Anisotropic properties of riddled basins

19 February 2001

Physics Letters A 280 (2001) 139–145www.elsevier.nl/locate/pla

Anisotropic properties of riddled basins

Peter Ashwin ∗,1, Michael Breakspear 2

Department of Mathematics and Statistics, University of Surrey, Guildford GU2 7XH, UK

Received 12 September 2000; accepted 15 January 2001Communicated by A.P. Fordy

Abstract

We consider some general properties of chaotic attractors with riddled basins of attraction (basins with positive measure butopen dense complements) in dynamical systems with symmetries. We investigate how a basin of attraction can be riddled insome directions and not in others if the attractor is contained in the intersection of several invariant subspaces. This means thatthe extreme sensitivity to added noise (bubbling) associated with riddled basin attractors is in fact strongly dependent on thenature of the noise. We discuss examples of this for systems of globally coupled maps. 2001 Elsevier Science B.V. All rightsreserved.

1. Introduction

Suppose that f :M → M is a smooth map of someM = Rm to itself. If A ⊂ M is a compact invariant set,we define the basin of A to be the set of points whoseω-limit sets are contained within A; i.e., B(A) = {x ∈M: ω(x) ⊂ A}. We say A is an attractor if B(A) islarge in some sense; either an open set (as, for exam-ple, in the case of asymptotically stable attractors) orif B(A) has positive m-dimensional Lebesgue measure(in the case of Milnor attractors [12]).

If an attractor has a basin that has positive measurebut there are no open sets possessing full measuresubsets contained in the basin, then the basin is saidto be riddled. In a remarkable paper [1] it was shownnot only that there exist dynamical systems with

* Corresponding author.E-mail address: [email protected] (P. Ashwin).

1 Address from October 2000: School of Mathematical Sciences,Laver Building, University of Exeter, Exeter EX4 4QE, UK.

2 Also: Brain Dynamics Centre, Department of PsychologicalMedicine, Westmead Hospital, Sydney, NSW 2145, Australia.

invariant subspaces but also that these are in somesense ‘typical’ for dynamics with invariant subspaces.For systems possessing symmetries, such invariantsubspaces are forced to exist and so riddled basins arecommon for chaotic attractors in symmetric systems,especially for attractors that are have lost asymptoticstability but that are still Milnor attractors.

The study of such riddled basin attractors and re-lated phenomena has been undertaken from theoreti-cal and numerical viewpoints [1,4,11,15,16,20]mostlyusing transverse Lyapunov exponents and skew-product models. In particular the bifurcations to rid-dled basins [10,14], the sensitivity of such attractorsto noise (bubbling) [3,19] and the distinction of localand global riddling have been studied by several re-searchers [4,5,11].

In this Letter we turn our attention to properties ofriddling that can appear in the presence of more thanone invariant subspaces; we observe that riddling mayoccur in some directions but not others, we say thatriddling is typically anisotropic in transverse direc-tions and this gives rise to anisotropic sensitivity tonoise. We show there are internal riddling transitions

0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0375-9601(01) 00 04 3- 3

140 P. Ashwin, M. Breakspear / Physics Letters A 280 (2001) 139–145

where the number of directions in which riddling oc-curs changes. This is supported with some observa-tions from numerical simulations of coupled map sys-tems.

1.1. Symmetries of maps

We consider a smooth iterated mapping f :M →M for M = Rm some Euclidean space. We writeN(·) to denote Lebesgue measure on N any linearsubspace of M . Suppose that M = Rm commutes(is equivariant) with the action of some finite matrixgroup Γ acting on M (we only consider finite groupsin this Letter). This means that for any x ∈ M and forany g ∈ Γ we have that

gf (x) = f (gx).

