Attractors stuck on to invariant subspaces

25 December 1995

cQ& __ __ @

PHYSICS

ELSEWIER

LETTERS

A

Physics Letters A 209 (1995) 338-344

Attractors stuck on to invariant subspaces Peter Ashwin lnstitut Non-L&are

de Nice, 1361 Route des Lucioles, 06560 Valbonne, France

Received I I July 1995; accepted for publication 30 October 1995 Communicated by A.P. Fordy

Abstract This note considers some attractors for maps with invariant subspaces. An example is presented with a family of attractors (displaying on-off intermittency) that intersect their reflections along a reflection plane. This is a robust example of (a) an attractor that is “stuck on” to its basin boundary and (b) two attractors in a symmetric system that collide at a reflection plane without merging. A further example with IID3symmetry having attractors stuck on to more than one reflection plane is presented.

1. Introduction It has been recognised that the existence of invariant subspaces can have surprising consequences for the dynamics generated by mappings and flows. For example, coupled systems can have complicated invariant sets in any neighbourhood of a synchronised attractor [ 11. In particular, the phenomenon of “riddled basins” has been identified [ 2-51 where an attractor within a dynamically invariant subspace can have a basin of attraction that is large in the sense of Lebesgue measure but small topologically, namely its complement can be open and dense in the whole phase space. Another closely related phenomenon is that of “on-off intermittency”, a dynamical state that exhibits intermittent large deviations away from an invariant subspace [ 61. Ott and Sommerer have shown that these are connected by being two types of dynamics near what they call a “blowout bifurcation” [ 71. This is a bifurcation of an attractor contained within an invariant subspace causing it to lose transverse stability. This bifurcation has been investigated in the

context of parametrised families that vary the transverse dynamics while preserving the dynamics in the invariant subspace [ 831. Both “riddled basins” and “on-off intermittency” have been found in several numerical and physical experiments and scaling behaviours have been analysed (mostly within phase space): see for example Refs. [g-12]. As detailed e.g. in Refs. [ 8,5] it is possible for invariant subspaces to arise in a number of ways; for example, in dynamics on manifolds with boundary [ 31, or in dynamics of driven oscillator systems. Notably, symmetry can cause a subspace to be invariant in the following way. Suppose the mapping f : W” --+ R” commutes with the action of a compact Lie group r. The so-called fixed point spaces of the subgroups Z of r, Fix(Z)

= {X E R” : ax=xforallaEZ}

are necessarily Elsevier Science B.V. SUN 0375-9601(95)00857-8

f-invariant.

P. Ashwin/

A matter of interest for attractors of symmetric maps is to what extent they intersect their images under the action of the group. It has been shown by Chossat and Golubitsky [ 131 that for certain definitions of an attractor (notably for attractors that are asymptotically stable) an attractor A satisfies the following se? dichotomy. For all elements p E I either or

A f~ pA = 8

AnpA=A.

(1)

on average 2~ of the attractor A corresponds to precisely all p such that A fl pA = A; The symmetry

this clearly forms a subgroup of I. For attractors that satisfy this condition, it is possible to say a lot about the admissible symmetries [ 14,151 and their generic changes [ 161. The definition of attractor can be weakened to a certain extent (see, e.g., Ref. [ 17] ) such that this dichotomy relation still holds. Since attractors with riddled basins are never asymptotically stable but are nonetheless observable in numerical experiments, one can weaken the definition in the following way. Suppose A is a compact invariant transitive set. Define B(A) to be the basin of A, i.e. all points whose w-limit sets are contained in A, Following Milnor’s definition [ 181 one can say A is an attractor if 1(B( A) ) > 0 and any proper compact invariant subset A’ has f( B( A’) ) = 0. (1( ) denotes Lebesgue measure.) One could ask whether ( 1) holds for attractors under this weaker definition. This note shows that this is not always the case. Proposition 2.1 exhibits a family of Milnor attractors A of a map with reflection symmetry K such that An KA =Ao,

339

Physics Letters A 209 (1995) 338-344

(2)

where Aa is a chaotic invariant set embedded in Fix( (K) ) . In particular, A0 is neither A nor the empty set. Moreover, the basin of attraction of A has open interior, i.e. the basin is not riddled. We prove in Proposition 2.2 that there are no attractors of nearby maps that are symmetric and not contained in the invariant x-axis; in this sense, the example gives a family of attractors satisfying (2) in a robust way. We give an example of a ID)3equivariant mapping with attractors that are stuck on to two invariant subspaces. We also observe a blowout bifurcation to a

state that is intermittent between three invariant subspaces, reminiscent of the “cycling chaos” observed recently by Dellnitz et al. [ 191. However, in our case all connecting trajectories are within one invariant subspace. The discussion in Section 3 distinguishes between two fundamentally different types of collision of attractor at an invariant subspace; those involving linear approach to a nonnormally repelling chaotic saddle and those involving the image of critical point of the map hitting a normally repelling chaotic saddle. Only the former are robust and do not cause merging of attractors.

