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ELSEVIER Physica D 100 (1997) 71-84

PHYSICA

Detection of symmetry of attractors from observations II. An experiment with $4 symmetry

P e t e r A s h w i n a, 1, J 6 r g T o m e s b

a Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK b Institutfiir Theoretische Physik, Universitiit T~bingen, K6stlinstrasse 6, D-72074 Tiibingen, Germany

Received 28 June 1995; revised 19 April 1996; accepted 10 June 1996 Communicated by H. Flaschka

Abstract

In part I of this paper we adapt the detectives of Barany et al. (1993) for symmetries of attractors of symmetric dynamical systems to apply to experiments. As an illustration of this theory we consider the dynamics of a system of four forced coupled electronic oscillators with full permutation symmetry ($4). On varying parameters in the circuit we obtain attractors with many different symmetry types and detect these using the methods described in part I.

1. Introduct ion

The symmetry of an attractor is a natural classifica-

tion tool for dynamical systems with symmetry. Ad-

ditionally, it can provide us with information about,

for example, multistabil i ty via the size of the conju-

gacy class of the symmetry. In this paper we study

the symmetries of dynamical states of a system of

four coupled nonlinear oscillators forced by an exter-

nal periodic signal. The system consists of four elec-

tronic van-der-Pol type oscillators coupled and forced

in such a way that they have $4 symmetry (the group

of all permutations of four objects). This system has a

rich variety of behaviours on changing a small num-

ber of parameters and in particular it has examples of

attractors with many symmetry groups.

1 Present address: Institut Non Linraire 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.

0167-2789/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PH S0167-2789(96)00176-5

Most examples of so-called symmetry detectives

have until now either been numerical experiments (see

e.g. [4,9]) or have had relatively simple symmetry

groups where one can essentially see what the sym-

metries of attractors are (for example, the Faraday ex-

periment of Gluckman et al. [6]). This experiment is

a physical system with a 24-element symmetry group.

A similar system has been studied numerically by

Tchistiakov [9].

In part I of this paper [3] we propose a definition of

detectives for symmetric dynamical systems in terms

Of equivariant observables, following ideas in [4, Sec-

tion 10]. This is a natural setting for discussion of

experimental systems with symmetry and it is the pur-

pose of this paper to provide an example for the ap-

plication of the methods discussed in part I. In part I :

(a) We consider detection as being a two-stage pro-

cess where one first examines an equivariant ob-

servation and then averages in a distinguishing

representation.

72 P. Ashwin, J. Tomes/Physica D 100 (1997) 71-84

(b) We give sufficient conditions that a function is a

detective in our sense. This is independent of the phase space; we merely need to know the irre-

ducible representations of the group on the phase

space. (c) We use averaging methods, working on the 'ob-

served' attractor. In particular we introduce the

'integrated observed' method. This greatly allevi-

ates problems with memory storage of integrated averaging methods. The storage requirement is es-

sentially related to the complexity of the group rather than the dimension of the phase space.

(d) We consider necessary conditions for detectives to

work on parametrised families of attractors. (e) We show that by measuring intersections with an

appropriate Poincar6 section we get the correct

symmetry types for attractors of the original

system. In. Section 2 we discuss the symmetry group $4,

its action by permuting a set of four oscillators and

some detectives for this action. For a system of electronic oscillators with forcing described in Section 3

we measure samples with a variety of different sym-

metries and give examples of several methods of detecting the symmetries. In the discussion in Section 4

we discuss the strengths and weaknesses of the vari-

ous tested methods. We refer the interested reader to part I [3] and [4]

for details of definitions and theorems, and here briefly summarise the notation used. Suppose we have a map- ping F : M --+ M that is equivariant under an action

of a finite symmetry group F. If A is an attractor for the map F, we define I;(A) to be the subgroup of F that fixes A setwise, and T ( A ) the subgroup that

fixed A pointwise. We say the actions of F on S and M are isotropy equivalent if the isotropy subgroups of the actions are the same. A representation space W is a distinguishing representation if all subgroups of F are isotropy subgroups of points in W. A detective (as defined in part I) is a map q~ : S --+ W such that for

any attractor A C M there is a generic set of contin- uous equivariant observations 7t such that