This will cause [8] a number of linear subspaces of Mto be f -invariant. More precisely, given any subgroupΣ of Γ the fixed point subspace

Fix(Σ) = {x ∈ M: gx = x for all g ∈ Σ}is f -invariant. Not all subgroups give rise to distinctfixed point subspaces; however those that do can becharacterised by the isotropy subgroups of the actionof Γ on M; these are the subgroups Σ such thatthere is an x ∈ M with Σ = {g ∈ Γ : gx = x}; theseare precisely the possible symmetries of points in M .We therefore obtain, for a given group action, a finitenumber of linear subspaces Ni ⊂ M, i = 0, . . . , n, andisotropy subgroups Σi � Γ such that Ni = Fix(Σi)

and so f (Ni) ⊂ Ni . Observe that for any Ni and Nj

then Ni ∪ Nj = Nk for some k also invariant. Thisimplies that Σk is the smallest isotropy subgroup thatcontains both Σi and Σj . Observe that Ni ⊂ Nj if andonly if Σi � Σj . Two subgroups Σ1, Σ2 are conjugateif there is a g ∈ Γ such that Σ1 = g−1Σ2g. It is a basicresult that conjugate subgroups have fixed point spacesthat are mapped onto each other by the group. Moreprecisely, we have the following elementary resultstated with proof for completeness.

Lemma 1. Suppose that Σ is an isotropy subgroupand g ∈ Γ . Then g−1Σg is an isotropy subgroup andFix(g−1Σg) = g−1 Fix(Σ).

To see this, compute

Fix(g−1Σg

) = {x ∈ M: g−1hgx = x for all h ∈ Σ

}

= {g−1gx ∈ M: hgx = gx

for all h ∈ Σ}

= g−1 Fix(Σ).

2. Symmetries and Lyapunov exponents

Now suppose that we have a compact invariant setA ⊂ M . We assume that A ⊂ N0 (without loss ofgenerality). Within N0 we assume that A supportsan ergodic f -invariant measure µnat that is a naturalmeasure, i.e., there is a positive measure set (w.r.t.Lebesgue measure on N0) such that points in these setshave ergodic averages determined by µ; i.e., such that

limk→∞

1

k

k−1∑j=0

φ(f j (x)

) =∫y

φ(y) dµ(y)

for any continuous φ :N0 → R and a positive measureset of x ∈ N0. We also define

Erg(A) = {µ: ergodic measures supported on A}.With respect to any ergodic measure µ ∈ Erg(A) wedefine the Lyapunov exponents (LEs)

λ(x, v) = limn→∞

1

nlog

|Df n(x)v||v|

which take one of the values λi(µ), i = 1, . . . ,m, forµ-almost all x and any v. Recall that given any ergodicmeasure with support contained within an invariantsubspace N , the Lyapunov exponents (LEs) can besplit into two groups; the tangential LEs λ

‖j and the

transverse LEs λ⊥j with the former corresponding to

perturbations v within N . We assume that the LEs areordered greatest first: λ1 � λ2, etc.

For nested invariant subspaces N and P with A ⊂N ⊂ P ⊂ M we define{λN,Pi (µ): i = 1, . . . ,dimP − dimN

}to be the set of possible λ⊥(x, v) attained for v ∈ P

where x is typical w.r.t. the measure µ. We also define

ΛN,Pmax = sup

µ∈Erg(A)

λN,P1 (µ)

and

ΛN,Pnat = λ

N,P0 (µnat).

P. Ashwin, M. Breakspear / Physics Letters A 280 (2001) 139–145 141

In the case of symmetries we often obtain an attrac-tor within an invariant subspace N that is nontriviallycontained within several distinct invariant subspacesP1,P2, etc. The LEs in different directions Pi can,however, in certain circumstances be related:

Theorem 1. Suppose that N = Fix(T ) and N ⊂Pk = Fix(Σk), k = 1, . . . , l, where Σk are conjugateisotropy subgroups, i.e., Σk = g−1

k Σ1gk for somegk ∈ T and k = 2, . . . , l. Let N = ⋂

k Pk and supposethat A ⊂ N is an attractor. Then

λN,Pk

i , ΛN,Pkmax and Λ

N,Pknat

are independent of k.

Proof. Consider P1 = Fix(Σ) and g = g2 so thatP2 = Fix(g−1Σg) = g−1 Fix(Σ) = g−1P1. Nowgx = x (and |g| = 1) and equivariance of f im-plies equivariance of the derivative (Df (gx)gv =gDf (x)v). Hence for any x ∈ N and v ∈ Σ1 we have

λ(g−1x,g−1v

) = limn→∞

1

nlog

|Df n(gx)gv||gv|

= limn→∞

1

nlog

|g||Df n(x)v||g||v|

= λ(x, v).