2. Example of an attractor invariant

“stuck on” to an

subspace

Consider the mapping f defined by

where (x, y) E 0 is the phase space and A E R is a parameter. Fix 1.1768 > S > 1, R > 0 and 0 = (-&a) x (-R, R) a rectangle in W2. R is chosen large enough so that the map satisfies f (i2)c Q in turn implying the existence of a global attractor for the dynamics. The map g is g(x) = ;&x(X2

- l),

the cubic logistic map such that there is a unique attractor Aa = [-l,l] with basin (-1.1768,1.1768) (the limits correspond to unstable fixed points of g) and a unique absolutely continuous natural measure supported on Aa [ 201. A closely related map was studied in Refs. [ 8,5]. Some relevant facts from this previous study are summarised below: (1) There is a chaotic invariant set A0 within the invariant subspace. This supports an absolutely continuous invariant measure ,u.e that is generic for a set of initial conditions of full measure in an open subset of the x-axis containing Ao. (2) For 0 < A < 1 the set A0 is asymptotically stable. (3) For 1 < A < 1.430 (= eKsRB)the set A0 has an open basin of attraction but is not Lyapunov stable. Moreover, the basin is locally riddled.

340

P. Ashwin / Physics Letters A 209 (1995) 338-344

(4) For 1.430 < A < 1.850 (= eKmln)the set A0 is a chaotic saddle, i.e. not an attractor. (5) For A > 1.850 the set A0 is a normally repelling chaotic saddle, i.e. all dynamics in A0 are transversely unstable. These transitions can be seen in terms of loss of transverse stability of various ergodic invariant measures supported in A0 (see Refs. [ 851). The transition at A = 1.430 where m loses stability is the so-called blowour bifurcation. We claim that any attractor in case (4) must contain Ao. More precisely, let r( X, y) = (x, 0) be projection onto the x-axis.

R

Lemma 2.1. For E = 0, any Milnor attractor A ( A) hasrA=Ao.

c

ProojI Note that f has skew product structure for E = 0. Suppose A has a positive two-dimensional Lebesgue measure basin of attraction. Then T(A) must be an invariant set with a positive onedimensional Lebesgue measure basin of attraction for the map g. Since A0 is the unique attractor for g in (-1.1768, 1.1768) it follows that T(A) = Ao. 0 Proposition 2.1. For E = 0 and 1.430 < A < 1.850 any Milnor attractor A( A) c R must contain Ao. Proofi For such a value of A the invariant set A0 is unstable (i.e. the I( a( AC,)) = 0) but there is also a dense set of points in A0 that are transversely stable. We pick a transversely stable invariant set C contained in A0 (with transverse Lyapunov exponent strictly less than unity) and show that the set of stable manifolds of points in C must intersect A, by Lemma 2.1. Since A is f-invariant, it follows that A0 is contained in A. To see that the stable manifolds of C intersect A, note that for x E [-l,l], f,(x,y) is a monotonic map with monotonically decreasing derivative in y E W+. The skew product structure of f means that ( fn)Y ( X, y) will be a composition of such maps, and this property is preserved under composition. For x E C, the transverse Lyapunov exponent implies that there are copstants K > 0 and p < 1 such that a ( f n(0, y) ),/ay < Kp”. Monotonicity of the derivative implies that all points on C x II%+converge to points in C. Cl

Note that we have not proven the existence of the Milnor attractor, only that it contains A0 if it does exist. The above proposition implies: Corollary 2.1. The attractors A (A) of Proposition 2.1 intersect their basin boundaries at a set of codimension one in the phase space, i.e. they are stuck on to their basin boundaries. We now show that this behaviour is robust to perturbations in a certain sense. Note that this includes perturbations that break the special skew-product structure of the map for E = 0. In general, the attractor may collapse onto a smaller attractor, but numerical examples presented in Section 2.1 suggest that there may be a large measure set of nearby parameter values such that there are attractors A satisfying (2). By Cd (s1, W2) we denote the space of continuously differentiable maps from fl to R2 preserving the x-axis. Proposition 2.2. There is a neighbourhood N of f in Cd (a, lR2) such that no mappings in this neighbourhood have an attractor A with points in the interior of both upper and lower half planes. ProoK This follows from the fact that f maps the upper and lower half planes into themselves in a robust manner. More precisely, for all (x, y) E 0 we have afy/ay > K for some uniform K > 0. Hence for any K > K’ > 0 there is a neighbourhood in the Ci topology such that all nearby g have ag,/ay > K’. If we restrict to g that leave the x-axis invariant, this implies that g, (x, y) > 0 if and only if y > 0. Thus no orbit can visit the interior of both the upper and lower half planes. Cl If we assume that there is a unique attractor for each value of A, this implies that the basin of attraction is a full measure subset of the upper half-plane intersected with a. Note also that by the above argument, for A > 1.850 the attractor “lifts off” the invariant subspace and an open neighbourhood of the attractor is contained in its basin of attraction. Numerical observations suggest that the Hausdorff dimension of the attractor decreases in A from 2 at the point of “lift off’.

