Detection of symmetry of attractors from observations I. Theory

ELSEVIER Physica D 100 (1997) 58-70

PHYSlCA D

Detection of symmetry of attractors from observations I. Theory

Peter Ashwin 1, Matthew Nicol 2 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

Received 28 June 1995; revised 29 April 1996; accepted 10 June 1996 Communicated by H. Flaschka

Abstract

Barany et al. (1993) proposed a method for determining the symmetries of attractors of equivariant systems by averaging certain classes of equivariant maps. We use an idea in Barany et al. to re-cast definitions of symmetry detectives assuming that we only have access to (equivariant) observations from the system. Detecting from observations allows one to perform averaging in spaces that may have much lower dimension than the phase space. This paper generalises and develops their suggestion.

Among the generalisations we consider is the use of nonpolynomial detectives, and we show using the notion of"prevalence" of Hunt et al. (1992) that our detectives and the detectives of Barany et al. (1993), Dellnitz et al. (1994), and Golubitsky and Nicol (1995) give the correct symmetry of attractors "almost certainly" in a measure-theoretic sense. We show that detectives can persistently give incorrect symmetries at isolated points in parametrised systems and discuss how to overcome this. We show how one can find the symmetry of an attractor from examination of a Poincar6 section.

In Part II of this article, Ashwin and Tomes apply these results to find symmetries of attractors in a physical system of four coupled electronic oscillators with $4 symmetry.

1. Introduct ion

The recent development of symmetry detectives [5]

(see also [10,13,15,22,23]) makes it theoretically pos-

sible to detect symmetries of attractors. However, it is

difficult to apply the definitions and theorems of these

papers to physical systems because one usually does

not have access to the phase space, but rather to ob-

1 Present address: Institut Non Lin6aire de Nice, 1361 route des Lucioles, 06560 Valbonne, France. Permanent address: De- partment of Mathematical and Computing Sciences, University of Surrey, Guildford GU2 5XH, UK.

2 Permanent address: Department of Mathematics, UMIST, Manchester M60 1QD, UK.

servations taken from the phase space. The purpose

of this paper is to re-cast the definition of detectives

along lines suggested by Barany et al. [5, Section 10]

so that detective ideas can be applied rigorously not

just to numerical or theoretical examples but also to

practical experiments.

Suppose we have a system with a symmetry group

/ ' . We define a notion of detective that will give the

correct answers for generic symmetric observations

from the phase space of the system M into a low

dimension Space S. The detectives are then map from

S into another representation space W of the symmetry

group; note that we have introduced an intermediate

space S and consider a composition of maps:

0167-2789/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PII S0167-2789(96)00 175-3

M 7r>S ~ W .

P Ashwin, M. Nicol/Physica D 100 (1997) 58-70

Roughly speaking the map q~ will be called a detective

if the isotropy of the average of ~b o ~ gives the sym-

metry of an attractor of a dynamical system in M for

"almost all" symmetric observations ~p. We address the following problems: How do

we rigorously characterise detectives? How do we characterise detectives that will work not only for one

attractor but also for a d-parameter family of attrac-

tors? How do we detect symmetries from a Poincar6

section? The paper is organised as follows. In Section 1.1

we discuss instantaneous and setwise symmetries of attractors, establish notation and give a brief

discussion of restrictions on possible symmetries for

attractors. Section 2 gives definitions of detectives

(Definition 2.6) in our setting and proves some results about a large class of detectives (Theorem 2.8). In

Section 3 we present results pertaining to the preva-

lence of detectives, and in particular we show that detectives correctly determine the symmetries of at-

tractors "almost certainly". In Section 4 we turn our

attention to d-parameter systems where detectives

may fail to give the correct symmetries at points of higher codimension in the parameter space. We give

necessary conditions that a detective works for a d- parameter system. Section 5 shows that we may cor-

rectly determine the symmetries of attractors from an

appropriately chosen Poincar6 section. We conclude with a discussion in Section 6. For completeness we

include an appendix covering some fundamental ideas necessary to apply prevalence results.

1.1. Symmetries o f attractors: Notation

Attractors in symmetric systems often have a symmetry that is a subsymmetry of the system. Local bi-

furcation of solutions with symmetry typically give

rise to solutions with lower symmetry through a pro- cess called spontaneous symmetry breaking. Ideas of local bifurcations work well when one restricts to discussion of fixed points or periodic orbits, but generally fail to say much about more complicated behaviour.

59

Chossat and Golubitsky [9] conjectured the exis-

tence of symmetry increasing bifurcations of chaotic

attractors and studied some examples for Dn symmetric maps of the plane. Since then, work has been

done to classify admissible symmetries of attractors

[3,12,19] and some progress has been made on discov- ering mechanisms by which this symmetry can change

[2,11].

We shall consider a dynamical system defined by a

map Xn+l = f ( x n ) or an ordinary differential equa- tion :t = F ( x ) on ~n. The defining functions are as-

sumed to be equivariant (i.e. y f ---- f y for all y c F,

similarly for F) under an action of a finite group F on M. We define an attractor A to be a Liapunov stable to-limit set.

We define a metric drn on compact subsets of ~n

such that sets close in this metric are close both pointwise and in the sense of Lebesgue measure. Given two

(Borel measurable) bounded subsets A and B of R n

we define dl(A, B) = £(AAB) where A is the setwise

symmetric difference. The metri cdm is defined by

din(A, B) ----- dH(A, B) q- dl(A, B).

with dH the usual Hausdorff metric.