~V(K~(~(A))) = 2S(A)

for some averaging operator K~. We apply two differ-

S 4

A4 D3 D4

Id

Fig. 1. Conjugacy classes of the subgroups listed in Table 1 are ordered by containment as shown.

ent averaging methods, ergodic averaging K~ (as de-

fined in [4,7]) and thickened observed averaging K~,

a new method proposed in part I of this paper [3] (in

fact we practically do this by performing discretised

observed averaging K~ 'e for some positive e).

2. Group action and detectives

We introduce some notation for 84, its subgroups and action on the phase space of the oscillators. We discuss the detectives that will be used in Section 3 to examine samples measured from the analogue elec-

tronic experiment.

2.1. The symmetry group 84

Writing permutations in disjoint cycle form, Table 1 gives a list of conjugacy classes of subgroups of S4 with generating set, order, size of conjugacy class and containment relations. Note that there are 33 possible subgroups arranged into 11 conjugacy classes so it is clear that some computational help is necessary to decide which subgroup corresponds to the symmetry of a particular attractor. The containment of the conjugacy classes is shown in Fig. 1. We aim to detect the conjugacy class of the relevant symmetry groups rather than the actual subgroup.

Table 1 Representative subgroups

P. Ashwin, J. Tomes/Physica D 100 (1997) 71-84

of 84 with generating set, order, size of conjugacy class and contained subgroups

73

Group Generating set Order Size of Contains conjugacy class

S 4 {(12), (13), (14)} 24 1 A4, D3, D4 A4 {(123), (234)} 12 1 D2, Z3 04 {(1234), (13)} 8 3 Z4, Z2 × Z2, D2 D 3 {(123), (12)} 6 4 Z3, Z b Z4 {(1234)} 4 3 Z a 2 D2 {(12)(34), (13)(24)} 4 1 Z~ 22 x 22 {(12), (34)} 4 3 Z~, Z2 b Z 3 {(123)} 3 4 1 Z~ {(12)(34)} 2 3 1

Z2 b {(12)} 2 6 1 1 {} 1 1

Table 2 Isotropy subgroups of the permutation action of 84 on R 4

Isotropy group Fix dim fix

84 (a, a, a, a) 1 D3 (a, a, a, b) 2 Z 2 × Z 2 (a, a, b, b) 2 Z2 b (a, a, b, c) 3 1 (a, b, c, d) 4

The action consists of a trivial one-dimensional representation and an irreducible three-dimensional representation. Note that all points in the phase space for four oscillators have isotropy equal to one of these subgroups and so T(A), the instantaneous symmetry of an attractor A must be one of these subgroups.

2.2. The action for four coupled identical oscillators

We assume that the dynamics of the four oscillators

are governed by an ODE on a phase space

M = X 4 ~ y

with individual oscillators whose phase space is X =

R ~ coupled by a network whose degrees of freedom

are parametrised by Y = R t. The group $4 acts or-

thogonally on this phase space by permuting the com-

ponents in X 4 and leaving them in Y invariant.

Clearly, the action of S 4 decomposes into dim X

copies of the standard representation of the full tetra-

hedral group on R 3 and dim X + dim Y copies of the

trivial representation on R. Note that though we do

not know k or l, the representation structure is clear.

Thus the isotropy types of points in the phase space

are in one-to-one correspondence with isotropy types

of the action of $4 on R 4 as listed in Table 2.

We shall consider the observation space R 4 isotropy

equivalent to M ($4 acts by permutation), and consider

equivariant observations of the form

= (~/tl, ~2 , ~r3, 1/t4) C R 4, (1)

where ~Pi is some generic observation from the ith

oscillator.

