodn saddlenode bifurcations and the homoclinic tangency of the period1 saddle manifolds.
6 Y
4 2 0 2 2
I
0
I
2
x
3.2. Crossing of the stable manifoM of simple Newhouse orbits with the unstable period1 manifold For example, the saddlenode bifurcation of the simple Newhouse orbit of period3 occurs at A =1.16 and the final crisis of the resulting chaotic attractor (which consists of three pieces) occurs at A  1.25. Fig. 13a shows a schematic diagram of the basin of attraction for the period3 attractor at A = 1.25. The actual numerically determined period3 basin appears in fig. 13b. Also shown in
Fig. 13. (a) Schematic view of the basin for the period3 attractor (cross hatched), the period1 attractor (white), and the attractor at infinity (shaded). The unstable manifold of the period1 saddle is also shown. (b) Numerically determined basin of attraction (black) for the period3 attractor for A = 1.25 and J = 0.3.
fig. 13a are the basin of attraction for attractor, the unstable manifold of period1 saddle, and the location of saddle. The basin boundary for the
the period1 the original the period3 period3 at
296
C Grebogi et al. / Basin boundary metamorphoses
tractor is the stable manifold of the period3 original saddle, and the basin boundary of infinity is the stable manifold of the period1 saddle. When the period3 saddlenode bifurcation occurs, it immediately creates a basin for itself by cutting a chunk out of the period1 basin of attraction*. It is important to note from fig. 13a that the stable manifold of the period3 saddle (the basin boundary for the period3 attractor) cuts across the unstable manifold of the period1 saddle. In fact, this is true for all values of A for which the period3 saddle exists, including those exceeding the crisis value for the three piece chaotic attractor (which evolves via a period doubling cascade from the period3 node). In this range, A~ 1.25, of course, the stable manifold of the period3 saddle is no longer a basin boundary. (In fact, we shall be most interested in what happens near A~'  1.315). Furthermore, we observe an analogous situation to hold for all simple Newhouse orbits. That is, for any simple Newhouse orbit the stable manifold
always cuts across the period1 saddle's unstable manifold. (While at A = 1.25 the period3 saddle has one branch of its unstable manifold becoming tangent and crossing its stable manifold, we are interested in what the other unstable branch does (cf. next subsection).)
3.3. Crossing of the unstable periodN manifold with
the stable periodN + 1 manifold We have numerically examined the stable and unstable manifolds of the simple Newhouse orbits, and we have observed the following sequence of events: 1) After A  A~, 5  1.308 the unstable rnanifold of the period4 saddle crosses the stable manifold of the period5 saddle. * The area of the period1 basin of attraction in the region shown in fig. 13 decreases discontinuously at the period3 saddlenode bifurcation. Note, however, that the total area of each basin must be infinite. This follows immediatelyfrom the facts that (1) contracts areas by J = 0.3 and that the basins are invariant under (1). The area of the set of points in figs. 13 which remains bounded appears to change continuously.
2) After A = A~. 6 ~ 1.310 the unstable manifold of the period5 saddle crosses the stable manifold of the period6 saddle. 3) After A = A*6,7  1.311 the unstable manifold of the period6 saddle crosses the stable manifold of the period7 saddle. Fig. 14 shows the crossings of the numerically determined period4 unstable (4u) and period5 stable (5s) manifolds. One can argue rigorously that an infinite number of such crossings A,,,+~ occur as A )A~': 4u X 5s
5u x 6s 6u x 7s 7u x 8s
(3)
8u x 9s
N u X ( N + 1)s
where N u and Ns stand for the periodN unstable (stable) manifold, and x signifies crossing of the two relevant manifolds*. 3.4. Heterocfinic manifoM crossings We now review the significance of the crossing of a stable and an unstable manifold when each are associated with different saddle periodic orbits. Consider two saddles, one of period M and one of period N. Say, that as a parameter of the system is varied the unstable manifold of the periodM saddle crosses the stable manifold of the periodN saddle. Also assume that the eigenvalues associated with these saddles are positive (as for simple Newhouse orbits). The situation is as illustrated in fig. 15. Evidently the unstable manifold of the * Note. however, that the period3 saddle does not appear in the sequence (3). Its crossing, 3u×4s, occurs later, immediately following A =A~ =Aft> Asf. This is implied by analogy with the lu x ls event at A~' for which we show (section 3.5) that the lu x ls and 4u x Is events are simultaneous.
C Grebogi et al. / Basin boundary metamorphoses
297
(stable) manifold (both branches included) of the simple Newhouse saddle of period N. Eqs. (4) have strong implications for the structure discussed in sections 3.2 and 3.3. According to section 3.2, the stable manifold of any simple Newhouse orbit crosses the unstable manifold of the period1 saddle. Thus, from (4),
I0 8 6 4 2 0 2 I
o
i
l"u_ Nu,
(5a)
N's_Dls
(5b)
for any periodN simple Newhouse orbit. Similarly sequence (3) of section 3.3 implies that as A ~ A sf from below, we have
2
Fig. 14. Numericallyobtainedperiod4 unstablemanifoldand period5 stable manifoldat A = 1.3087, J = 0.3.
4u_5u~6u_D
...,
(6a)
4s ___5s ___6s ___ • . . .
(6b)
Combining (5) and (6) we have
MI Fig. 15. Heteroclinic crossings for a periodM saddle orbit and a periodN saddle orbit. Onlyone point of each periodic orbit is shown.
lu_D4u_5u_~6u_ . . . ,
(7a)
ls__.4s___5sc_6sc_ . . . ,
(Tb)
for A > A~'. Thus, for example, eq. (7a) implies that the unstable manifold of the period1 saddle is extremely complex and convoluted, since its closure includes all the unstable manifolds of all simple Newhouse orbits. 3.5. Homoclinic crossing for the period1 saddle
periodM saddle limits on the periodN unstable manifold, and the periodN stable manifold limits along the entire periodM stable manifold. Thus the closure of the periodM unstable manifold contains the periodN unstable manifold, and the closure of the periodN stable manifold contains the periodM stable manifold, Mu D Nu,
(4a)
Ns ~ Ms,
(4b)
where the overbar denotes closure. In statements like (4), we write Nu(Ns) to denote the unstable
We now claim that when the stable and unstable manifolds of the period1 saddle cross, then simultaneously the 4u, 5u .... manifolds also cross the period1 stable manifold. That is, the relationships ls x l u and ls × Nu, N > 4, are valid immediately following A = A~. In order to see how this arises consider, for example, the period4 unstable manifold. By eq. (7a), and as illustrated in fig. 16, 4u comes arbitrarily close to all the simple Newhouse orbits of period 5,6 . . . . . As A ~ A ~ an infinite number of such orbits are created by saddle node bifurcations. The locations of these orbits are closer and
C. Grebogi et al. / Basin boundary metamorphoses
298
t4
unstable manifolds become equal and similarly for their stable manifolds. In particular, the basin boundary for the attractor at infinity will be (Basin boundary) = ls = 4s.
(9c)
Since the period1 and the period4 saddles are the only accessible saddles, all the other simple Newhouse orbits ( N > 4 ) are "buried" in the boundary.
3.6. Basin boundary jumps Fig. 16. Schematicillustration of the implications of Nu x (N + l)s. It is seen that the left branch of 4u comes arbitrarily close to the simple Newhouseorbits of periods 5,6, 7.....
closer to the stable manifold of the period1 saddle as the period of these orbits gets higher and higher. In fact, in a suitable sense we can regard the tangency points of ls and lu at A A~' (cf. fig. l l b ) as the limit of the elements of a simple Newhouse saddle of period N as N, ~ (cf. ref. 6). Thus, as A ~ A*, the closure of the period4 unstable manifold touches the period1 stable manifold. As soon as A exceeds A~', a set of tongues, as in fig. 5, come shooting up along the formerly smooth basin boundary. These tongues must cross the period4 unstable manifold. Hence, ls x lu and ls x 4u become true simultaneously and similarly for Is x Nu, N > 4. Since the ls x Nu, N > 4 relationships become true simultaneously, we see that eq. (4) implies that N~_lu
and
NsC_ls
(8)
for A > A~', N >_ 4. (Note that the period3 simple Newhouse orbit does not experience this type of event at A 1.315.) Eqs. (8) and (7) imply that for A >AI*=1.315 we have for the simple Newhouse orbits lu = 4 u = 5 u = 6 u . . . . Is=4s=5s=6s
....