This means that any LE λN,P1i is also a LE λ

N,P2i ,

etc. ✷The previous result requires that the conjugating

element g is in T . More generally, we require thatg maps Fix(T ) to itself which in turn implies thatg ∈ Norm(T ) where Norm(T ) = {h ∈ Γ : hT = T h}is the normalizer of T . In this case the result abovecan be adapted if the measure µnat has a symmetry g

(a symmetry on average [7]). More precisely,

Theorem 2. Suppose that N = Fix(T ) and N ⊂ Pk =Fix(Σk) for k = 1, . . . , l, where Σk = g−1

k Σ0gk forsome gk ∈ Norm(T ). Suppose moreover that µnat isinvariant under gk , N = ⋂

k Pk and A ⊂ N is anattractor. Then

λN,Pk

i , ΛN,Pkmax and Λ

N,Pknat

are independent of k.

Proof. This follows as for the previous result onnoting that there is a gk-invariant set of x with fullµnat-measure that has the same LEs at each point. ✷2.1. Directions of riddling

Recall that a basin of attraction B(A) is riddled(in M) [1] if for any x ∈ A and δ > 0 we have

M(B(A)∩Bδ(x)

)M

(B(A)c ∩ Bδ(x)

)> 0,

where M(·) denotes Lebesgue measure on M . Simi-larly, we say the basin of attraction of A ⊂ N0 is rid-dled in the direction Ni for any invariant subspace Ni

where N0 ⊂ Ni if

Ni

(B(A)∩Bδ(x)

)Ni

(B(A)c ∩Bδ(x)

)> 0.

Observe that it is necessary for B(A) ∩ Ni to havepositive Ni -measure in order to get riddling. Considerany open neighbourhood U of A and define the basinof A relative to U to be

BU(A) = {x ∈ M: ω(x) ⊂ A and f n(x) ⊂ U

for all n � 0}.

We say the basin of A is locally riddled if there is aneighbourhood U of A such that BU(A) is riddled [5](this is a stronger assumption than that given in [4]).

We say a Milnor attractor is regular if for any openneighbourhood U of A then M(BU(A)) > 0; thismeans that a regular Milnor attractor attracts a positivemeasure set of nearby points. A result in [1] impliesthat any Milnor attractor that is a uniformly hyperbolicwithin N and whose natural measure has all transverseexponents negative will be regular.

Proposition 1. Suppose that A ⊂ N is a regularMilnor attractor that contains a periodic orbit p suchthat (a) Ws(p) is dense in B(A) and (b) p has apositive transverse Lyapunov exponent and no zeroLyapunov exponents. Then the basin of A is locallyriddled.

Proof. Consider the unstable manifold Wu(p); thisis locally an embedding of Rk where k is the num-ber of unstable eigenvalues of p. As this manifold istransverse to N at p we can find an open neighbour-hood U of N (and hence of A) such that Wu(p) in-tersects an open set V that is not in U . Now consider


Vn = ⋃ni=0 f

−i (V ); this is an open set that for all n

is contained in BU(A)c and such that⋃

n>0 Vn con-tains p. Since it contains points transverse to Wu(p),it must also approach preimages and points on the sta-ble manifold of p on increasing n. Hence we pickany x ∈ B(A) ∩ U that is a point of Lebesgue den-sity of B(A), and any open neighbourhood O of x .By (a) O contains a point q in Ws(p) and so O mustintersect VN in an open set P for some N . Hence(BU(A) ∩ O) > 0 (as A is assumed regular) and(BU(A)c ∩ O) > 0 (as BU(A) ⊃ ⋃

n>0 Vn ⊃ P ) andso the basin of A is locally riddled. ✷

Note that for uniformly hyperbolic A (within N ) allergodic measures on A are limit points of sequencesof periodic measures supported within A [17] and sofor such sets the existence of a measure with positivetransverse Lyapunov exponent implies the existence ofperiodic points with LEs that are arbitrarily close tothat of the natural measure. From hereon we assumethat any Milnor attractor A is regular, all ergodicmeasures on A are limits of hyperbolic periodic meas-ures in A and moreover that the stable manifolds ofperiodic points in A are dense in B(A).