2.3. Detectives; polynomial and smooth

We propose two different detectives from R 4 to a

distinguishing representation W. We use W = Rs4,

the group ring introduced to this problem by Tchis-

tiakov [9] (this can be seen as the vector space of

R-valued functions on the group). We enumerate the

elements in the group S4 in some (arbitrary) way:

7Z" : 84 --+ { 1 , ' - ' , 24}.

There is a natural left action on the group ring given

by the action on the basis. It is easy to check that

this is a distinguishing representation of $4, moreover,

for any finite group the group ring is a distinguishing

representation.

The polynomial detective qSP. Tchistiakov [9] pro-

poses polynomial detectives by enumerating the elements in the group $4 as {cri: i = 1 . . . . . 24} and then

defining

2 3 f P ( x l , X2, X3, X4) = XlXzX 3 •


The detectives are then given by the functions

(aiP (x) = fP(xai (1) , Xai(2 ), Xai(3 ), Xai(4)).

f o r / = 1 . . . . . 24. The smooth detective (a s . As an alternative and to

demonstrate that one is not limited to polynomial de-

tectives, we propose the following smooth detective:

f s (Xl, x2, x3, x4) = sin 3x2 sin 6x3 sin 9x4.

The factors are arbitrary but chosen in this range be-

cause the voltages measured are of the order of 1V

peak-to-peak. We show that both ~bP and (as satisfy the require-

ments to be detectives in Theorem 2.8 of part I [3]. This result strengthens [9, Theorem 2.6] to show that

the hypotheses of [9, Theorem 2.5] also apply.

Proposition 2.1. For all isotropy subgroups G of the action of $4 on S = R 4 and all open neighbourhoods

N C Fixs(G) we have for (a = (aP, (as

Span{Dx(av:x ~ N, v c Fixs(G)} = Fixw(G) (2)

and thus they are detectives.

Proof Note that (aP can be seen as a complex analytic function from C 4 to C 24. If (2) does not hold

for an open set N then by analyticity it does not hold

for all x 6 Fixs(G). Thus we can consider N to be a

neighbourhood of the origin. (Note that Fixw (G) corresponds to linear combinations of cosets of G).

Define the orthogonal projection pS : S--+ Fixs(G)

by

1 pS(v) = -~1 E g.v

' ' gcG

and likewise pW on W. Note that (apS = pW(a by

equivariance, and because the group action is orthogonal, the projections are as well. By Tchistiakov's result

Span{(aP(x):x ~ S} = W.

Suppose that for some G there exists a 0 5~ y E Fixw(G) with yW(aP(x) = 0 for all x E Fixs(G). Then for all x c S we have that p W y = 0 imply-

ing that y = 0. This contradiction means that (aP is a

detective.

Note that (as can be written as a complex analytic

function

f s = g(h(xl) , h(x2), h(x3), h(x4))

with g (Yl . . . . . Y4) = (Yl -- Y l 1 ) (y2 _ 2 2 2) (23 _ y 3 3 )

and h(x) = e3iX/2i. Because g is dominated by f P

for Yi large we have that (as also satisfies (2). []

Computation of the symmetry groups. Given an at-

tractor A we can find the symmetry on average E ( A )

by computing

E(K¢(ap(A))) ,

i.e. by averaging the image under (a with respect to A.

The instantaneous symmetry T(A) can be found by

computing

T(A) = r~ E((a 0P(x))) xEA

or equivalently,

0 ~:(7~(x)). xEA

Thus we need to examine the isotropy E (w) of points

w = K4~Op(A)) ~ Rs4. We do this by examining the presence or absence of given group element ak in E (w). This is found by computing the quantity

24

t~ k = ~ . ~ ( w~k i - t0 i ) 2.

i=1

This can be interpreted as the distance of w from the fixed point subspace F = Fix((ak}) in a (non- standard 2) metric. Note that w E F if and only if

waki - wi = 0 for all i. To distinguish all subgroups of $4 it is necessary to test for the presence or absence

of 16 of the 24 elements; a representative sample to

test for is

{(12), (13), (14), (23), (24), (34), (12)(34),

(13)(24), (14)(23), (123), (234), (124), (134),

(1234), (1243), (1324)}

2 The standard Euclidean metric can be used weighing the sum by the inverses of the cycle lengths as detailed by Tehistiakov [9].