(9a) .
Thus we see from eqs. (9) that all the simple Newhouse orbits of period greater than or equal to 4 lie in the basin boundary. Furthermore, equalities (9a) and (9b) imply a striking discontinuity in the boundary at A~e. The period4 saddle is certainly not on nor is it especially near the boundary at or before A~f; yet we see it is part of the boundary immediately after A st. Since this orbit necessarily moves smoothly and continuously, it follows that the boundary discontinuously jumps inward to the orbit. This also holds for the period5 (simple Newhouse) saddle and period 6, etc. But the lower the period is, the more striking the conclusion since the orbits of lower period are further from the boundary. For other values of J, the period 4 must be replaced by some appropriate period and the accessible orbit might not be a simple Newhouse orbit. Based upon the ideas developed in this paper, the following rigorous result [61 has been obtained for the smooth fractal transition:
(9b)
Thus at the smoothfractal basin boundary bifurcation the closures of all the simple Newhouse
Theorem. Consider a typical map with a saddle fixed point or periodic orbit P that has a transition value Asf as A increases (where the stable and unstable manifolds of P cross for the first time). Assume the absolute value of the determinant of the Jacobian at P is less than one (that is, the Jacobian of the Nth iterate of the map where N is the period of P). Then there will be a periodic saddle Q that is in the closure of the stable manifold of P for all A slightly greater than A~r but is not in it at A~r. Furthermore, P will be in
C Grebogi et al. / Basin boundary metamorphoses the closure of the stable manifold of Q for all A slightly greater than A a. S. Hammel and C. Jones [7] were the first to prove a result with these hypotheses concluding that the basin boundary jumps inward. Their version makes no mention of the orbit Q. Their techniques are totally different from the ideas in eqs. (9), though the ideas in eqs. (9) are a sufficient foundation to give a rigorous proof of this theorem. In addition, note that this theorem also applies to the A ff transitions, since for the period4 saddle the first crossing of its righthand (inward) unstable manifold with its stable manifold follows A ff.
299
complex sequence of events which precede the basin boundary metamorphosis (section 3). In particular, a saddle set lying in the basin grows via a chain of crossings of stable and unstable manifolds. Eventually this inflated saddle set collides with the boundary, resulting in the metamorphosis described above*. vi) The saddle which a chaotic attractor collides with at a boundary crisis is the boundary saddle accessible to that attractor. As a parameter is varied, a change in the accessible saddle at the crisis boundary is called a crisis transfer (section
1.3). Acknowledgements
4. Summary In this paper [8] we have investigated sudden large scale changes in basin boundaries with variation of a system parameter. Our conclusions are as follows: i) The basin boundary can jump in size and change its character as the system parameter passes through certain critical values, and we call these changes basin boundary metamorphoses. ii) Metamorphoses can occur as a result of homoclinic intersections of the stable and unstable manifolds of a saddle periodic orbit on the basin boundary. iii) The structure of the basin boundaries which we investigate is, to a large extent, determined by the accessible saddles which lie on the basin boundary (section 2). iv) The character changes referred to in (i) are changes in the accessible saddle orbits on the basin boundary and sometimes a change of the boundary from being smooth to being fractal (as in the transition from fig. 2a to fig. 2b but not from fig. 2b to fig. 2c). v) The way in which saddles get buried in the basin boundary (thereby becoming inaccessible) and the means by which basin boundaries can jump in size have been investigated, and it has been shown that these are accomplished by a
We would like to thank ShenTeng Yang for making the pictures of fig. 1 and BaeSig Park for numerically determining the dimension of the basin boundary corresponding to fig. 2b. This work was supported by the Department of Energy (Office of Basic Energy Sciences), DARPA under NIMPP, and the Office of Naval Research.
Appendix Finding accessible boundary saddles In fig. 2, we have indicated accessible saddles on the basin boundary. Here we describe our numerical technique for the determination of these accessible periodic boundary saddle orbits. Say we have two attractors, attractor A and attractor B, and we wish to find a boundary saddle orbit that is accessible from the basin of attractor B. We follow the following steps: 1) Choose two points close to the basin boundary that go to the two different attractors, say, point PA goes to attractor A and PB goes to attractor B.
*Alternatively, the attractor may collide with this inflated saddle set before this set has collided with the boundary (an interior crisis). This is what is happening at the period three interior crisis of fig. 3a.
300
C. Grebogi et al. / Basin boundary metamorphoses
2) Take 64 points equally spaced in the segment
References
PAPa and n u m b e r them sequentially so that PA = 1 and PB = 64. (We choose 64 points because 64 is the vector length in Cray computers.) 3) Iterate the 64 points and determine the highest n u m b e r e d point which goes to the attractor A; call this the new point PA4) M o v e the point PB a fraction (say, a fifth) of the distance PAPa towards PA, and make sure that the new point PB is in the basin of B. If it is not, m o v e back in small steps (in the direction away f r o m PA) until it is. 5) G o back to step 2 and keep repeating the p r o c e d u r e until the distance PAPB is less than, say 1012 6) Iterate PA and PB and print the first, say, 50 points of the trajectories. 7) Extract the period and the elements of the accessible b o u n d a r y saddle orbit from the printed trajectories. F o r the cases in figs. 2, for different choices of the initial points P^ and PB, this procedure always yields the indicated b o u n d a r y saddles accessible from the black and white regions, and we thus believe that these are the only accessible b o u n d a r y saddles.
[1] C. Grebogi, E. Ott and J.A. Yorke, Physica 7D (1983) 181. [2] S.W. McDonald, C. Grebogi, E. Ott and J.A. Yorke, Physica 17D (1985) 125. [3] C. Grebogi, S.W. McDonald, E. Ott and J.A. Yorke, Phys. Lett. 99A (1983) 415. [4] C. Grebogi, E. Ott and J.A. Yorke, Phys. Rev. Lett. 50 (1983) 935; S.W. McDonald, C. Grebogi, E. Oft and J.A. Yorke, Phys. Lett. A107 (1985) 51. C. Mira, C.R. Acad. Sc. Paris 288A (1979) 591; I. Gumowski and C. Mira, Dinamique Chaotique (Cepadues, Paris, 1980). F.T. Arecchi, R. Badii and A. Politi, Phys. Rev. A32 (1985) 402. M. Iansiti, Q. Hu, R.M. Westervelt and M. Tinkham, Phys. Rev. Lett. 55 (1985) 746. R.G. Holt and I.B. Schwartz, Phys. Left. A105 (1984) 327; I.B. Schwartz, Phys. Lett. A106 (1984) 339; I.B. Schwartz, J. Math. Biol. 21 (1985) 347. E.G. Gwinn and R.M. Westervelt, Phys. Rev. Lett. 54 (1985) 1613. Y. Yamaguchi and N. Mishima, Phys. Lett. A109 (1985) 196. O. Decroly and A. Goldbeter, Phys. Lett. A105 (1984) 259. S. Takesue and K. Kaneko, Progr. Theor. Phys. 71 (1984) 35. F.C. Moon and G.X. Li, Phys. Rev. Lett. 55 (1985) 1439. [5] S.E. Newhouse, Publ. Math. IHES 50 (1979) 101. N.K. Gavrilov and L.P. ShJlnikov, Math. USSR Sbornik 17 (1972) 467. [6] K. Alligood, L. TedeschiniLalli and J.A. Yorke, private communication. [7] S. Hammel and C. Jones, Jumping stable manifolds for dissipative maps on the plane, preprint.. [8] A preliminary version of this work is contained in C. Grebogi, E. Ott and J.A. Yorke, Phys. Rev. Left. 56 (1986) 1011.
Deskripsi
Nuclear Physics B (Proe. Suppl.) 2 (1987) 281300 NorthHolland, Amsterdam
281
BASIN BOUNDARY METAMORPHOSES: CHANGES IN ACCESSIBLE BOUNDARY ORBITS* Celso G R E B O G I , a E d w a r d O T T a,b and James A. Y O R K E c
University of Marylan~ CollegePark, MD 20742, USA Received 5 November 1985 Revised manuscript received 28 April 1986
Basin boundaries sometimes undergo sudden metamorphoses. These metamorphoses can lead to the conversion of a smooth basin boundary to one which is fractal, or else can cause a fractal basin boundary to suddenly jump in size and change its character (although remaining fractal). For an invertible map in the plane, there may be an infinite number of saddle periodic orbits in a basin boundary that is fractal. Nonetheless, we have found that typically only one of them can be reached or "accessed" directly from a given basin. The other periodic orbits are buried beneath infinitely many layers of the fractal structure of the boundary. The boundary metamorphoses which we investigate are characterized by a sudden replacement of the basin boundary's accessible orbit.