Remark 1. Suppose that A is a Milnor attractor suchthat A ⊂ N0 ⊂ N1 ⊂ M . If A is riddled in M then itneed not be riddled in either Ni . In fact the examplesdiscussed in [1,4] have attractors that are riddled inM but asymptotically stable and therefore unriddledin the largest linear subspace N that contains theattractor. What we emphasise here is that it may beunriddled in a larger invariant subspace.

Remark 2. The same holds even if A is not anattractor but a chaotic saddle in M . If A is a Milnorattractor relative to some subspace it may or may notbe riddled in that subspace.

Remark 3. Suppose that a Milnor attractor A in M

obeys the hypotheses of Proposition 1 for the systemrestricted to some invariant subspace N ′ with N ⊂N ′ ⊂ M . Then the basin of A is riddled in M . Notethat the hypotheses of on A with respect to the systemon N ′ also apply to the system on the whole of M .We need the extra hypothesis that A is an attractorin M .

Remark 4. One can find systems f :M → M withan invariant set A contained in an invariant subspaceN such that A is an attractor in M but not in N . Forexample, consider the flow induced by the vector field(x, y) = (y2x−x3, y3 −x2y). This has an equilibriumat (0,0) that has the open basin of attraction givenby y2 > x2. However, all points with y2 < x2 arerepelled away to infinity. Thus the origin is an attractorin R2 but has trivial basin of attraction in the invariantsubspace given by the x-axis. In this example thisis a degeneracy caused by nonhyperbolicity of thefixed point at the origin. We expect that this behaviourcannot occur in sufficiently hyperbolic systems.

3. Anisotropic riddling and sensitivity to noise

A traditional model for noise in iterated maps is theaddition of uniformly distributed random variable atevery iteration. For attractors with riddled basins, thishas been recognised to give rise to discontinuous be-haviour in the support of attractors as the noise goes tozero; this was called bubbling in [3]. In applications,however, the noise may be highly directional and rid-dled basin attractors will have a degree of sensitivityof an attractor to anisotropic noise that is dependent onwhether the noise is in directions in which the basin isunriddled or not.

More precisely, suppose that A ⊂ N ⊂ M with A anattractor for f :M → M and the basin of A is locallyriddled (say BU(A) is riddled with U compact in M)but which is asymptotically stable in the invariantsubspace N . Consider a perturbed map

(1)xn+1 = g(xn,n) = f (xn) + σξn,

where ξn is a uniformly distributed in some compactregion R ⊂ M . If R ⊂ N and we can find an attractorAσ for (1) such that Aσ → A in the Hausdorff metricas σ → 0. Conversely, if R /⊂ N (for example if R is aneighbourhood of 0) at each point in A, the noise cancause us with positive probability to exit from BU(A).This means that Aσ /→ A in the Hausdorff metric asσ → 0 and we get bubbling of the attractor; examplesof this are illustrated below.

3.1. Internal riddling transitions

Given any attractor A with a riddled basin B(A)

we characterise this basin by examining the dimension


d of the largest invariant subspace N containingA such that the basin of B(A) ∩ N is unriddled.Note that dim(A) � d � dim(M). Any change in d

we term an internal riddling transition. It is clearthat such transitions will occur in higher-dimensionalsystems; as A loses asymptotic stability in moredirections the index d will decrease. Similarly one canapply the standard riddling bifurcation criteria in eachsubspace to predict parameter values when internalriddling transitions will occur; see [5,10,11,14,19].Such transitions will typically be rather unclear onvarying system parameters if the chaotic attractorsare not structurally stable; only by examining normalparameters of the system such that the dynamics onthe attractor is left unchanged can one hope to findinternal riddling transitions appearing as codimensionone transitions. Similar transitions where riddlingbifurcations are replaced by blowout bifurcations [15]will also occur in such systems; see, for example, thesystems studied in [2].