P. Ashwin, J. Tomes/Physica D 100 (1997) 71-84

Table 3 Zeros of di indicating the presence of these group elements in the isotropy of w

k Group elements present if dk = 0

1 At least one of (12), (13), (14), (23), (24), (34) 2 At least one of (12)(34), (13)(24), (14)(23) 3 All of (12)(34), (13)(24), (14)(23) 4 At least one of (123), (234), (124), (134) 5 At least one of (1234), (1243), (1324)

Table 4 The conjugacy class of ZT(w) distinguished by the functions dl to d5

Group dl d2 d3 d4 d5

S 4 0 0 0 0 0 A 4 1 0 0 0 1 D 4 0 0 0 1 0 !) 3 0 1 1 0 1 Z 4 1 0 1 1 0 1)2 1 0 0 1 1 Z2 x Z 2 0 0 1 1 1 Z 3 1 1 1 0 1 Z~ 1 0 1 1 1 Z b 0 1 1 l 1 1 1 l 1 1 1

A '1' indicates that di is non-zero while a '0' indicates that it is zero. Note that 21 of the 32 possible combinations are prohibited.

Ramp generator

interface 10k '

IIEEE488 ~1 ~ 2 ~

Sine generator

[Ok

but this is not unique. For example, we need detect the

presence of one of the four-cycles (1234) or (1432)

but do not need both. It is possible to classify the con-

jugacy class of a subgroup into one of the 11 possible

classes in Table 1 by defining dl to d5 in the following

way:

dl = min(t02 ), t(13), to4 ), t(23), t(24), t(34)),

d2 = min(t(12)(34), t(13)(24), t(14)(23)),

d3 : max(t(12)(34), t(13)(24), t(14)(23)),

d4 = min(t(123), t(124), t(134)),

d5 = min(t(1234), t(1243), t(1324)).

Zeros of these functions can be interpreted as in

Table 3. The isotropy of the point w is then given by

Table 4.

3 . T h e e l e c t r o n i c e x p e r i m e n t

The system investigated was a network of all-to-

all coupled identical van-der-Pol type oscillators with

75

) Fig. 2; Schematic set-up for the electronic experiment of four identical coupled oscillators with $4 symmetry forced by a signal generator.

sinusoidal forcing. The circuit is that described in [1].

Each oscillator consists of an LC series resonator in

parallel with a nonlinear resistance. This is synthesised

using a negative impedance converter; a network of

diodes provides an asymmetric nonlinearity. The four

oscillators are coupled by resistors to a node that is

earthed via a variable capacitor C. This capacitance

acts as a coupling parameter such that at large capac-

itance, a rotating wave or splay phase state is stable

while for small capacitance the in-phase solution is

stable. This is discussed at length in [2]. The forcing

is through a 10k resistance and a 2.2 txF capacitance

in series from a 'Wavetek ' signal generator producing

sine-wave output with zero offset voltage (the effect

of the 2.2 IxF capacitor is to decouple the offset), fre-

quency f and amplitude A. Fig. 2 shows the schematic

layout of the experiment. Quasi-static variation of the

forcing frequency is achieved by feeding a very low

frequency ramp generator into the frequency modu-

lation input of the signal generator. The equations of

motion for this circuit are given in [2].

Care was taken to ensure that the oscillators are

close to identical by providing the operational ampli-

fiers with offset null trimmers, using low tolerance

components and by trimming the ferrite cored induc-

tors until identical frequencies are obtained when un-

coupled. The ambient temperature was kept stable,

though this was not found to be a major factor affect-

ing reproducibility. Samples ( x l , x2, x3, x4) are taken

using a 16 bit analogue to digital interface sampled at


the frequency of forcing (i.e. we measure an equivariant observable from a Poincar6 section).