I. Introduction 1.1. Preliminary metamorphoses
discussion o f basin boundary
Basin b o u n d a r i e s * for dynamical systems with multiple attractors can be either smooth or fractal; they can a p p e a r to be very convoluted; or they can be simple in appearance. As an example illustrating this, fig. 1 shows some computergenerated pictures of basins of attraction for the d a m p e d driven p e n d u l u m , 0 + v~ + to2 sin 0 = f c o s t. Given the variety of basin b o u n d a r y structure observed in fig. 1, it is natural to ask how basins of attraction c h a n g e as a system parameter varies continuously. In particular, are there qualitative changes o f basin boundaries as the parameter passes
t h r o u g h certain critical values? This paper is devoted to a study of such changes, which we call basin boundary metamorphoses. In particular, basin b o u n d a r y m e t a m o r p h o s e s can occur as a result of a homoclinic tangency of the stable and unstable manifolds o f a saddle orbit on the basin boundary. Perhaps the simplest nontrivial system exhibiting the p h e n o m e n a of interest here is the H t n o n map, and so our numerical investigations are devoted to this case. More generally we believe that the type o f m e t a m o r p h o s e s we investigate are the typical large scale metamorphoses of maps in two dimensions and in m a n y ordinary differential e q u a t i o n systems (e.g., the forced d a m p e d p e n d u lum). T h e H t n o n m a p is Xn+ 1 =
A  x ,2 _ j y . ,
Y,,+l = x,,, ~Laboratory for Plasma and Fusion Energy Studies. band Departments of Electrical Engineering and of Physics and Astronomy. c Institute for Physical Science and Technology and Department of Mathematics. • *The basin of attraction of an attractor is the set of initial conditions which approach the attractor as time approaches positive infinity. The boundary of the closure of this region is invariant under the dynamics and is called the basin boundary. *Reprinted from Physica 24D (1987) 243
09205632/87/$03.50 ©Elsevier Science Publishers B.V. (NorthHolland Physics Publishing Division)
(la) (lb)
where the p a r a m e t e r J is the determinant of the J a c o b i a n matrix of the map. (Hence J is the net c o n t r a c t i o n ratio for any finite area in the x  y plane u n d e r the action of the map.) F o r eqs. (1), as A increases past the value A x =  ( J + 1)2/4, a s a d d l e  n o d e bifurcation occurs in which one at
C. Grebogi et al. / Basin boundary metamorphoses
282
(a)
(b) r
3
I
3
3
I
I
0
0
(c)
(d)
3
5 3
3
I
"*.;L'"

.
....
". . . . . . .
x
I I
3
5
3
I
I
3
3
8
I
I
3
8
Fig. 1. Basins of attraction for the damped driven pendulum, 0"+ v0 + 2 sin 8 = j cos t. To generate each one of these pictures, we choose over 1 000000 initial conditions in a grid. The differential equation is then integrated for each initial condition until the orbit is close to one of the possible attractors. For a given picture, if an orbit goes to the attractor indicated below, a black dot is plotted corresponding to that initial condition. Thus, the black region in each picture is essentially the basin of attraction for that attractor. (a) o~= 1, v = 0.1, f = 1.2; attractor in the black basin: O = 2.2055, 0'= 0.3729. (b) w = 1, v = 0.1, f = 2.0: attractor in the black basin: 0 = 0.8058; # = 0.9375. (c) oa = 1, u = 0.2, f = 1.8; attractor in the black basin: 0 = 2.9320; 0 = 0.5200. (d) ~ = 1, ~, = 0.3, f = 1.85; attractor in the black basin: 0 = 0.4186, 0 = 0.5744.
t r a c t i n g f i x e d p o i n t a n d o n e s a d d l e fixed p o i n t a r e
A~ t h e f i x e d p o i n t a t t r a c t o r m o v e s a w a y f r o m t h e
c r e a t e d (cf. t a b l e I). F o r A j u s t s l i g h t l y l a r g e r t h a n
s a d d l e f i x e d p o i n t . T h e s a d d l e fixed p o i n t lies o n
A 1 there are two attractors, one at
(Ixl, lYl)= ~ ,
the boundary
of the basin of attraction of the
a n d t h e o t h e r t h e a t t r a c t i n g fixed p o i n t ( f o r A < A~,
f i x e d p o i n t a t t r a c t o r . I n fact, t h e s t a b l e m a n i f o l d
i n f i n i t y is t h e o n l y a t t r a c t o r ) . A s A i n c r e a s e s p a s t
o f t h e s a d d l e is t h e b a s i n b o u n d a r y (cf, fig. 2a). A s
C Grebogi et al. / Basin boundary metamorphoses Table I Notation for special values of the Hdnon map parameter A
Notation Event signified A,,
Asf A rf
A,*
A,,. ,, ÷ ~
saddlenode bifurcation of a periodn orbit. smoothfractal metamorphosis fractalfractal metamorphosis chaotic attractor has a crisis by colliding with an accessible periodn saddle on the basin boundary tangency of the stable and unstable manifolds of an accessible periodn saddle on the basin boundary tangency of the unstable manifold of a periodn saddle with the stable manifold of a periodn + 1 saddle
Where notation first appears section 1.1 section 1.1 section 1.1 section 1.3
section 2.1
section 3.3
A increases further, the fixed point attractor undergoes various bifurcations, most prominently a perioddoubling cascade. We shall be interested in following the evolution of the basin boundary that is created at A = A 1 as the parameter A increases. Figs. 2 show the basin of attraction for the point at oo in black for three parameter values: (A, J ) = (1.150,0.3) (fig. 2a); (A, J ) = (1.395,0.3) (fig. 2b); and (A, J)=(1.405,0.3) (fig. 2c). This figure was obtained by starting with a grid of 640 × 640 initial conditions. Initial conditions in the basin of attraction of infinity were determined by seeing which ones yield orbits with large x and y values after a large number of iterations. Those which did are plotted as dots. These dots are so dense that they fill up the black region in the figure. For the case of fig. 2a the boundary of the basin of infinity (i.e., the black region) is apparently a smooth curve (the stable manifold of the period1 saddle). All initial conditions in the white region generate orbits which remain bounded and are generally asymptotic to the fixed point attractor labeled in the figure. For the case of fig. 2b the basin boundary shows considerably more structure, and, in fact, magnifications of it reveal that it is fractal (cf. refs. 14 for discussion of fractal basin boundaries). Initial conditions in the white region are generally asymptotic to a period
283
two attractor (labeled by two dots in the figure) which results from a period doubling bifurcation of the fig. 2a fixed point attractor*. One of our purposes in this paper will be to describe how a smooth basin boundary, as in fig. 2a, can become a fractal basin boundary, as in fig. 2b, as a system parameter is varied continuously (e.g., as A is varied from 11150 to 1.395 with J fixed at 0.3). It is found that the boundary becomes fractal following a homoclinic tangency of the period one saddle. Furthermore, we find that this is preceded by a sequence of heteroclinic intersections of stable and unstable manifolds of an infinite number of unstable periodic saddles (section 3). It will be shown that this sequence of events, preceding the actual metamorphosis, determines to a large extent the boundary structure following the metamorphosis to fractal. Henceforth, we shall denote by A u the value of A for which the boundary becomes fractal as A increases through Asf, and we shall call the accompanying conversion a smoothfractal basin boundary metamorphosis. We conjecture that the dimension of the boundary is greater than one following Asr. We also find another type of basin boundary metamorphosis in which the extent of a fractal basin can grow discontinuously by suddenly sending a Cantor set of thin fingers into the territory of another basin. This is illustrated by a comparison of figs. 2b and 2c which have slightly different values of the parameter A and the same J value: A1.395 in fig. 2b and A=1.405 in fig. 2c. Comparing these two figures we see that the basin *The "basin boundary" under study is usually the boundary of the basin of the attractor at infinity. Note that saddlenode bifurcations in the white region do occur and can lead to the presence of more than one attractor there. The white region would be the union of the basins of the attractors not at infinity. Most commonly, however, additional attractors created by saddlenode bifurcations are only seen over a comparatively small range in A before they are destroyed by a. crisis [1] (for example, see section 3.1). Thus it is common for the white region to be the basin of a single attractor. We have not observed any saddlenode bifurcations occurring in the black region.