3.2. A numerical example

We now examine an example mapping where wecan investigate the effects of riddling in differentdirections. Consider the map on R4 defined by

(2)x ′i = (1 − ε)f (xi) + ε

n

n∑j=1

f (xj ) + σiηi

with n = 4, where at each timestep each ηj is an inde-pendent random variable that is uniformly distributedon [−0.5,0.5]. The noise in the ith component hasamplitude that can be controlled with σi . This map hasbeen studied by several authors including Kaneko [9],although the only noise perturbations considered havebeen isotropic. Note also that Taborev et al. [18] haverecently looked at this system with 3 cells in some de-tail.

This map has the symmetry of all permutations on n

objects; in this case S4. There are a large number of in-variant subspaces corresponding to isotropy subgroupsthat can be characterised by partitions of {1 . . .4} into1 � m � 4 groups of identical cells, namely all sub-groups of the form S4

1 , S2 × S21 , S2 × S2, S3 × S1

and S4.In such a system at parameter values that are

intermediate between strong and weak coupling one

may find a wide variety of attractors of differentsymmetries that are multi-stable. One can also find,for example, (i) chaotic saddles that have basins thatare riddled within certain invariant subspaces, (ii) at-tractors that are riddled in some directions but notothers, and hence (iii) anisotropic bubbling responseto anisotropic noise.

As a specific example we consider an attractor in

Fix(S2 × S2) = {(p,p, q, q): p,q ∈ R

}that occurs when a = 1.71 and ε = 0.15. Fig. 1(i)shows a timeseries on this attractor for an initial condi-tion very close to (0.00292,0.00292,0.8004,0.8004)with no added noise; σj = 0 for all j . On addition ofnoise Fig. 1(iia) shows x1–x2 for the same initial con-dition but with σ1 = 10−5. The large deviations awayfrom x1 = x2 indicative of bubbling in the direction(1,0,0,0) and hence indicate that the basin of attrac-tion of this attractor is riddled to perturbations into theinvariant subspace (p, q, r, r). By contrast, Fig. 1(iib)shows x3–x4 for for the same initial condition but withσ3 = 10−5. In this case there is apparently stable re-sponse indicating that the basin is not riddled intothe invariant subspace (p,p, q, r). Note that Fig. 1(i)shows that the statistics of x1 and x3 are quite differ-ent; the natural invariant measure associated with thisattractor is not invariant under the transformation

(x1, x2, x3, x4) �→ (x3, x4, x1, x2).

If this symmetry did leave the attractor invariant thenby applying Theorem 2 either both or neither of thedirections (1,0,0,0) and (0,0,1,0) would be rid-dled. Numerical simulations indicate that the naturaltransverse LE in the direction (1,0,0,0) is approx-imately −7.8 × 10−4 whereas it is approximately−0.2842 in the direction (0,0,1,0); this agrees withthe anisotropic bubbling observations.

3.3. Discussion

In this Letter we demonstrate that the presence ofbasins of attraction that are riddled in some direc-tions but not in others is in fact typical in higher-dimensional systems. This can give rise to a numberof new phenomena in such symmetric systems.

This anisotropic riddling in only certain directionsleads to sensitivity under perturbation that can vary alot between different directions. Work in progress [6]


(i)

(ii)

Fig. 1. Anisotropic bubbling behaviour caused by anisotropic riddling of the basin of an attractor for a system of four globally coupled maps (2).(i) shows timeseries of response of the noise-free system; the circles show the values of x1 and x2 while the crosses show the values of x3and x4. In (ii) the attractor in (i) is subject to very low amplitude noise. In (iia) the noise is added in the x1 direction only giving rise to bubblingin this direction. In (iib) it is added in the x3 direction giving a stable response, consistent with the attractor not being riddled in this direction.


indicates this gives a good model for the sensitivityof neural systems to small perturbations, for exampleallowing the brain to classify smells given very minorinputs. Note that basin riddling has already been foundin the learning dynamics of neural networks [13].

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Anisotropic properties of riddled basins