3.1. Results for constant periodic forcing

Several points in parameter space (C, f , A) were measured, with f the frequency (in Hz), A the amplitude (in V) of the forcing signal and C the coupling capacitance. The system shows a rich variety of dynamical behaviour with periodic solutions of many symmetry types, quasi-periodicity with two and three frequencies (see [1]) and symmetric chaos apparent from visual examination of the time series or projected Poincar6 sections. The precise symmetry type of the attractors is often very difficult, if not impossible to judge from mere visual examination: The symmetry detectives provide us with a way to settle the ques- tion of precisely what symmetry a dynamical state possesses.

We compute the symmetries of several time series using the observed integral and ergodic average methods for the detectives 4, s and ~b f as discussed in Section 2.3 and part I. For the integral method, we compute the discretised observed average of 4) using a rectangular grid with spacing 0.122 V. A total of 254 ----- 390 625 grid points were stored. In different circumstances, different detectives were found to pro- duce better results but the most consistently good re- suits were found for the smooth detective ~b s and the ergodic averaging method.

Fig. 3 shows some measurements projected into the

(xl -- x2, x3 - x4)-planes. It is obvious that there are several different symmetry types and no single two- dimensional projection has a chance of showing all of them. As discussed in [1] this projection faithfully reproduces a D4 subgroup of $4 and one can for example recognise six different Z4 periodic solutions. Fig. 4 shows the convergence of detectives for these samples. For each sample, we have shown the convergence of detectives

d i = symmetry on average, ds+i = instantaneous symmetry,

which by Table 4 gives the symmetries Z(A) and T (A) of the observed attractor. The methods tried are

(b)

(c)

qt-

I

(d)

+

+

d

(e)

Fig. 3. Example time series taken from the experiment, shown as Poincar6 sections projected into the plane (Xl - x 2 , x3 - x 4 ) . In (d) and (e) the points are plotted as crosses for emphasis. Scale is in volts, and the forcing frequency is f = 6.158 kHz in all cases. (a) A = 6.20V; this appears to have full symmetry on average when projected into this plane. (b) A = 5.77V; a chaotic attractor sitting in a fixed-point subspace. On increasing A, this attractor undergoes a blowout bifurcation to give rise to the attractor in (a). (c) A = 0.647 V; this is clearly a quasi-periodic state with Z4 symmetry on average. (d) A = 4.80V; in this projection, the orbit appears to have full symmetry. A closer inspection of the time series reveals that it is a periodic orbit with S ( A ) = T(A) = Z2 x Z2 symmetry. (e) This periodic point has Z(A) = D4 and T(A) = Z 2 × Z 2.

P. Ashwin, J. Tomes/Physica D 100 (1997) 71-84 77

(i)

"i 5 0 0 ~ 10

n O 0 i

(ii)

1 t - I 0

5000 I 0

n O 0 i

(iii)

t -4

~-6

10

n O 0 i

- 5 0 0 0 " ~

n

(iv)

10

0 0 i

Fig. 4. Graphs showing convergence of detective i for average symmetry di and instantaneous symmetry d5+ i as a function of sample length n. In all the samples investigated, the final values are approximated very closely after only a few hundred sample points. The samples correspond to those shown in Figs. 3(a)-(d). The methods of calculating the average symmetries used in each case are (i) K~p, (ii) K~, (iii) KI~p and (iv) KIts . For (a)all four methods of measuring the detectives converge to the same answer, i.e. E(A) = S4, T(A) = 1.

all combinations of ergodic average and observable in-

tegral averaging with the detectives ~bP and q~s; namely

(i) K~p, (ii) K~s, (iii) K~p and (iv) K~s. For the com-

puted groups, we write

,~E,p = ~,(K~p ( 0 ( A ) ) ) ,

etc. As discussed in the figure captions and at length

in Section 4, there are several factors at work which

contribute to which (if any) of the methods ( i)-( iv)

provide good detectives. The converged values of the

detectives for the time series in Fig. 3 and the inferred

the minimum and maximum value of zero threshold

necessary to obtain the correct answers are shown in

Table 5.