284
C. Grebogi et al. / Basin boundary metamorphoses
I0
8 6
4
2'
O' ~).
2
{
0
J
2
2
f
X
2
r
0
f
2
X
0
r
2
X
Fig. 2. Basin of infinity in black for the Hdnon map with J = 0.3 and (a) A = 1.150, (b) A = 1.395 (the numerals 1,2, 3, 4 denote the accessible period4 saddle in the sequence in which the points are visited), and (c) ,4 = 1.405 (1,2, 3 denote the accessible period3 saddle).
of attraction of infinity (the black region) has enlarged by the addition of a set of thin filaments, some of which appear well within the interior of the white region of fig. 2b (in particular, note the region  1.0 7z x ~  0.3, 2.0 7zy 7z 5.0 in fig. 2c). We call the change exemplified by the transition from fig. 2b to fig. 2c a fractalfractal basin boundary metamorphosis, and we denote the value of A at Which it occurs Art. As A is decreased toward A ff, the filaments become even thinner,
their area in the frame of the figure going to zero, but they remain in position, not contracting to the position of the basin boundary shown in fig. 2b. Hence, the position of the boundary changes discontinuously although the area of the wlaite and black regions apparently changes continuously. (This type of discontinuous jump in the boundary also occurs at A = A~f.) To conclude this subsection we note that Moon and Li [4], in work done at the same time as ours,
C Grebogi et al. / Basin boundary metamorphoses have also noted that basin boundaries can become fractal as a result of a homoclinic crossing of stable and unstable manifolds, and they have demonstrated how Melnikov's criterion can be used to find the smoothfractal metamorphosis condition for a periodically forced Dufling equation. Our analysis, on the other hand, reveals several basic underlying aspects not discussed by Moon and Li [4]: the essential role of accessible orbits, the fractalfractal metamorphosis, jumps in the location of the boundary at metamorphoses, crisis transfers, and the phenomena discussed in section 3.
1.2. Preliminary discussion of accessible boundary
points Both types of metamorphoses lead to a change in the accessible saddle orbits on the basin boundary. The concept of accessible boundary points and its importance for understanding the structure of fractal basin boundaries will be discussed in section 2.
285
saddle and its stable manifold*. (No other points are accessible. The stable manifold is dense in the boundary but most boundary points are not accessible.) In particular, the transition from (i) to (ii) is considered in section 3 and leads us to a study of the macroscopic collision processes that take place as a system parameter is varied. Considering this transition, as well as the transition from (ii) to (iii), we find that a nonattracting saddle set lying in the closure of the basin grows via a complex chain of crossings of stable and unstable manifolds. (This saddle set is a saddle plus the closure of all the crossing points of the stable and unstable manifolds of that saddle.) Eventually this inflated saddle set collides with the boundary, resulting in the basin boundary metamorphosis described above. Alternatively, the saddle set may collide with the attractor (an interior crisis). These virtually invisible saddle sets play a key role in the observed metamorphoses. In the appendix we describe our numerical techniques for finding accessible boundary saddle orbits.
Definition. A boundary point p is accessible from a region if there is a curve of finite length connecting p to a point in the interior of the region such that no point of the curve lies in the boundary except for p. As we will describe later (sections 2 and 3), we find that i) For A 1 < A < Asf, the set of boundary points accessible from the white region is the saddle fixed point and its stable manifold (and this is the entire boundary). ii) For Asf < ,4 < Aff, the set of boundary points accessible from the white region is a period4 saddle and its stable manifold. (No other points are accessible. The Stable manifold is dense in the boundary but most boundary points are not accessible.) iii) For A > A n, the set of boundary points accessible from the white region is a period3
1.3. Crises transfers One reason for studying basin boundary metamorphoses is that, as a system parameter is varied, a chaotic attractor can be destroyed by colliding with an unstable orbit on its basin boundary (a boundary crisis [1]). Basin boundary metamorphoses affect boundary crises. We find that basin boundary metamorphoses can lead to changes in the type of crisis which ultimately destroys the attractor, and we call this phenomenon a crisis transfer. To limit the scope of the problem of describing these metamorphoses we examine a particularly simple scenario. A typical saddlenode bifurcation *The filamentsmentionedabove are the stable manifoldof the period3 saddle. The period3 saddle came into existence much earlier (namelyat A ' 1.16) but, beforeArt, it was in the interior of the whiteregion.
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C Grebogi et aL/ Basin boundary metamorphoses
creates a new attractor (namely the node), and the saddle is on the basin boundary. Often the attractor takes a perioddoubling route to becoming chaotic. The chaotic attractor then grows as a parameter is varied, collides with the boundary (a boundary crisis), and the attractor and its basin are suddenly destroyed. We wish to describe the major basin boundary metamorphoses that are seen in this scenario, from saddlenode bifurcation to the final boundary crisis. If one consideres the onedimensional quadratic map (i.e., eqs. (1) with J = 0 ) , then it is well known that the attractor in txl < ~ no longer exists for A > 2. What happens is that, as A increases, the chaotic attractor widens, until, at A = 2, it touches the unstable period one point on the basin boundary (a boundary crisis [1]). For J, a nonzero fixed but small value, the J = 0 boundary crisis which occurs as A is increased is qualitatively unchanged. But for J larger a qualitative change does occur. This is illustrated in figs. 3a and 3b. These figures are obtained at fixed values of J ( J = 0.05 in fig. 3a and J = 0.30 in fig. 3b) by plotting a large number of consecutive x coordinates for an orbit generated by the map, eqs. (1), as the parameter A is raised in small increments (vertical axis). For figs. 3 the map is iterated 103 times for each value of A. Fig. 3a shows two clear crisis events: one, an interior crisis, at A = A~1.78, in which the chaotic attractor suddenly widens due to a collision with a period3 saddle orbit which is inside the attractor's basin; and another, a boundary crisis at A = A~ = 1.8874, in which the chaotic attractor is destroyed due to a collision with a saddle fixed point on the basin boundary. The crisis at A = A~ is essentially the same event as occurs for J = 0 and A = 2 in the 1 D quadratic map. As J is raised above the value 0.05 (which applies to fig. 3a), the values A~ and A]~ move closer together and eventually a crisis transfer occurs. The result is that the final boundary crisis killing the chaotic attractor becomes a period3 crisis, in which the attractor collides with the unstable period three orbit, which is now on the basin boundary. Figs. 4a and 4b show the
(a) PERK BOUN CRISI
PEF INT CRI
2
I
0
I
2
x
(b) i
I.O
0.5
J
0.0
I
0.5
I
I.O
i
I.5
X
Fig. 3. Bifurcation diagrams (A versus x) for the H6non map with (a) J = 0.05 and (b) J = 0.3. For a given value of A, an initial condition (x, v) is chosen and the x coordinate of its orbit is plotted after the first several thousand iterates are discarded.
chaotic attractor, its basin boundary, and the boundary saddle with which it collides (labeled by arrows) for a case ( J = 0.05, A = A~) in which the collision is with a period1 saddle (fig. 4a) and for a case ( J = 0.3, A = A3c) in which the collision is with a period3 saddle (fig. 4b). As the above discussion implies, for J = 0.05, the period3 sad
C. Grebogi et al. / Basin boundary metamorphoses
(a)
287
(b)
2l
Y
O
I22
I
0
I
2
x
2
I
0
I
2
x
Fig. 4. (a) The chaotic attractor and its basin at the crisis, A = 1.8874, for J = 0.05. The period1 saddle on the basin boundary is indicated by an arrow. (b) The chaotic attractor and its basin at the crisis, A = 2.12467, for J = 0.3. The elements of the period3 saddle on the basin boundary are indicated with arrows.
die is not on the basin boundary, but, for J  0.3, it is.