The usual definition of the symmetry on average is the subgroup

,F,(A) =-- {a c Nz dH(crA, A) = 0}.

An important subgroup of Z(A) is

T ( A ) ~ {a c F : a x = x f o r a l l x ~ A } ,

the instantaneous symmetry that fixes all points of A.

As noted in [19], T ( A ) is a normal subgroup of Z:(A) and is an isotropy subgroup of the action corresponding to the intersection of the isotropies of points in A.

Other notation. For a subgroup G acting on X we define the fixed-point subspace of G to be Fixx (G) =

{x ~ X: a x =- x for all a 6 G}. We write Fix(G) if the particular space X is clear from the context.

For a single point y 6 M the isotropy of y is Z (y) ---- Z({y}) =- T({y}). For a vector space M write

.A(M) = {compact A C M: either y A fq A = 0

o r g A = A f o r g c F ] .

60 P Ashwin, M. NicOl/Physica D 100 (1997) 58-70

Note that Liapunov stable attractors are sets in .A(M)

[19, Proposition 4:8]. In addition, define

B ( M ) = {A 6 .A(M): points with

isotropy T ( A ) are dense].

If A 6 A ( M ) is an attractor with a dense orbit, then

A c B(M). Finally, Ckr(M, S) denotes the space of

F-equivariant maps from M to S which are k times continuously differentiable (k > 1).

1.2. Attractors with nontrivial instantaneous

symmetries

gent restrictions if we allow the dynamical system to be a flow or a diffeomorphism rather than just a map.

The two cases are described in [3,12,19]. Effectively

the only restrictions are imposed by the existence of

reflection hyperplanes of the group action, i.e. codi-

mension one surfaces fixed by some subgroup. For

the system of oscillators investigated in Part II we

note that there are no such invariant hypersurfaces and so there are no a priori restrictions on possible

average symmetries of attractors in this system.

2. Detecting the symmetry of attractors

Suppose that F acts faithfully on the space V. To take proper account of all symmetries of a subset A C

V it is necessary to consider those that may be setwise

symmetries as opposed to instantaneous symmetries,

along the lines discussed in [15]. For example, sup-

pose that T ( A ) = G is nontrivial; then there may be symmetries of A that are not contained in G but are

contained in the normaliser N r (G) = 22 (Fixv (G))

of G in F . For each subgroup G C F we define

G t ~ N F ( G ) / G .

When G ~ is trivial, A cannot have any additional av-

erage or setwise symmetries and thus detecting further symmetries beyond instantaneous symmetries is

unnecessary. Therefore we make the following defini-

tion:

Barany et al. [5] considered the following question:

Given a trajectory from an equivariant dynamical sys-

tem that converges to an attractor, how can one detect the symmetry of the attractor? They introduced

the idea of a detective, a (vectorvalued) equivariant

function that when averaged over the attractor enables

one to read off the symmetry. The theory of detectives has been generalised and discussed by several workers since; see [8,10,15,23].

Previous work has mostly concentrated on cases

where one has full access to the phase space and only

polynomial detectives have been used. By introduc- ing an intermediate observation space (suggested by

Barany eta l . [5, Section 10]) one can overcome the

first restriction, whilst by adapting an idea of Tchisti- akov one can overcome the second.

G ( V ) = {G C F: G is an isotropy

subgroup such that G t 5& 1 }.

Note that 1 (the trivial group) is always in G ( V ) if V itself is nontrivial. We may sometimes assume that there exists an SBR (Sinai-Bowen-Ruelle) measure

PSBR on A; that is, an ergodic invariant measure that is generic for Lebesgue a.e. point in some neighbourhood U of A. We define an SBR attractor to b e a n attractor A with a dense orbit and an SBR measure. For further discussion see [6,7,10].

With a suitable definition of attractor, one can characterise the permissible symmetries, i.e. those symmetries of attractors that can be realised by a F - equivariant dynamical system. There are more strin-

2.1. Equivariant observables

King and Stewart [18] have considered the prob-

lem of reconstructing the phase space of a symmetric system and prove an equivariant version of the Tak- ens embedding theorem [21]. They show that in order

to reconstruct the dynamics one must consider equivariant observables that carry a "complicated enough" representation of the group.

More precisely the group action on S must satisfy the representation-theoretic condition that M is subor- dinate to S. This means that (a) S contains an isomorphic copy of every irreducible representation of any isotropy type of points in M and (b) every orbit type in M embeds equivariantly into some space associated

P. Ashwin, M. Nicol/Physica D 100 (1997) 58-70 61

with S. Note that (b) is a statement about the global

geometry of M.

In this paper we shall assume that the phase space M = ~m and the observation space S = R n for some

m and n, with F a finite group acting orthogonally.

We do not consider phase-space reconstruction but

instead merely wish to find observables such that we

can detect the symmetries of the attractors. To this

end, we consider equivariant observables that allow us

to distinguish all isotropy types of the action of F on

M. The following is a weaker equivalence than lattice equivalent considered in [5, Definition 4.2]; we do not

require that the fixed-point subspaces are isomorphic, just that they are in 1-1 correspondence.

Definition 2.1. We say two orthogonal representations of f ' , W1 and W2 are isotropy equivalent if H is an

isotropy subgroup for the action of F on W1 if and

only if it is an isotropy subgroup for the action on W2.

/4i > G and so Fixw(Hi) C Fixw(G). Therefore

we have Fixw(H1) = Fixw(H2), contradicting the assumption that W is a distinguishing representation.