3.2. Results f o r parameter space scans

As a further investigation into the various differ-

ent methods of detecting the symmetries, we have

performed slow quasi-static scans through parameter

space. To achieve this, a voltage controlled frequency

modulation input (denoted FM in Fig. 2) to the sig-

nal generator was fed by a slow ramp generator. The

rise rate of the frequency was kept constant during

the sample. Fig. 5 shows an illustration of projections

into the (X1 --X2, X3 --x4)-plane for such a scan, start-

ing at f = 7.78 kHz and finishing after 64k samples

at f = 9.258 kHz. If we are away from a bifurca-

tion point, the system quickly approaches an attrac-

tor, and by examining small blocks of the scan, we

can spot changes in symmetry. This method has been

applied by Tchistiakov [9] to numerical experiments

on a system of four oscillators with $4 symmetry.

Note that unlike in numerical experiments, we need

not worry about being 'caught ' in a fixed-point sub-

space. Imperfections and intrinsic noise in the sys-

tem are enough to perturb away from any fixed-point

space.


Table 5 Minimum and maximum zero thresholds necessary to correctly classify the symmetry types of samples shown in Fig. 3, by method used

rE,p rE,s •I,p v?I,s TE, P TE,s TI,p TI,s

(a)

max * * * * 2 e - 2 2el 3 e - 2 le l min 3 e - 6 5 e - 3 4 e - 7 6 e - 3 * * * *

(b) max 8 e - 4 1 e0 5 e - 4 4e0 l e - 2 le 1 3 e - 2 le 1 min 5 e - 8 3 e - 3 3 e - 4 3 e - 2 5 e - 6 2 e - 1 6 e - 4 3e0

(c) max 5 e - 6 4 e - 1 2 e - 6 l e - 1 5 e - 4 le l 6 e - 4 le l min 4 e - 6 2 e - 2 l e - 6 2 e - 2 * * * *

(d) max l e - 4 le0 2 e - 4 2e -1 4 e - 3 le l l e - 2 le l min 3 e - 8 4 e - 3 2 e - 7 6 e - 2 2 e - 6 2 e - 2 l e - 5 4e0

(e) max 1 e - 5 6 e - 2 2 e - 5 3e0 6 e - 7 8e0 4 e - 6 9e0 min 2e--7 2 e - 2 * * 4 e - 7 7e -1 * *

The ' , ' indicates that there is no limit to the threshold. Note that it is not possible to choose a single threshold that gives consistent results for r I 'p, T E'p or T I'p, and the window of choice of possible threshold for z~ 7E'p is very small. By contrast, the smooth detective methods SE,s and T E's give the correct answers for a larger range of zero thresholds.

(i) (ii)

~1-4] -2 .~1.11 1

,oo ,o ,oo ,o n O 0 i n O 0 i

- 2

~1-4

.9o~-6

n O 0 i

(iii) (iv)

10 5 0 0 0 ~ ~ / ~ - 5 10

n O 0 i

Fig. 4. Continued. (b) There appears to be agreement between the methods that 2~(A) = D3; however, it is not clear whether the instantaneous symmetry is trivial or D 3. Visual examination shows that T(A) = D3, and so the smooth detectives give more accurate results than the polynomial detective in this case.

P Ashwin, J. Tomes/Physica D 100 (1997) 71-84 79

(i) (ii)

- J

5000 10 5 0 0 0 ~ ~ ~ - ~ - 10

n O 0 i n O 0 i

(iii) (iv)

5000 10 10

n O 0 i n 0 0 i

Fig. 4. Continued. (c) All methods show quite clear convergence to 2~(A) = 2 4 , T(A) = 1.