2. Accessible points 2.1. Accessible points for the Hbnon map As mentioned earlier, a point p on the basin boundary is accessible from the white (black) region if one can draw a finite length curve from a point in the interior of the white (black) region to p in such a way that the curve touches the boundary only at that one accessible boundary point. As an illustration, consider the stable and unstable manifolds of the period1 saddle after they have crossed, i.e., A > A~' where A~' denotes the value of A at which these manifolds become tangent. We say l u x ls holds when the period1 unstable manifold crosses the period1 stable manifold. (The test will make it clear which saddle is under discussion and also
which branch of an unstable manifold is under investigation.) A schematic illustration is given in fig. 5. This figure shows that, as a result of the homoclinic intersection, lu x ls, a series of progressively longer and thinner tongues of the basin of the attractor at infinity (black regions in fig. 2b) accumulate on the outer portion of the stable manifold through the period1 saddle. [The (n + 1)st tongue is the preimage of the nth tongue.] A finite length curve connecting a point in the interior of the white region and the period1 saddle would have to circumvent all the tongues accumulating on the stable manifold of the period1 saddle. This, however, is not possible because the length of the nth tongue approaches infinity as n ~ oo. Thus the period1 saddle is not accessible from the white region. By contrast the period1 saddle is always accessible from the black region. This behavior is evident from fig. 2b. In fig. 2b we have also labeled a period4 saddle orbit. Evidently this saddle lies on the basin boundary and is accessible from the white region. We also note that there are tongues accumulating on the stable manifold of the period4 saddle but they do so on the side away from the finite attractor (cf. fig. 6a).
C Grebogi et al. / Basin boundary metamorphoses
288
TONGUE
ATTRACTOR,,~
PERtOD I
~'~
~'TONGUE 2 ~TONGUEI
SADDLE
\
After the metamorphosis, however, the boundary includes the period4 saddle. In this sense we can say that as A increases past A~ the basin boundary suddenly jumps inward into the white region. Note, however, that the area of the basin of infinity in any finite regien of the plane, numerically, at least, appears to change smoothly as A increases through A~'. As soon as 1u x 1s holds, the basin boundary is fractal, i.e. A~ = A s f .
TONGUE 4
Fig. 5. Schematic illustration of tongues accumulating on the stable manifold of the period1 saddle.
Hence, a white region is left open for a curve to join the period4 saddle to the interior of the white region. There are also infinitely many other saddle periodic orbits lying in the basin boundary (cf. section 2.2 and section 3). These, however, are not accessible either from infinity or from the finite attractor because the tongues of the two basins accumulate on both sides of those orbits. This is illustrated in fig. 6b where we show successive blowups of the basin structure in fig. 2b in a small region around one element of the period5 saddle. Thus as A increases past A~' the boundary saddle which is accessible from the inner region suddenly changes from being the period1 saddle to being the period4 saddle. Just before the metamorphosis the period4 saddle is in the interior of the white region. As A approaches A~' from below, the distance between the basin boundary and the period4 saddle does not approach zero.
Indeed, the fractal dimension of the boundary for the case shown in fig. 2b has been numerically determined to be d = 1.530 + 0.006. We have done this by using the numerical technique of "uncertainty exponent measurement" introduced in ref. 3. Next consider the transition from fig. 2b to fig. 2c. We find that the essential difference between these two figures is that, for the parameters of fig. 2b, the accessible saddle is the period4, while for fig. 2c the period4 saddle is no longer accessible from the finite attractor, but a period3 attractor is, as indicated in fig. 2c. Furthermore, the transition between the two cases occurs instantaneously at A = A~" when the stable and unstable manifolds of the period4 saddle are tangent*. Thus A~ = A~. Fig. 7 shows a schematic illustration of the tangency of the numericaly determined period4 stable and unstable manifolds at A = A~ = 1.396.
*One branch of the unstable manifold of the period4 saddles slices through the basin boundary at a Cantor set of points. Here, we are interested in the other branch of the unstable manifold (i.e., the branch which is entirely in the white region in fig. 2b). When it becomes tangent and then crosses the stable manifold of the period4 saddle, the period4 saddle will no longer be accessible.
289
C Grebogi et al. / Basin boundary metamorphoses
(o)
(b)I
i.45785
I.45810

....

1.45835 I.22940
I.22915
I.22890
I.98
1.88
1.78
X
1.68
1.58
x
(b)2
(b)3
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 1.63150
I.627
y
 1.632
y
I.63175
I.637
1.642 I.792
I.787
I.782
I.777
I.772
 1.63200 I.78300
x
I.78275
I.78250
x
Fig. 6. (a) Blowup of the basin structure in a small region around one element of the accessible period4 saddle orbit for J = 0.3 and A = 1.395 (magnification  105). (b) Three blowups around one element of the period5 saddle orbit for the same parameter values. The sequential blowups illustrate how an inaccessible orbit has tongues of both basins accumulating on it from both sides.
As expected, this value is between those applying for figs. 2b and 2c. Fig. 8a shows a blowup around an element of the period3 saddle illustrating that it is accessible. Fig. 8b shows blowups of the basin structure in a small region around one element of the period4 saddle for a case where A > A ~. The period4 saddle, which was accessible for A < A~' (fig. 6a), no longer is when A > A~' (fig. 8b).
2.2.
A simple example
In this subsection we describe a model onedimensional map, x.~: 1 = F(x.), which illustrates the concept of accessible boundary points in a
particularly simple way. The specific map* which *This m a p is similar to the onedimensional version of a twodimensional m a p treated in ref. 2 (i.e., set J = 0 in ref.. 2), a n d is similar to 1D maps treated by Mira [4], and Takesue and K a n e k o [4].
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C. Grebogi et al. / Basin boundary metamorphoses
Fig. 7. Schematic diagram of the stable and unstable manifolds of the period4 saddle orbit at tangency A = A,~ = 1.396. This diagram shows how the unstable manifold of one element of the period4 saddle orbit accumulates on the unstable manifold of the next element of the same orbit, which in turn accumulates on the next, and so on cyclically.
we investigate is piecewise linear and is illustrated in fig. 9a. This map has only two attractors, namely, x = + oo and x =  o o . In addition, as we shall show, it also has the following additional properties: i) There is a C a n t o r set basin b o u n d a r y separating the basins of the two attractors. ii) The only periodic orbit on the basin b o u n d a r y which is accessible from the attractor basin of x = + ~ ( x =  o o ) is the unstable fixed point at x= +1 (x=l). iii) Each b o u n d a r y point which is accessible f r o m the x = + m (x =  m ) basin is mapped to x = + 1 (x =  1) in a finite n u m b e r of iterates. iv) The C a n t o r set basin b o u n d a r y contains an infinite n u m b e r of unstable periodic orbits, and, except for the points x = _+1, all of these are inaccessible f r o m either basin. T o d e m o n s t r a t e (i)(iv) we first note that, for any point x , > 1, the map may be expressed ( x , + ~  1) = 5 ( x , 
1).
(2)
T h u s any initial condition in x > 1 generates an orbit which tends to + ~ , and x > 1 is part of the basin for the x = + ~ attractor. By symmetry,
x + 1). Hence, 0.2 < x < 0.6 is in the  ~ basin, and  0 . 6 < x <  0 . 2 is in the + ~ basin. This is illustrated in fig. 9b [the symbol 1 + (1  ) signifies that the designated interval maps to x > +1 (x <  1) in one iterate of the map]. Continuing the construction, one finds three open intervals which map to 1 + interval in one iterate and hence to x > 1 in two iterates (these intervals are labeled 2 + in fig. 9c), and three more open intervals which map to x <  1 in two iterates (labeled 2  ) . The 2 + ( 2  ) intervals are in the basin of + ~ (  ~ ) . We see that the basin b o u n d a r y must lie in [  1, + 1] and does not contain any of the eight intervals labeled 1 + , 1  , 2 + , or 2  . This is clearly the first two stages in a standard C a n t o r set construction; this demonstrates (i). [The fractal dimension of this C a n t o r set basin b o u n d a r y is ( l n 3 ) / ( l n 5 ) = 0.6826 . . . . ] Further. we see that the b o u n d a r y points accessible from 1 + (1  ) are x =  0 . 6 and x =  0 . 2 (x = 0.6 and x = 0.2), and these map in one iterate to x = +1 ( x =  1 ) . Similarly, the b o u n d a r y points accessible from the 2 + ( 2  ) intervals are seen to all map to x = + 1 (x =  1) in two iterates. Continuation of the Cantor set c o n s t r u c t i o n implies point (iii) and hence point (ii). Since the map takes three subintervals of [  1 , + 1 ] and linearly stretches them into the whole interval (the subintervals are [  1 ,  0 . 6 ] , [  0 . 2 , +0.2], and [0.6,1], there are an infinite n u m b e r of unstable periodic orbits* in [  1 , + 1]. These points cannot lie in either basin and hence must lie on their boundary. However, by (iii) they are also not accessible. Hence (iv) follows.