P. Ashwin / Physics Letters A 209 (199s) 338-344

341

2.1. Numerical examples Using the dynamical system simulator DSTOOL the map (3) was explored numerically. Fig. 1 shows attractors for A = 1.5 for the case (a) E = 0 (skew product structure) and (b) E = -0.1. For E small and positive, the attractor breaks into two connected components (this is related to the fact that the attractor for the cubic logistic map can break into two components under small perturbations). As can be seen, both trajectories contain a segment of the x-axis in their closure (see Fig. 2). To investigate the same phenomenon in a different, possibly less contrived setting where we are no longer close to a skew-product mapping, the following lIDs equivariant mapping of the plane was investigated numerically, [ 211

Y

-1.5

I 1.5

-1.5 X

(4

(4) This is llD3equivariant as it is a linear combination of invariant functions multiplying equivariant vectors. For A = 7.2, LY- 1.1 and /I = -0.21, Fig. 3 shows an attractor that is “stuck on” to two distinct invariant sets contained in reflection planes; (a) shows a single trajectory while (b) shows six symmetrically related trajectories. This is observed to be stable to small changes in the parameters. On detailed examination, the group orbit of the attractor avoids the origin. This is a state that is on-off intermittent between two invariant subspaces. If different parameters are used, attractors are found that cross over the invariant planes. On loss of stability of an attractor in a reflection plane by a blowout, one can get a state similar to the “cycling chaos” of Dellnitz et al. [ 191. This is shown in Fig. 4 for A = -7.2, a = 3 and (a) p = -0.96 before the blowout bifurcation, and (b) p = -0.98 the cycling state. This state is on-off intermittent between three invariant subspaces.

3. Discussion We have demonstrated that attractors can stick on to their basin boundaries if there is an invariant subspace

-1.5 -1.5

1.5 X

(b)

Fig. 1. Examples of trajectories for (3) with A - 1.5 (a) E - 0 and (b) E - -0.1. For both of these systems there are two symmetrically related attractors, one in each of the upper and lower half planes; only the attractor of an initial condition in the upper half plane is shown. Both attractors arc “stuck on” to the x-axis in a robust manner.

342

P. Ashwin / Physics Letters A 209 (I 995) 338-344

1.5

4

Y

Y

-4

1.5

-1.5

-4 X

X

Fig. 2. Trajectories of two initial conditions for (3) and E - -0. in the

I. To

with A -

I .5 (4

distinguish the attractors 50000 points are shown

upperattractor

and 5000 in the lower attractor.

on the basin boundary. Moreover, this is a robust occurrence in that it persists under small perturbations that preserve the invariant subspace. These attractors have previously been observed in work on on-off intermittency, but this consequence for the basin boundary has, to the author’s knowledge, not been remarked upon. For more general perturbations one cannot expect the invariant subspace to be preserved because, as discussed in Ref. [ 51, they are not normally hyperbolic in a neighbourhood of the blowout bifurcation. Similarly, on introduction of (e.g. additive) noise to the system, any attractors that are not isolated will be joined together to yield a single attractor. All in all, the addition of noise will cause attractors between the loss of asymptotic stability and the gain of normal repulsion of an at&actor to be qualitatively similar to the “bubbling” attractor observed in Ref. [ 81. One might ask why such robust “stuck on” attractors have not been previously observed in numerical experiments on symmetric systems. For example, Refs. [ 221 and [ 161 only report collisions of conjugate attractors that give rise to “fused” attractors with more symmetry. To address this point, it seems apparent there are two distinct types of collision of attractor at invariant planes; (i) There exists a nonnormally repelling chaotic

-4

I -4 X

(b)

Fig. 3. Examples of trajectories for the !&-equivariant with A = 7.2, cx = 1.1 and p -

-0.21.

(a)

100000

map (4) points for

an attractor that contains line segments in two separate reflection lines (but in fact does not contain the fixed point at the origin on closer examination).