(i) (ii)

i i "-I 0 GI-4 ~ - I

10 5 0 0 ~ 10 5000

n O 0 i n 0 0 i

(iii) (iv)

0 0-

"--I ~l -o -5 t= m - 5 - _o o

-10 -10,

10 5 10

n O 0 i n O 0 i

Fig. 4. Continued. (d) This sample has 22(A) = T(A) = Z 2 × 22 according to all detective methods.

80 P.. Ashwin, J. Tomes/Physica D 100 (1997) 71-84

(i) (ii)

1 t

5 0 0 0 10 10

n O 0 i n O 0 i

(ii i) (iv)

0 0.

"-I "-I "O - 5 "(3 m - 5 . o .go

- 1 0 -10 •

10 ~ 10

n O 0 i n O 0 i

Fig. 4. Continued. (e) This sample has ~?(A) = D4 and T(A) = Z 2 x Z 2 according to the smooth detective, although the polynomial detective is less clear.

x

h "//}~ ..~ ~, ';~":.~T~,~I/ ,," ~ ~-~-)'."~..~~

• ~ .~?-.:

¢ • . .~.~ J . ~,~

Fig. 5. An example of a scan through parameter space. This shows a series of 64 slices, each of which shows a block of 1024 points projected onto the plane (xl - x2, x3 - x4). The amplitude A ---- 1.5 V of forcing is constant while the frequency f varies from f ---- 7.78 kHz on the left up to f = 9.258kHz on the right. The total sample is of length 65536 observations.

Fig. 6 shows the va r i a t ion o f de tec t ives for ave rage

and i n s t a n t a n e o u s s y m m e t r i e s as we go a long the scan.

Table 6 shows the i m p l i ed symmet r i e s ; the de tec t ives

go f r o m gene ra l a g r e e m e n t tha t the s y m m e t r y is $4

to a state tha t is c lear ly Z 4 symmet r i c . The samples

s tud ied separa te ly ind ica te tha t the m o s t t rus twor thy

p red ic t ions are l ike ly to c o m e f r o m the c o l u m n s Z 'E,s and T E,s .

P. Ashwin, J. Tomes/Physica D 100 (1997) 71-84 81

"-, If,"+"' ',,.,.t ,, i i

o lliIhVlL v ~,i.,, ' , , .J I . 1 o l Y ~ i . ~ !L ' , , iL / . I

0 2 4 6

n x 10 4

. i~ I ~" ~ ] " I " --"

~ - 5 ' a-' / '~r~'~'

- 10 [~ . l ~ . ~ .

" 0 2 4 6 n x 10 4

Ol . . " = " , " ' . ' ] I," ~' V/~u~..~g~+ - . . . . . . . ,.I

I I ,- t i l . I

- : , . . . . . , i 6'

n x 10 4

0t . , - "~ ( • l • A " , , I r " -" ; .

I U' - ~"c' ~/.t ~ ...... 1 ~ [ ~ . ~ ) ~ I I | II ] -~ I I I

II. I II _.i~.~- _o

_1olg . !~6.~,~ . I 0 2 4 6

n x 10 ~'

ok.--"-"~--,"-.~...;.'.,.... L =

_, I ,. ,: ~a ~k r , a . / ~ ~0 "I l ~ "",.al} ¢ ~ - K I " I I

- - -1 2

n x 10 4

Fig. 6. Values of di for detectives of the average symmetries of attractors for the sample shown in Fig. 5. The series was averaged in 64 blocks of 1024 points. The lines correspond to K~p (solid), K~s (dotted), K~p (dot-dashed)and K~s (dashed). Observe that there is a general agreement, although the jumps can take place on a wide range of different scales. The number n refers to the position of the first sample in the current block of 1024 samples.