*To see this, note that each of the three subintervals themselves have three subintervals (cf. fig. 9c) which are mapped onto [  1, + 1] in two iterates, and so on. Thus, the n times iterated map has 3" subintervals which are linearly stretched and mapped to [ 1, + 1]. Each of these thus contains a fixed point of the n times iterated m a p . Hence, there are 3"/n periodic orbits of period n.
C. Grebogi et al. / Basin boundary metamorphoses
(o)
1.44095
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Fig. 8. (a) Same as fig. 6a b u t showing an element of the period3 saddle orbit and A = 1.5. (b) Same as fig. 6b b u t showing an element of the period4 saddle orbit and .4 = 1.5.
As an example of an inaccessible boundary point, consider the fixed point x = 0. From the construction of the Cantor set, we see that this point certainly is on the boundary but that any e neighborhood of it contains an infinity of alternating segments of the two basins. [Note that the accessible points are those points which are end points o f one of the intervals deleted as one con
structs the Cantor set. In addition, they are precisely those points that map exactly onto one or the other fixed points at x =_+1. Such points could be considered the (generalized) stable manifold of the fixed points.] The example just given is analogous to the fractal basin boundary in fig. 2b in the following ways. First, a straight line cutting across the basin
C. Grebogi et al. / Basin boundary metamorphoses
292
Xn+l
/
(a)
/ Xn
f 2
(b)
,
, I+ ,
0 I ~
'
(c)
Fig. 9. (a) The map x,+ 1 = F(x,). (b) The interval 1 + (1  ) maps on one iterate to x > 1 (x <  1). (c) The intervals 2 + (2  ) map on two iterates to x > 1 (x <  1).
boundary of fig. 2b intersects it in a Cantor set along that line. Second, there are two accessible periodic orbits in the boundary in both cases: the period one saddle and the period four saddle, for fig. 2b; and the two unstable fixed points x = + 1, for the onedimensional map model. Third, all points accessible from the white (black) region of fig. 2b are on the stable manifold of the period four (one saddle); while all accessible points of the + oo(  oo) attractor for the onedimensional map model m a p to x = + l ( x   1) in a finite number of iterates. Fourth, both boundaries contain an infinite n u m b e r of inaccessible unstable periodic orbits, which, in the case of fig. 2b, are saddles. Section 3 will, in fact, show how an infinite number of saddles get buried in the boundary as a result of the basin boundary metamorphosis.
Now we consider how the one dimensional map with a fractal basin boundary, given in fig. 9, might come about with variation of a system parameter. That is, we consider a basin boundary metamorphosis for a onedimensional noninvertible map. As shown in fig. 10, we imagine that as some parameter p is raised from Pl to P2 the lower minimum of the map moves from F ( x ) >  1 at P = P l [fig 10a] to F ( x ) =  1 at P=Psf [fig. 10b] to F ( x ) <  1 at P=P2 [fig. 10c]. For P = P l the basin boundary is simply the single point x = 1, but there is also an invariant Cantor set in 1 > x >  1 (i.e., entirely in the + oo basin) generated due to the fact that the positive maxim u m of F(x) is greater than one. For fig. 10c there is a fractal basin boundary (as in fig. 9), and the conversion to fractal occurs at p = Psf. When this occurs the basin boundary dimension changes discontinuously. In particular, it is zero for p < psf (the boundary is the single point x = 1), but as p approaches Par from above it is the dimension of the invariant Cantor set in the limit as p approaches p~f from below. Furthermore, as p increases through p~f the basin boundary jumps far into the interior of the + oo basin. Indeed, for p just slightly bigger than p~f, there are thin interval pieces of the  o o basin near all the elements of the Cantor set which, for p < Pat, existed entirely in the + oo basin. Just as the thin filaments which j u m p into the white basin in fig. 2c approach zero width as A~Aef from above, so too do the abovementioned interval pieces of the  oo basin approach zero as p , p~f from above. F r o m the point of view of directly observable consequences, perhaps the most significant aspect of the basin boundary structure we have described here in section 2 is its implications for the final crisis in which the chaotic attractor is destroyed as it collides with a saddle on the basin boundary. Although there are an infinite number of saddles embedded in the basin boundary, the attractor can collide only with that one which is accessible to it. Thus, the type of crisis is, in a sense, determined by the accessible boundary saddle at the crisis (cf. figs. 4). Further, a crisis transfer may be thought
C Grebogiet al. / Basin boundarymetamorphoses
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of as a change of the accessible boundary saddle at the crisis.
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:~n
i
(c)p:p 2 Fig. 10. Metamorphoses. in onedimensional noninvertible maps. (a) F(x)> 1 at p=p]; (b) F(x)= 1 at P =Psf; (C) F(x)< 1 at P=P2.
It is evident from the discussion of figs. 2 in section 2.1 that the two events occurring at A = Asf and at A = A u are closely analogous. Both involve homoclinic crossings of accessible boundary saddle orbits: l u x ls at Asr and 4u × 4s at Aff. (At those values the relevant stable and unstable manifolds are tangent.) Also, both transitions are accompanied by a jump of the basin boundary inward into the white region. For the case of the smoothfractal basin boundary metamorphosis, the inward jump occurs because the period4 saddle is a finite distance from the basin boundary w h e n A < Asf , while for A > Ast there are parts of the basin of the attractor at infinity which are arbitrarily close to the period4 saddle. A similar statement applies for the period3 saddle at the fractalfractal basin boundary metamorphosis. With this in mind, we study the events leading up to the metamorphosis at A u (since Art is similar). In particular, we are interested in determining why the basin boundary suddenly jumps inward to the period4 saddle at the onset of l u x ls, and in determining how an infinite number ofother inaccessible saddles get buried in the basin boundary. We do not have a complete understanding of the answers to these questions. For example, we shall show how a particular class of saddles get buried in the basin boundary. We call saddles in this class simple Newhousesaddles*. While there are an infinite number of simple Newhouse saddles they are a comparatively small subset of all the buried saddles. Nevertheless, our study will be.sufficient for understanding why the basin boundary jumps inward at Asp *Newhouse [5] was one of the first to give an extensive picture of how saddlenodebifurcationsoccur near homoclinic tangencies, and he developedan extensivetheory of them. See also references in [6].
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C. Grebogi et aL / Basin boundary metamorphoses (a)
3.1. Simple Newhouse orbits We begin by describing in more detail the sequence of events for J = 0.3 as A is increased. As A increases a saddlenode bifurcation occurs at A t =  ( 1 + J ) 2 / 4 =  0 . 4 2 2 5 in which a period1 attractor and a period1 saddle are created. In addition, an infinite number of other saddlenode bifurcations of other higher periods also occur for larger A. F o r these higher period orbits the stable orbit created is generally observed to proceed rapidly (with increase of A) through a perioddoubling cascade to a final crisis that destroys the attractor and its basin. The remnant of such a sequence following a periodN saddlenode bifurcation (i.e., what is left following the final crisis) is a horseshoe (or at least it rapidly becomes a horseshoe) containing an infinite n u m b e r of saddle orbits. In particular, one of these unstable orbits will be the periodN saddle created at the original saddlenode bifurcation. We call these saddles original saddles. These saddles have the property that their associated eigenvalues of the linearized m a p are b o t h positive (since J > 0). Furthermore, we shall concentrate our discussion on a special class of these original saddles. In order to delineate this class, consider fig. l l a which schematically illustrates the stable and unstable manifolds of the original period1 saddle for a value of A below that at which the manifolds are tangent. As A increases, a homoclinic tangency, as shown in fig. l lb, occurs at some value of A which we d e n o t e d A~'. (This value is also the value after which the basin b o u n d a r y of infinity becomes fractal, A~' = Asf. ) For now, however, we wish to discuss the occurrence of saddlenode bifurcations (and hence the creation of original saddles) for A < A~' as A ~ A~'. In particular, we observe an infinite sequence of saddlenode bifurcations of period 3, 4, 5, 6, 7 . . . . . which occur at parameter values A 3, A 4, A 5. . . . . where A3 < A4 < As < A 6 < A 7 < A s " " " < A ~'. The particular bifurcations which we refer to above are those which create original saddles that proceed once around a circuit following the unstable maplf,~!d, as sche
(b)
Fig. 11. Schematic illustration of the stable and unstable manifolds of the period1 saddle for (a) A < A~' (A~' is the value of A at homoclinic tangency) and (b) A = A?. matically illustrated in fig. 12 for a period5 case. W e call these orbits simple Newhouse orbits*. It can be shown [5] that for large n A~'  A , ,  X
n,
*Actually [5, 6] there is a much larger class of saddlenode bifurcations than the ones we concentrate on here. For example, in general, a high period orbit may proceed several times around the loop before returning to its original value.