[]

2.2. Relating 27(A) and ZT(Tr(A))

As the following example shows, given an arbitrary

compact set A and observable ~ we can get the wrong

answer for an open set of observations close to ~p, even if ~p works for other sets A!

Example. A and ~ such that all fit ~ close to ffs satisfy Z (~ ' (A) ) ¢ Z(A) : We take M = S = R and consider A = [ -1 .1 , 1], 7t = sin 2zcz. Then i t is clear that 7t(A) = [ - 1 , 1] and since the extrema of ~p are

attained inside A, any small enough perturbation of

7e will still have that ~y(A) has Z2 symmetry, even

though A does not.

In order to distinguish all possible subgroups of a particular symmetry group we need to consider representations that satisfy the following definition.

Definition 2.2. (5, equivalent to Definition 4.1). An orthogonal representation W of F is a distinguishing representation if all subgroups of f" are isotropy subgroups.

In [5, Theorem 4.3] it is shown that there exist

distinguishing representations, namely W that contain all nontrivial irreducible representations of F at least

once. As in [15] we will prove that the notion of de-

tective we introduce will correctly determine the symmetries of attractors regardless of their instantaneous

symmetries. We shall make use of the following result which is a simplification of [15, Lemma 2.3].

Lemma 2.3. Suppose that W is a distinguishing representation for F . Then for any G, the action of G I =

Z(Fixw(G)) /G on Fixw(G) is a distinguishing representation for G'.

For attractors in /3(M), the next proposition shows that for a generic set of equivariant observables taking

values in a large enough representation of F we have 27(~(A)) = 27(A).

Proposition 2.4. For any A 6 /3(M) and any S iso-

tropy equivalent to M there is an open dense set of observations 7t 6 C~(M, S) such that 27(Tr(A)) =

Z(A) .

Proof Since ak is f '-equivariant we have 27(A) C ZOP(A)). We will show that for each/9 ~ F - Z(A) there is an open dense set Xp of observations such

that p~(A) ~ ~(A). The set N(pzF-Z(A)) Xp is then

an open dense set of observations in C k (M, S) such

that Z 0 P ( A ) ) = Z(A) . Openness is clear because dr t (~(A), p~(A)) varies continuously with gt. Now let p ~ F - - 27 and suppose there exists ~ ~ C~ (M, S) such that p~(A) = ~(A) . We will show that given

> 0 there exists an observation ~kz in C~(M, S) which is within ¢ of ~ in the C k topology such that p~E(A) • ~t(A). []

Proof Consider any two subgroups HI = Hi~ G for i = 1, 2 of G ~. Assume that Fixw(H1) F) Fixw(G) = Fixw(H2) n Fixw(G). Note that Hi satisfies f ' >

We first claim that there is a unit vector v with isotropy T(A), and an a 6 A with isotropy T(A) such that ~ ( a ) = rv ~ ~(A) for all r > 0.

62 P Ashwin, M. Nicol/Physica D 100 (1997) 58-70

To prove the claim, pick v such that 0P(a) +rv) has isotropy T(A) for all r > 0. This is possible because

the set of points with isotropy T(A) is a finite union

of convex cones and 7t (a) is in the closure of these

cones. Since ~ (A) is closed, we can find r0 >__ 0 such

that ~r(a)+rov ~ 7t(a) but (~(a)+(ro+r)v) ¢ ~r(a) for all r > 0. Since ~ ( a ) + roy has isotropy T(A) it must be the image of a point with isotropy contained

in T(A). Given such an a and v choose 8 > 0 such that

Bs(p(a)) intersects A if and only if p E Z ( A ) (we can do this because A E/3(M) and F is finite). Define

zl c Ck(M, R) supported on B~(a) with r/(a) = 1 and define ~p~ ~ C k (M, S) by

7~(x ) = ~p(x) + ~ ~'vr~(y-lx),

FEF

Thus I1~ - ~ l lc~ < EIT(A)I]I~IIC~ which can be taken arbitrarily small (the C k norm of q depends

upon 8 but is independent of e). Note also that

~-~.×c/~ y v ~ ( y - l a ) = v. If p ¢ Z ( A ) and pC~(A) = ¢~(A) then

pape(A) = ~E(pA) = ~ ( p a ) = ~(a) .

Thus for any e > 0 we have

min d(p(C'(a) + Ev), x) > 0 xE~ke(A)

and so dH(p~e(A), ~e(A)) > 0. This implies that

there are ~PE arbitrarily close to ~h (in the Whitney C k topology) such that E0PE(A)) = Z(A) . []

Remark 2.5. If A is an attractor with a dense orbit then this orbit must have isotropy T(A) and so A c /3(M). However, there are Liapunov stable invariant sets A (for example heteroclinic cycles) that can be

attracting and even robust in systems with symmetries but which possess no points with isotropy T(A).

M ~>S ¢>W.

Note that if S = M and ap is required to be a diffeo-

morphism then we have the definition of Barany et al.

More precisely, they define the observable ¢ to be a

detective if for each subset A ~ ,A(M) (of positive Lebesgue measure) there is a residual set of diffeomor-

phisms apc Diff~(E n) (the space of f '-equivariant k

times continuously differentiable diffeomorphisms of ~n) such that

f ¢ (~(y) ) dE(y) A

has symmetry S (A) .

Two methods of averaging are proposed by Barany

et al. One method (see also [10,15]) involves taking the time average of an equivariant observable along a

trajectory. This method can be shown to work if we

assume there exists an SBR measure supported on the attractor (see [10,15]). Numerical evidence suggests

that this assumption is often justified; see the discussion in [10]. The convergence of this method may be

very slow, but it has the advantage that it requires little computer memory to be practically implemented.