(b)

The trajectory in (a)

for five other symmetrically related trajectories.

and 1000 points

P, Ashwin / Physics Letters A 209 fI 995) 338-344

4 \

“1 Y

\

\ J

-4 -4 X

(4 4

Y

343

saddle in the invariant subspace. This can cause linear approach to the invariant subspace along stable manifolds of dynamics embedded in the chaotic saddle and is the case investigated in this paper. Small changes in the map will adjust the rate of approach to the chaotic saddle but as long as the chaotic saddle has transversely stable manifolds one can expect to see this persist. (ii) An image of a critical point of the map within an attractor intersects the invariant subspace. For this case small perturbations will cause the critical point to fall on different sides of the invariant subspace and so there will typically be nearby systems with attractors that are “fused” (cf. second iterate of the map (3) on changing E through zero). Note that this can happen whether the chaotic saddle is transversely repelling or not. Moreover, attractors which collide at a periodic point must undergo this type of collision. If the chaotic saddle is not normally repelling then this corresponds to two “stuck on” attractors fusing together. Although chaotic saddles that are not normally repelling are as structurally stable as the chaotic dynamics (i.e. they can be expected for a positive measure set of parameters), they are quite difficult to find as the spectrum of changes from asymptotic stability to normally repelling chaotic saddle may take place over a small parameter range. References [ I] AS. Pikovsky and P. Grassberger, [2]

-4

[ 31 [4]

-4 X

[5]

(b) [6] Fig. 4. Trajectory of a single initial point for the D3-equivariant map (4) with A - -7.2, a - 3 (a) p - -0.96 and (b) p - -0.98. Between (a) and (b) there is a blowout bifurcation that gives rise to chaotic state visiting close to all invariant lines. Closer examination shows that both attractors avoid a small neighbourhcod of the origin. Before the blowout there is presumably a transitive invariant set with full symmetry close to the observed attractor in (b).

[7] [8] 191

[IO] [II

]

J. Phys. A 24 ( 1991) 4587. J.C. Alexander, 1. Kan, J.A. Yorke and Zhiping You, Int. J. Bifurc. Chaos 2 (1992) 795. 1. Kan, Bull. Am. Math. Sot. 31 ( 1994) 68. E. Ott, J.C. Sommerer, J.C. Alexander, 1. Kan and J.A. Yorke, Physica D 76 ( 1994) 384. P. Ashwin, J. Buescu and IN. Stewart, From attractor to chaotic saddle: a tale of transverse instability, Warwick preprint 7411994 ( 1994). N. Platt, E.A. Spiegel and C. Tresser, Phys. Rev. Lett. 70 ( 1993) 279. E. Ott and J.C. Sommerer, Phys. Lett. A 188 (1994) 39. P. Ashwin, J. Buescu and IN. Stewart, Phys. Len. A 193 (1994) 126. P.W. Hammer, N. Platt, SM. Hammel, J.F. Heagy and B.D. Lee, Phys. Rev. Lett. 73 ( 1994) 1095. J.F. Heagy, T.L. Carroll and L.M. Pecora, Phys. Rev. Lett. 73 (1994) 3528. J.F. Heagy, N. Platt and SM. Hammel, Phys. Rev. E 49 (1994) 1140.

344

P. Ashwin / Physics Letters A 209 11995)338-344

[ 121 N. Platt, S.M. Hammel and J.F. Heagy, Phys. Rev. Lett. 72 ( 1994) 3498. [ 13 ] I? Chossat and M. Golubitsky, Physica D 32 ( 1988) 423. [ 141 l? Ashwin and 1. Melbourne, Arch. Rat. Mech. Anal. 126 ( 1994) 59.

[ 1.5J 1.Melbourne, M. Dell&z and M. Golubitsky, Arch. Rational. Mech. Anal. 123 ( 1993) 75. and C. Heinrich, Admissible symmetry increasing bifurcations, University of Houston Mathematics Department Research Report UH/MD- 187 ( 1994). [ 171 1. Melbourne, Generalizations of a result on symmetry groups of attractors, to appear in: Proc. Fields Institute 1993.

[ 161 M. Dellnitz

[ 18) J. Milnor, Commun. Math. Phys. 99 (. 1985) 177. [ 191 M. Dellnitz, M. Field, M. Golubitsky, J. Ma and A. Hohmann, lnt. J. Bifurc. Chaos 5 ( 1995) 1243. [ 201 A. Lasota and J.A. Yorke, Trans. Am. Math. Sot. I86 ( 1973) 481. [ 211 J. Guckenheimer, M.R. Myers, F.J. Wicklin and PA. Worfolk. Dstool: a dynamical systems toolkit with an interactive graphical interface, user’s manual, Center for Applied Mathematics, Cornell University ( 1991). [22] P.J. Aston and M. Dellnitz, Symmetry breaking bifurcation of chaotic attractors. Int. J. Bifurc. Chaos, to appear ( 1995).