C. Grebogi et at / Basin boundary metamorphoses
295
(a)
.
.
.
.
.
.
.
.
.
Fig. 12. Schematic illustration of a simple Newhouse orbit of period5. Dashed arrows indicate the sequence in which points on the orbit occur.
and, furthermore, the width zan of the window is
(b)
[6] I0 8
where ~, > 1 is the unstable eigenvalue at the original period1 saddle for A =A~', and A denotes the difference between A n and the value of A at the final crisis of the attractor created at A n. Thus we see that, for large n, the range of A over which the attractor exists is small compared to the range of A between the initial periodn saddlenode bifurcations and the homoclinic tangency of the period1 saddle manifolds.
6 Y
4 2 0 2 2
I
0
I
2
x
3.2. Crossing of the stable manifoM of simple Newhouse orbits with the unstable period1 manifold For example, the saddlenode bifurcation of the simple Newhouse orbit of period3 occurs at A =1.16 and the final crisis of the resulting chaotic attractor (which consists of three pieces) occurs at A  1.25. Fig. 13a shows a schematic diagram of the basin of attraction for the period3 attractor at A = 1.25. The actual numerically determined period3 basin appears in fig. 13b. Also shown in
Fig. 13. (a) Schematic view of the basin for the period3 attractor (cross hatched), the period1 attractor (white), and the attractor at infinity (shaded). The unstable manifold of the period1 saddle is also shown. (b) Numerically determined basin of attraction (black) for the period3 attractor for A = 1.25 and J = 0.3.
fig. 13a are the basin of attraction for attractor, the unstable manifold of period1 saddle, and the location of saddle. The basin boundary for the
the period1 the original the period3 period3 at
296
C Grebogi et al. / Basin boundary metamorphoses
tractor is the stable manifold of the period3 original saddle, and the basin boundary of infinity is the stable manifold of the period1 saddle. When the period3 saddlenode bifurcation occurs, it immediately creates a basin for itself by cutting a chunk out of the period1 basin of attraction*. It is important to note from fig. 13a that the stable manifold of the period3 saddle (the basin boundary for the period3 attractor) cuts across the unstable manifold of the period1 saddle. In fact, this is true for all values of A for which the period3 saddle exists, including those exceeding the crisis value for the three piece chaotic attractor (which evolves via a period doubling cascade from the period3 node). In this range, A~ 1.25, of course, the stable manifold of the period3 saddle is no longer a basin boundary. (In fact, we shall be most interested in what happens near A~'  1.315). Furthermore, we observe an analogous situation to hold for all simple Newhouse orbits. That is, for any simple Newhouse orbit the stable manifold
always cuts across the period1 saddle's unstable manifold. (While at A = 1.25 the period3 saddle has one branch of its unstable manifold becoming tangent and crossing its stable manifold, we are interested in what the other unstable branch does (cf. next subsection).)
3.3. Crossing of the unstable periodN manifold with
the stable periodN + 1 manifold We have numerically examined the stable and unstable manifolds of the simple Newhouse orbits, and we have observed the following sequence of events: 1) After A  A~, 5  1.308 the unstable rnanifold of the period4 saddle crosses the stable manifold of the period5 saddle. * The area of the period1 basin of attraction in the region shown in fig. 13 decreases discontinuously at the period3 saddlenode bifurcation. Note, however, that the total area of each basin must be infinite. This follows immediatelyfrom the facts that (1) contracts areas by J = 0.3 and that the basins are invariant under (1). The area of the set of points in figs. 13 which remains bounded appears to change continuously.
2) After A = A~. 6 ~ 1.310 the unstable manifold of the period5 saddle crosses the stable manifold of the period6 saddle. 3) After A = A*6,7  1.311 the unstable manifold of the period6 saddle crosses the stable manifold of the period7 saddle. Fig. 14 shows the crossings of the numerically determined period4 unstable (4u) and period5 stable (5s) manifolds. One can argue rigorously that an infinite number of such crossings A,,,+~ occur as A )A~': 4u X 5s
5u x 6s 6u x 7s 7u x 8s
(3)
8u x 9s
N u X ( N + 1)s
where N u and Ns stand for the periodN unstable (stable) manifold, and x signifies crossing of the two relevant manifolds*. 3.4. Heterocfinic manifoM crossings We now review the significance of the crossing of a stable and an unstable manifold when each are associated with different saddle periodic orbits. Consider two saddles, one of period M and one of period N. Say, that as a parameter of the system is varied the unstable manifold of the periodM saddle crosses the stable manifold of the periodN saddle. Also assume that the eigenvalues associated with these saddles are positive (as for simple Newhouse orbits). The situation is as illustrated in fig. 15. Evidently the unstable manifold of the * Note. however, that the period3 saddle does not appear in the sequence (3). Its crossing, 3u×4s, occurs later, immediately following A =A~ =Aft> Asf. This is implied by analogy with the lu x ls event at A~' for which we show (section 3.5) that the lu x ls and 4u x Is events are simultaneous.
C Grebogi et al. / Basin boundary metamorphoses
297
(stable) manifold (both branches included) of the simple Newhouse saddle of period N. Eqs. (4) have strong implications for the structure discussed in sections 3.2 and 3.3. According to section 3.2, the stable manifold of any simple Newhouse orbit crosses the unstable manifold of the period1 saddle. Thus, from (4),
I0 8 6 4 2 0 2 I
o
i
l"u_ Nu,
(5a)
N's_Dls
(5b)
for any periodN simple Newhouse orbit. Similarly sequence (3) of section 3.3 implies that as A ~ A sf from below, we have
2
Fig. 14. Numericallyobtainedperiod4 unstablemanifoldand period5 stable manifoldat A = 1.3087, J = 0.3.
4u_5u~6u_D
...,
(6a)
4s ___5s ___6s ___ • . . .
(6b)
Combining (5) and (6) we have
MI Fig. 15. Heteroclinic crossings for a periodM saddle orbit and a periodN saddle orbit. Onlyone point of each periodic orbit is shown.
lu_D4u_5u_~6u_ . . . ,
(7a)
ls__.4s___5sc_6sc_ . . . ,
(Tb)
for A > A~'. Thus, for example, eq. (7a) implies that the unstable manifold of the period1 saddle is extremely complex and convoluted, since its closure includes all the unstable manifolds of all simple Newhouse orbits. 3.5. Homoclinic crossing for the period1 saddle
periodM saddle limits on the periodN unstable manifold, and the periodN stable manifold limits along the entire periodM stable manifold. Thus the closure of the periodM unstable manifold contains the periodN unstable manifold, and the closure of the periodN stable manifold contains the periodM stable manifold, Mu D Nu,
(4a)
Ns ~ Ms,
(4b)
where the overbar denotes closure. In statements like (4), we write Nu(Ns) to denote the unstable
We now claim that when the stable and unstable manifolds of the period1 saddle cross, then simultaneously the 4u, 5u .... manifolds also cross the period1 stable manifold. That is, the relationships ls x l u and ls × Nu, N > 4, are valid immediately following A = A~. In order to see how this arises consider, for example, the period4 unstable manifold. By eq. (7a), and as illustrated in fig. 16, 4u comes arbitrarily close to all the simple Newhouse orbits of period 5,6 . . . . . As A ~ A ~ an infinite number of such orbits are created by saddle node bifurcations. The locations of these orbits are closer and
C. Grebogi et al. / Basin boundary metamorphoses
298
t4
unstable manifolds become equal and similarly for their stable manifolds. In particular, the basin boundary for the attractor at infinity will be (Basin boundary) = ls = 4s.
(9c)
Since the period1 and the period4 saddles are the only accessible saddles, all the other simple Newhouse orbits ( N > 4 ) are "buried" in the boundary.