The other method involves "thickening" the attrac-

tor so that the resulting set has positive Lebesgue measure, and then performing numerical integration of an

observable over this thickened attractor with respect

to Lebesgue measure. It is clear that this method has severe limitations if the dimension of the phase space

is large. In practice, it is impossible to perform this averaging method exactly; usually coarse-graining of the underlying space has been used (see Section 2.5).

With reference to the first method, the ergodic average o f ¢ over a sequence {xi: i ~ N} is defined to be

1 N-1 K~({xi}) : N--+~lim ~ ~-~ ~(Xj).

j=0

2.3. Detection from equivariant observables

Our definition of detectives involves composing an observable 7' with a detective ¢ in the following

manner:

If A is an attractor in M supporting an SBR measure

PSBR we say that A is an SBR attractor and define

K~,g~(A ) = f ~(~(x)) dpSBR(X).

A

P. Ashwin, M. Nicol/Physica D 100 (1997) 58-70

By the definition of an SBR measure, for a positive

(Lebesgue) measure set of initial conditions x we have

K~({~(x) , ~ ( f ( x ) ) , ~ ( f 2 ( x ) ) . . . . . })

= K~,Ts(a).

2.4. Averaging the observed attractor

63

Z,(K~,7,(A)) = E(A) .

(b) ¢ is an integrated observed detective if for all

compact sets C C S there is an open dense set

in a neighbourhood (in the dm-topology) Af of C such that if B ~ A/" and Z ( B ) = E ( C ) then

E(K~(B) ) = Z(C) .

The integrated observed method of averaging in-

volves taking an image of the attractor in an observa-

tion space and then averaging over Lebesgue measure

on a fixed-point subspace in this observation space:

~(A)

where £ is Lebesgue measure 3 on Fixs(T(A)) . As noted by Barany et al. it is typical that A may

have zero Lebesgue measure, for example if A is a periodic orbit. In this case KI0p(A) ) = 0 and so it is necessary to equivariantly "thicken" the attractor

while keeping the symmetry constant to have a chance

of detecting the correct symmetry. We take the image ~p(A) of the attractor in the ob-

servation space S and consider sets B with the same symmetry as A that are close both pointwise and mea-

surewise. We then compute K~(B). The advantage of

this method of averaging is that one needs only integrate in relatively low-dimensional spaces where the

dimension is dependent upon the group representation. Even partial differential equations with infinite-

dimensional phase spaces can he considered using this

method. We can now define our detectives.

The open dense set of observations will be depen-

dent on the attractor A c ,A(M) and the map ~b.

Remark 2.7. By applying Proposition 2.4 for A

/3(A), (b) above implies that generically an integrated

observed detective will give the correct symmetry of the underlying set in M. Note that we do not require

that C 6 .A(S) to be satisfied or that the boundary is piecewise smooth for (b) and in this sense our defini-

tion is slightly weaker than that of [5].

We now state a theorem generalising [5, Theo-

rem 5.2] to give a large class of continuously differ-

entiable detectives. We do not require the detectives to be polynomials or anything more than just contin-

uously differentiable.

Theorem 2.8. Let ~b 6 C1F(S, W) with S isotropy

equivalent to M and W a distinguishing represen-

tation. Suppose that for each isotropy subgroup G for M (or equivalently S) and all neighbourhoods

N 6 Fixs(G) we have

Span{Dx49V:X ~ N, v ~ Fixs(G)} = Fixw(G).

(2.1)

Then ~b .is a detective.

Definition 2.6. A detective for M is a F-equivariant map ~b : S -+ W between two representations o f / ~

such that for all k c ~ we have: (a) q~ is an SBR detective if for all SBR attractors

A C M there is an open dense set of observations 7t c C k (M, S) such that

3 Note that we must know T(A) in order to know on which subspace we define the Lebesgue measure £. Practically this is not a problem because for generic ~ we have T(A) = T((t(A)) and the latter is easily measured.

We relegate the proof to Appendix B. For polynomial equivariant maps, this theorem is equivalent to

that of [5,10,i5].

Proposition 2.9. Suppose that ~b : S -+ W is a

F-equivariant polynomial map into a distinguishing representation W. Write Fixw(G) = ~ W/~ as the isotypic decomposition of the action of G ' = ~ ( F i x w ( G ) ) / G on Fixw(G) and zri the orthogonal projection onto W/~. The following are equivalent:

64 P. Ashwin, M. Nicol/Physica D 100 (1997) 58-70

(a) For each G c G(S) and each i the projection

Jri e ~b(Fixs(G)) C W[ is nonzero.

(b) For each neighbourhood N C Fixs(G) we have

Span {Dx(aV:X ~ N, v E Fixs(G)} = Fixw(G).

Remark 2.10. This implies that on restricting to poly-

nomial equivariant observables, our sufficiency con-

ditions for an' equivariant observable to be a detective

are equivalent to those given in [15].

Proof of Proposition 2.9. Suppose that for some i,

Ygi oq~ is identically zero. Then the linearization 7g i Ox~) is identically zero for every x E S, a contradiction

to condition (b). Thus condition (b) implies condition

(a). Conversely, suppose that

Span{DxqbV:X E N, v c Fixs(G)} ¢ Fixw(G)

2.5. Discretised detectives

The method of thickening an attractor (either in the

original phase space or thickening its image in some observation space) and then averaging over Lebesgue

measure on this space cannot be done exactly. In gen-

eral we approximate it numerically.

A natural way to do this (and the one used, for

example in [5] and Part II of this paper) is to take

a discrete lattice whose E-neighbourhood covers the whole space and (piecewise continuously) project the

attractor onto this grid. A great advantage of this to thickening the attractor is that if the grid is chosen

to respect the invariant subspaces one does not have

to worry about first measuring T(A) and then thick-

ening in Fixs(T(A)). However, a sensible definition

for a discretised detective that enables results to be rigorously interpreted is still elusive.

for some neighbourhood N C Fixs (G). If q~ : En R m is a polynomial (smooth map) and if the images

DgaxV for all x E N, v E Fixs(G) lie in a proper subspace of ~m then modulo a fixed constant vector

the image of ~b : S --+ W (q~ : N ~ W) also lies in that subspace. If ~b is equivariant then the fixed

constant vector must be the zero vector. We also note

that 7 r i D f b x = DTrifb x so if Dggx is not onto then D J r i q ) x is contained strictly in a subspace of Wi for

some i and hence (as Wi is irreducible and zri~b is equ iva r i an t ) 7giq~ = 0. Thus condition (b) implies

condition (a). []

Using Theorem 2.8 and Proposition 2.9 we get the

following result.

Corollary 2.11. Let F be a finite group acting on V and let: W be a distinguishing representation. Define Pk to be the space of F-equivariant polynomial map- pings of S to W of degree at most k. We give P~ its usual topology. In light of the theorem above and [15, Theorem 1.2] we have that for each sufficiently large k, there exists an open dense subset 79 C Pk such that each cp E 79 is a detective. Moreover, for any k _> 1 there exists an open dense subset of C~.(S, W) that

consists of detectives.

3. Prevalence results

Since a topologically generic set can have small

measure and, conversely, a set of large measure may be nongeneric, it is desirable that our detectives give

the correct answer not only for a topological (generic)

large set of observations but also in a measure-

theoretic sense. The measure-theoretic notion of prevalence [17]

was developed to enable one to talk of a property hold-

ing "almost everywhere" on infinite-dimensional vec-

tor spaces - a generalisation of full Lebesgue measure to an infinite-dimensional setting. For further details

on the notion of prevalence see Appendix A. We show that a detective with the ergodic sum method "almost certainly" gives the correct symmetry of an SBR at-

tractor.

Theorem 3.1. Let ~b 6 C I ( s , W) with S isotropy equivalent to M and W a distinguishing representation. Suppose that for each isotropy subgroup G C G(M) of M and all neighbourhoods N ~ Fixs(G) we have

Span{DxcPv: x E N, v E TxN} = Fixw(G).

P. Ashwin, M. Nicol/Physica D 100 (1997) 58-70 65

Then for all SBR attractors (A, p) there is a preva,

lent set of observations 7t 6 C~ (M, S) such that

Z(K~,#(A)) = Z(A) .

N}. It is easy to see that for each g E G~ the set of

t 6 R such that g ÷ tp ~ GC~ has Lebesgue measure

zero. Hence for each integer k > 0

Remark 3.2. Thus detectives for the ergodic sum

method give the correct symmetry of an SBR attractor

not just generically but also "almost certainly".

We will make use of the measure transversality the-

orem [17, Lemma 3, p.230] which, adapted to our set-

ting, can be stated in the following way.

Lemma 3.3. Let B C ~t be an open set. Let F : B -+ R m be continuously differentiable and assume that the

derivative DF has full rank at every point of B (that is to say DF is onto). Suppose Z is a subspace of codimension one or greater in R m. Then for almost

every p in B, F(p) c Z c.

ZT(K~,~p (A)) = Z(A)

for a prevalent set of observations ~ E C~ (M, S). []

Remark 3.4. The same proof with minor modifica-

tions shows that the detectives of [5,10,15] "almost

certainly" give the correct symmetry for both the ergodic sum and Lebesgue integral method. In fact in [5,

Definition 5.1], [10, Definition 2.2] and [15, Defini-

tion 1.1 ] we may replace the topological notion 'open,

dense' in the definition of detective by prevalence and the corresponding theorems [5, Theorem 5.2], [10,

Theorem 3.3] and [15, Theorem t.3] still are valid.

Proof of Theorem 3.1. We will identify the space Pq (M, S) of F equivariant polynomials of degree at

most q with W for some t and consider, for a given

(A, p), the map

F(g) = f cp(g(x)) dp(x)

A

Remark 3.5. We have a somewhat weaker result in the

case of the Lebesgue integral method as applied in this paper. In the proof of Theorem 2.8 we may show that

for a prevalent set of g c C~ (M, S) the symmetry of

£(g) is equal to the symmetry of Z(A).

from Pq(M, S) to W. F is continuously differentiable. Furthermore for a generic set of g ~ Pq (M, S)

the differential has full rank if q is sufficiently large [15, Theorem 1.2]. Choose such a q and gP, i.e., so that DF has full rank at g ' and note that since this is

an open condition there is a neighbourhood B of gt such that DF restricted to this neighbourhood is onto.

Hence by Lemma 3.3 for almost every g ~ B, F(g) has symmetry group equal to 27(A). The fact that F is linear and prevalence is translation invariant im-

plies that almost every g E Pq (M, S) has the property that F(g) has symmetry group equal to Z(A).