3.6. Basin boundary jumps Fig. 16. Schematicillustration of the implications of Nu x (N + l)s. It is seen that the left branch of 4u comes arbitrarily close to the simple Newhouseorbits of periods 5,6, 7.....
closer to the stable manifold of the period1 saddle as the period of these orbits gets higher and higher. In fact, in a suitable sense we can regard the tangency points of ls and lu at A A~' (cf. fig. l l b ) as the limit of the elements of a simple Newhouse saddle of period N as N, ~ (cf. ref. 6). Thus, as A ~ A*, the closure of the period4 unstable manifold touches the period1 stable manifold. As soon as A exceeds A~', a set of tongues, as in fig. 5, come shooting up along the formerly smooth basin boundary. These tongues must cross the period4 unstable manifold. Hence, ls x lu and ls x 4u become true simultaneously and similarly for Is x Nu, N > 4. Since the ls x Nu, N > 4 relationships become true simultaneously, we see that eq. (4) implies that N~_lu
and
NsC_ls
(8)
for A > A~', N >_ 4. (Note that the period3 simple Newhouse orbit does not experience this type of event at A 1.315.) Eqs. (8) and (7) imply that for A >AI*=1.315 we have for the simple Newhouse orbits lu = 4 u = 5 u = 6 u . . . . Is=4s=5s=6s
....
(9a) .
Thus we see from eqs. (9) that all the simple Newhouse orbits of period greater than or equal to 4 lie in the basin boundary. Furthermore, equalities (9a) and (9b) imply a striking discontinuity in the boundary at A~e. The period4 saddle is certainly not on nor is it especially near the boundary at or before A~f; yet we see it is part of the boundary immediately after A st. Since this orbit necessarily moves smoothly and continuously, it follows that the boundary discontinuously jumps inward to the orbit. This also holds for the period5 (simple Newhouse) saddle and period 6, etc. But the lower the period is, the more striking the conclusion since the orbits of lower period are further from the boundary. For other values of J, the period 4 must be replaced by some appropriate period and the accessible orbit might not be a simple Newhouse orbit. Based upon the ideas developed in this paper, the following rigorous result [61 has been obtained for the smooth fractal transition:
(9b)
Thus at the smoothfractal basin boundary bifurcation the closures of all the simple Newhouse
Theorem. Consider a typical map with a saddle fixed point or periodic orbit P that has a transition value Asf as A increases (where the stable and unstable manifolds of P cross for the first time). Assume the absolute value of the determinant of the Jacobian at P is less than one (that is, the Jacobian of the Nth iterate of the map where N is the period of P). Then there will be a periodic saddle Q that is in the closure of the stable manifold of P for all A slightly greater than A~r but is not in it at A~r. Furthermore, P will be in
C Grebogi et al. / Basin boundary metamorphoses the closure of the stable manifold of Q for all A slightly greater than A a. S. Hammel and C. Jones [7] were the first to prove a result with these hypotheses concluding that the basin boundary jumps inward. Their version makes no mention of the orbit Q. Their techniques are totally different from the ideas in eqs. (9), though the ideas in eqs. (9) are a sufficient foundation to give a rigorous proof of this theorem. In addition, note that this theorem also applies to the A ff transitions, since for the period4 saddle the first crossing of its righthand (inward) unstable manifold with its stable manifold follows A ff.
299
complex sequence of events which precede the basin boundary metamorphosis (section 3). In particular, a saddle set lying in the basin grows via a chain of crossings of stable and unstable manifolds. Eventually this inflated saddle set collides with the boundary, resulting in the metamorphosis described above*. vi) The saddle which a chaotic attractor collides with at a boundary crisis is the boundary saddle accessible to that attractor. As a parameter is varied, a change in the accessible saddle at the crisis boundary is called a crisis transfer (section
1.3). Acknowledgements
4. Summary In this paper [8] we have investigated sudden large scale changes in basin boundaries with variation of a system parameter. Our conclusions are as follows: i) The basin boundary can jump in size and change its character as the system parameter passes through certain critical values, and we call these changes basin boundary metamorphoses. ii) Metamorphoses can occur as a result of homoclinic intersections of the stable and unstable manifolds of a saddle periodic orbit on the basin boundary. iii) The structure of the basin boundaries which we investigate is, to a large extent, determined by the accessible saddles which lie on the basin boundary (section 2). iv) The character changes referred to in (i) are changes in the accessible saddle orbits on the basin boundary and sometimes a change of the boundary from being smooth to being fractal (as in the transition from fig. 2a to fig. 2b but not from fig. 2b to fig. 2c). v) The way in which saddles get buried in the basin boundary (thereby becoming inaccessible) and the means by which basin boundaries can jump in size have been investigated, and it has been shown that these are accomplished by a
We would like to thank ShenTeng Yang for making the pictures of fig. 1 and BaeSig Park for numerically determining the dimension of the basin boundary corresponding to fig. 2b. This work was supported by the Department of Energy (Office of Basic Energy Sciences), DARPA under NIMPP, and the Office of Naval Research.
Appendix Finding accessible boundary saddles In fig. 2, we have indicated accessible saddles on the basin boundary. Here we describe our numerical technique for the determination of these accessible periodic boundary saddle orbits. Say we have two attractors, attractor A and attractor B, and we wish to find a boundary saddle orbit that is accessible from the basin of attractor B. We follow the following steps: 1) Choose two points close to the basin boundary that go to the two different attractors, say, point PA goes to attractor A and PB goes to attractor B.
*Alternatively, the attractor may collide with this inflated saddle set before this set has collided with the boundary (an interior crisis). This is what is happening at the period three interior crisis of fig. 3a.
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C. Grebogi et al. / Basin boundary metamorphoses
2) Take 64 points equally spaced in the segment
References
PAPa and n u m b e r them sequentially so that PA = 1 and PB = 64. (We choose 64 points because 64 is the vector length in Cray computers.) 3) Iterate the 64 points and determine the highest n u m b e r e d point which goes to the attractor A; call this the new point PA4) M o v e the point PB a fraction (say, a fifth) of the distance PAPa towards PA, and make sure that the new point PB is in the basin of B. If it is not, m o v e back in small steps (in the direction away f r o m PA) until it is. 5) G o back to step 2 and keep repeating the p r o c e d u r e until the distance PAPB is less than, say 1012 6) Iterate PA and PB and print the first, say, 50 points of the trajectories. 7) Extract the period and the elements of the accessible b o u n d a r y saddle orbit from the printed trajectories. F o r the cases in figs. 2, for different choices of the initial points P^ and PB, this procedure always yields the indicated b o u n d a r y saddles accessible from the black and white regions, and we thus believe that these are the only accessible b o u n d a r y saddles.
[1] C. Grebogi, E. Ott and J.A. Yorke, Physica 7D (1983) 181. [2] S.W. McDonald, C. Grebogi, E. Ott and J.A. Yorke, Physica 17D (1985) 125. [3] C. Grebogi, S.W. McDonald, E. Ott and J.A. Yorke, Phys. Lett. 99A (1983) 415. [4] C. Grebogi, E. Ott and J.A. Yorke, Phys. Rev. Lett. 50 (1983) 935; S.W. McDonald, C. Grebogi, E. Oft and J.A. Yorke, Phys. Lett. A107 (1985) 51. C. Mira, C.R. Acad. Sc. Paris 288A (1979) 591; I. Gumowski and C. Mira, Dinamique Chaotique (Cepadues, Paris, 1980). F.T. Arecchi, R. Badii and A. Politi, Phys. Rev. A32 (1985) 402. M. Iansiti, Q. Hu, R.M. Westervelt and M. Tinkham, Phys. Rev. Lett. 55 (1985) 746. R.G. Holt and I.B. Schwartz, Phys. Left. A105 (1984) 327; I.B. Schwartz, Phys. Lett. A106 (1984) 339; I.B. Schwartz, J. Math. Biol. 21 (1985) 347. E.G. Gwinn and R.M. Westervelt, Phys. Rev. Lett. 54 (1985) 1613. Y. Yamaguchi and N. Mishima, Phys. Lett. A109 (1985) 196. O. Decroly and A. Goldbeter, Phys. Lett. A105 (1984) 259. S. Takesue and K. Kaneko, Progr. Theor. Phys. 71 (1984) 35. F.C. Moon and G.X. Li, Phys. Rev. Lett. 55 (1985) 1439. [5] S.E. Newhouse, Publ. Math. IHES 50 (1979) 101. N.K. Gavrilov and L.P. ShJlnikov, Math. USSR Sbornik 17 (1972) 467. [6] K. Alligood, L. TedeschiniLalli and J.A. Yorke, private communication. [7] S. Hammel and C. Jones, Jumping stable manifolds for dissipative maps on the plane, preprint.. [8] A preliminary version of this work is contained in C. Grebogi, E. Ott and J.A. Yorke, Phys. Rev. Left. 56 (1986) 1011.
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