Fix an integer k > 0 and let Gk denote the subset

of C~F(M, S) defined by Gk = {g 6 C~F(M, S): F(g) has symmetry equal to 27(A)}. Finally note that for each k > 0 the subspace Pq(M, S) C CkF(M, S) serves as a probe for Gk. In fact given p E Gk, we may define a one-dimensional probe by P = {tp: t E

4. Detectives for parametrised families of attractors

The aim of a detective is for a given attractor A to obtain the correct symmetry for a generic set of equivariant observations. However, it may fail at isolated

points in a persistent way if we examine parametrised families of attractors.

For systems parametrised by some vector k c D open in ~d we want detectives that will work for parametrised families of attractors. This leads to the

following definition for SBR detectives; a similar definition holds for integrated observed detectives.

Definition 4.1. A d-parameter SBR detective is a function q~ • S --+ W such that given any continuous family of attractors A(k) parametrised by k ~ R d there exists an open dense set of observations

66

E CkF(M, S) such that for all £, if A()0 is an SBR

attractor then

Z(K~,~p(A()0)) = 27(A(Z)).

Note that the definition in Section 2.1 Corresponds

to the case d = 0. For larger d we need to con-

sider larger-dimensional observations and distinguish-

ing representations.


In part II we observe an example of isolated points

in parameter space that are assigned the incorrect

symmetry due to S not being d-nested. One can of

course use detectives in the sense of Definition 2.5

for parametrised systems, but one must be aware that

they can give incorrect answers on a subset of param-

eter space. The following characterisation of these

incorrect answers can be proved as for Lemma 4.3.

Definition 4.2. A representation W of the finite group

F is d-nested if d is the minimum integer such that

for all isotropy subgroups G < H,

dim F ix (G) > dim F i x ( H ) + d.

This means that any fixed-point spaces of higher

isotropy contained in F ix(G) are of condimension

strictly greater than d.

Lemma 4.3. A necessary condition that ~b : S --+ W

is a d-parameter detective is that the isotropy faithful

observation space S and the distinguishing represen-

tation W are d-nested.

Proof Under the assumptions of continuity of A()0

in )~ we note that KEoT~(A()O)_ is continuous in ~.. I f

W is not d-nested, we know that there are G < H

subgroups with F i x w ( H ) of codimension less than or

equal to d in F i x w ( G ) . Thus we can construct a fam-

ily of attractors (all with E(A) = G, T(A) = 1) such

that the image of D under K intersects Fixw (H) trans-

versely. There will then be an open set of ~p which

preserve this intersection implying that all observa-

tions in this set will have at least one point of )~ with

the isotropy of K ()0 equal to H.

Similarly if S is not d-nested, we assume that for

two isotropy subgroups G < H we have F i x w ( H ) of

codimension less than or equal to d in F ixw (G). We

construct a family of fixed point attractors A()0 =

{x(~.)} with T(A) = 2?(A) = G and a ~p such that

~(x(~.)) intersects F i x s ( H ) transversely. This intersection will be persistent under deformation of

7t implying that there is at least one point where

T(C ' (A()0) ) = H. []

Lemma 4.4. If we use a d-parameter detective ¢

(which is not a d ÷ 1-parameter detective) for a p -

parameter continuous family of attractors A 0 0 (p >

d), then there will be at best an open dense set of ob-

servations ~ such that the detectives give the correct

symmetries for a set of parameters whose complement

is a set of codimension p - d,

We finish this section with two examples of how

detectives in the standard sense do not give the correct

answers for one-parameter families of attractors.

Example. (Incorrect instantaneous symmetry). Con-

sider M = ~ with 77 2 acting by x w-~ - x . If a

dynamical system on M has a path of fixed points

x()0 with trivial isotropy then for S = R (which is

zero nested) and an equivariant observation ~ ( x ) =

sin x we cannot avoid giving the "wrong answer" at

points where sin x = 0. However by taking S =

77 2 ( two copies of the same representation) and e.g.

7t(x) = (s inx, sin q ~ x ) we avoid hitting points of

higher isotropy in a one-parameter family.

Example. (Incorrect average symmetry). Consider

aga in Z2 acting by x ~-~ - x , this time in S = W

= R. Suppose we have a dynamical system on

some isotropy equivalent M with a continuously

parametrised family of ergodic invariant measures

that project onto /z()~) measures supported on S, all

of which are asymmetric and a detective q~(x) ---- x.

Then it is a codimension one phenomenon that the

mean f x d # ( x ) can pass through zero on varying

)~ whereas it is codimension two if one considers W = ~2 and q~(x) = (x, x3).

P.. Ashwin, M. Nicol/Physica D 100 (1997) 58-70 67

5. Symmetries from Poincar6 sections 6. Discussion

We now discuss how one can relate the symmetries

of attractors for flows in a phase space M to those

of the intersection of the attractor with an invariant

Poincar6 section in M. Suppose we have a flow de-

fined on a phase space M with a continuous evolution

operator

~ ( x , t ) ' M x N + ~ M

equivariant under an action of F on M.

Definition 5.1. An invariant Poincard section for the

attractor A c A ( M ) is a subset P of M such that

(a) Z ( P ) = F and (b) A = ~ ( A n P , R+), i.e. A

is precisely the forward evolution of its intersection

with P .

Given such a section, we relate the symmetry of the

intersection of an attractor with the Poincar6 section to

that of the attractor in the following lemma. The same

symmetry group is obtained whether we measure the

symmetry of the attractor A or its intersection with

the Poincar6 section. Note that B ----- A fq P will be an

attractor for the return map on the invariant Poincar6

section P .

Lemma 5.2. Suppose A is an attractor for ~ and B =

A n P is its intersection with P . Then Z ( A ) = Z ( B ) .

Proof I f y c 27(A) then y A = A; also y P = P by the definition of invariance and so y ( A n P) = A n P and y c r ( B ) . Conversely, if y c 27(B) then

y ( A V1P) = A VI P and so y A = qO(y(AVI P), R +) -----

• ( ( A A P ) , R + ) = A . []

Remark 5.3. We do not require that A possess a dense

orbit or that the intersections are transverse.

Typically for automonous systems there is no

"global Poincar6 section", i.e. no section that works for all attractors. However for periodical ly forced sys-

tems there can be because all orbits pass transversely

through sections of constant forcing angle. We use

this property for the example in Part II of this paper.

In summary, we have made rigorous the suggestion

in [5, Section 10] that to apply detectives to experi-

ments one must take an equivariant observation and

then compute the average in this space. Vital for this

is Proposition 2.4 which shows that a generic observa-

tion in a large enough space will permit measurement

of Z ( A ) and T(A).

We have shown that using an observation space S

that is isotropy equivalent to M we can stiil generi-

cally detect the symmetries. If the action of the group

on M is either unknown or distinguishing for F we

may take a distinguishing representation as S, but in

general it can be much Smaller. For example, in a net-

work of n coupled cells where symmetries act by per-

mutation of the cells one may easily see that ~n is

isotropy equivalent; thus it is necessary only to take

one measurement from each cell and not to distinguish

all subgroups (for example, 7/4 cannot be the isotropy

of a point for the $4 example in Part II).

One of the difficulties associated with the imple-

mentation of detectives has been the memory require-

ments that the Lebesgue integral method places upon

computers. To overcome this the ergodic sum method

was developed, but to be made rigorous this method

requires stronger assumptions on the attractor dynam-

ics (namely the existence of an SBR measure). In ad-

dition, a difficulty associated with the ergodic sum

method is the often slow convergence of the ergodic

s u m .

The method of averaging over the discretised ob-

served attractor in an observation space has the advan-

tage of fast convergence coupled with relatively small

dimension of domain over which one needs integrate,

and this gives us an discretisation method that will

even work for partial differential evolution equations

with symmetry or experiments that have unknown

phase spaces.

Attractors for partial differential equations. In the case of partial differential evolution equations that

have finite-dimensional attractors, our results still ap-

ply. Notably, if M is a Banach space and attractors

68

are contained in an attracting invariant submanifold

M ° C M the method of Proposition 2.4 gives a generic

set of observables in the supremum norm topology on

C ° ( M , S).

Open problems. There remain many outstanding

questions, for example how to characterise sufficient

conditions for d-parameter detectives, and how to

proceed with attractors that are not in 13(A).

Several points concerning the notion of detective

are still in some sense unsatisfactory. For example,

perhaps a more natural definition of detective than

the one we have given would be "q~ is an integrated

observed detective if for all compact A there is an

open dense set B of observations ~ ~ C~F (M, S) and

a positive function e0(gt) such that

2 : (K~0p(A)E)) = Z ( A )

for all ~p c B and all 0 < e < e0". However it is

in general impossible to obtain this uniformly for e

once we have fixed an observable 7t. The following

example shows that as e ~ 0 there may be countably

infinite values of e for which 2? (K~ (Tt (A)~)) does not

give the correct answer.

Example. A set B such that ~(K~(ap(B)C) ) 5& 27(B)

for a sequence of ei --+ 0: Let 7/2 act on N. Note

that ~b(x) = x satisfies the conditions for a detective.

Choose a point Xl and for a fixed el construct intervals

(Xl - e l , Xl + e l ) and ( - X l - e l , - X l + e l ) . In the in-

terval (Xl - e , Xl + e ) place another point x2. Let B1 be

the set {-4-xl, x2}. Note that the integral of q~ over a 8

neighbourhood of S1 will be ?72 symmetric (i.e. equal

zero) i f6 > el but will be positive if3 < l e l . Suppose

that the value of the integral of ~p over a 3 neighbour-

hood of B1 has positive value a l if 8 = e2. Now take

X 3 C (X 1 - - E l , X 1 q - e l ) c a n d E 2 < 1¢ 1 s m a l l e n o u g h so

that the integral of cp (x) = x over an interval of length

e2 centred on x3 is less than oel. Construct intervals

(X3 - - e2, X3 -[- e2) and ( - x 3 - e2, - x 3 + e2). Note that if n is large enough and the points { - x i }, i = 3 . . . . . n,

lie in the interval ( - x 3 - e2, - x 3 + e2) and we define

B2 = {-4-x3, - - x 4 . . . . . --Xn} then the set B1 U B2 is

such that the integral of ~b over a 8 neighbourhood of


B1UB2 is zero i f8 > e l , positive i f6 = ½el and nega-

tive if 6 is sufficiently small. We may then take a point

in the complement of an el neighbourhood of B1 and

an e2 neighbourhood of B2 and repeat the construc-

tion to obtain a set B = [,.Ji Bi such that as 8 --+ 0, the

integral of ~b over a 6 neighbourhood of B oscillates

infinitely often between positive and negative values.

Acknowledgements

PA thanks the EPSRC and the EU (grant ERBCH-

BCT930503) for their support during this research.

MN thanks the DTI for their support. We gratefully

acknowledge some very helpful conversations with

Marty Golubitsky and Jim Swift.

Appendix A. Prevalence

The key idea behind the notion of prevalence derives

from the following observation [17, p. 219]: Let S C

Nn be a Borel set. If there exists a probabili ty measure

/X with compact support such that every translate of

S has /x-measure zero, then S has Lebesgue measure

zero. Suppose that V is a complete metric linear space.

We write S + v for the translate of a set S C V by

a vector v 6 V. The generalisation of this idea to

the infinite-dimensional setting leads naturally to the

following definition.

Definition A.1. A measure/X is said to be transverse

to a Borel set S C V if the following two conditions

hold:

(1) there exists a compact set U C V for which 0 <

/x(U) < ~ , (2) /X(S + v) = 0 for every v c V.

In R n those subsets S which have a measure

transverse to them are precisely the subsets of zero

n-dimensional Lebesgue measure. In an infinite-

dimensional setting subsets having a measure trans-

verse to them play the analogous role to subsets of

zero Lebesgue measure. These sets are called shy sets

- the complement of a shy set is called a prevalent set.

P. Ashwin, M. Nicol /Physica D 100 (1997) 58-70

Definition A.2. A Borel set S C V is called shy if

there exists a measure transverse to S. The complement of a shy set is called a prevalent set.

A useful way to show that a set is prevalent is to

use what is termed a probe.

Definition A.3. We call a finite-dimensional subspace P C V a probe for a set T C V if Lebesgue measure supported on P is transverse to a Borel set which

contains the complement of T.

Thus a sufficient condition for T to be prevalent is

that T has a probe.

Appendix B. Proof of Theorem 2.8

For the SBR method we give a constructive proof,

i.e. without using an implicit function theorem as in

[5,10,151. Define the map ~A : C k (M, S) -+ W by

~ A ( ~ ) : = f ( ~ o ~ d p , t t

A

where (A, p) is an SBR attractor in M and a projection Pff : W -+ W by

Proof of Theorem 2.8 (Ergodic sum method). We assume that (A, p) is an SBR attractor in M (recall that

A ~ ,4(M)). Suppose that G = T(A) and 27(A) = 27. We define the subset of observations in Ckr (M, S)

that gives the correct symmetry in the following way:

7 ) : = {~: E(K~,q(A)) = ~}

= {~/r: d ( ~ P A ( ~ ) , tOCrPA(~r)) = 0

if and only ifp c E }.

From the second line, by continuity of qSA with ~p C~ (M, S) it is apparent that 7 9 is open.

It remains to prove that 79 is dense. Choose any 7t E C k (M, S) and any a 6 A with isotropy G. Note that if WG is the stratum (set of all points) with isotropy

69

G in W then for all y c WG the isotropy of pW(y) is

precisely E .

Because of Condition (2.1) it is clear that the span

of the derivative D~bx intersects WG for a dense set of

points x c S. Thus it is possible to find ~0 arbitrarily

close to ~p such that the span of DC~o(a) intersects

WG. Choose any v such that DO~o(a) (v) is in WG. By continuity of D~b, for all small enough 3 there exists an open cone C in WG such that

{D~#(x)V:X c Bs(a)} C C.

Also for small enough 3 the fact that A c .A(M) means

that Z / G - o r b i t of Bs (a) consists of disjoint balls. We define

1 ~+(x) = 7]0(x) + 61-fl5 ~ (×v)~(×- l (x ) ) ,

ycF

where 77 ~ C k (M, ~) is a nonnegative function with

~(a) = 1 and ~(x) = 0 for x ~ Bs(a) c. Observe that for given 8 and ~](x)

= + + f D¢,o(x) * 2

A

( 0(6 2 ) X \lZl r~r ]

=*A(~o) +6P W f ( D(a~Po(x ) V )

x~Bs(a)

x rl(X ) dx + 0(6 2)

= ODA(I~O ) "-[- 6pWy + O(62) ,

where y 6 C c W6. We have used the fact that ~/ is nonzero only in a neighbourhood of B~(a) and that

the integral of a set of vectors in the cone C will also

be contained in that cone. Writing ~A (~k0) = PWz for some z, for small enough 6, ~A0P~) has isotropy E . Noting that [] 7r0 - 1/r e [] < 6 [[/7 [[ in the C k norms completes the proof of density of 79: []

Proof of Theorem 2.8 (Integrated observed method). Suppose C C S with T(C) = G. Let m denote Lebesgue measure on Fixw(G) and suppose C has positive m measure.

If ~r C = C then rr fc ¢ de = fc¢ de and thus fc¢ de lies in Fixw (27 (C)). Clearly the condition that

70 P Ashwin, M. Nicol/Physica D 100 (1997) 58-70

fc ~P de does not lie in a fixed-point subspace of some

group A ~ E(A) is an open condi t ion on the space of

sets B with respect to the dm topology. Thus we need

only show that it is dense. To this end define the map

Z~ from the space of F -equ iva r i an t d i f feomorphisms

of S with the C ~ topology Diff~,(S) to W by

/~(g) : f q~o glJac(g)l d~,

c

where g E D i f f , ( S ) . Note that by a change of vari-

ables f c ~ o g l Jac (g ) ldg is equal to fgcCde. Fur- thermore the l inearisat ion of F~(g) is the same as the

l inearisat ion of the map g --+ f~(a)~ ~o gd/~ and the

same l inearisat ion a rgument as that used in [5,10,15]

shows that this map is generical ly onto so there exists

a near- ident i ty d i f feomorphism of S, call it gl, so that

the isotropy subgroup of fg'c cp de precisely equals

E(C). Thus the condi t ion is also dense since if g / i s

a near- ident i ty d i f feomorphism then g~C is close to C

in the dm topology. []

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Detection of symmetry of attractors from observations I. Theory