The scan illustrated in Fig. 5 incidentally demon-

strates that the detective is not a one-parameter de-

tective as defined in part I. About half way through

the sample, the system remains on a periodic orbit

(at double the forcing frequency) with symmetry

T(A) : 22 × 2 2 , E ( A ) : D4. As this periodic orbit

slowly deforms, it passes through a point in parameter

space where it appears to have T(A) = Z ( A ) = S4

(the smooth detective give this incorrect answer over

a smaller range than the polynomial detective). The

particular observable xi that we have taken from

the circuit is not capable of distinguishing this. To

correctly measure the symmetries at all points in

this one-parameter scan one would need to take two

measurements from each oscillator, for example xi and xi at the Poincar6 section. In practice this would require a lot more effort and data and so we have not

at tempted this. This means that one should be suspi-

cious of codimension one lines of higher symmetries

on any parameter space scan.

4. D i s c u s s i o n

This work is to our knowledge the first demonstra-

tion of an application of detective ideas to an exper-

imental system with a symmetry that is not a planar

symmetry, i.e. where the observation space is forced

to have dimension greater than two.

The polynomial detectives introduced by Tchis-

tiakov can give reliable results for the average

symmeb'ies of attractors from this system but suf-

fer from the fact that they are high-order poly-

nomials and thus have extreme scaling behaviour

near the origin. Using a smooth detective over-

comes this to a certain extent and allows one to

reliably test many different symmetry states using

the same 'zero threshold' . The averages were com-

puted by the ergodic average and the ' thickened

observed' (or more precisely the 'discretised ob-

served') averaging methods outlined in part I of this

paper [3].

82 P Ashwin, J. Tomes/Physica D 100 (1997) 71-84

Table 6 Implied symmetries of attractors measured from the scan through parameter space illustrated in Figs. 5 and 6 n ,~yTE, p r,E,s ~7I,p .v,I,s TE,P TE,s TI,p TI,s

zero 3e-06 0.04 3e-06 0.01 3e-06 0.4 3e-06 0.4

0 24 24 04 Z4 1 1 1 1

1024 !)4 Z4 D4 Unk. 1 1 1 I

2048 84 Z2 x 22 S4 Z~ S4 Z 2 x 22 S 4 Z~

3072 $4 Z2 x Z 2 $4 Z2 x Z2 S4 Z2 x Z2 $4 Z2 x Z2

4096 S 4 Z~ $4 l 84 Z~ Z2 x Z2 1 512o s4 1 1 1

6144 Z~ Z~ Z~ 1 Z~ Z~ 1 1

7168 Z~ Z~ Z~ 1 Z~ Z~ 1 Z~

8192 Z~ Z~ Z2 ×Z2 Z~ Z~ Z~ Z~ Z~

9216 D4 Z~ Z 2 × Z 2 Z~ Z~ Z~ Z~ Z~

10240 Z 2 x Z 2 Z~ Z~ Z~ Z~ Z~ Z~ Z~

11 264 Z~ Z~ Z2 X Z2 1 1 Z~ Z~ 1 12288 Z~ Z~ 1 1 1 Z~ 1 1

13312 Z~ Z~ 1 1 1 Z~ 1 1

14336 Z2 X Z2 Z~ Z~ 1 1 Z~ 1 1 15360 1 1 1 1 l 1 1 1 16 384 1 1 1 1 1 1 1 1 17408 1 Z3 1 1 1 1 1 1 18 432 1 1 1 1 1 1 1 1 19456 Z4 Z~ Z~ 1 1 1 1 1 20480 24 24 O 4 24 l 1 1 1 21504 Z4 Z4 24 Z4 1 1 1 1 22528 Z4 Z4 Z4 Unk. 1 1 1 1 23552 24 24 24 Z~ 1 1 1 1 24576 Z4 Z4 Z4 1 1 1 1 1 25600 24 Z~ Z 4 Z 4 1 1 1 1

26624 D4 Unk. D4 Z~ 1 Z~ Z~ 1 27 648 O 4 1)4 $4 Unk. Z2 x Z2 Z2 x Z2 Z2 × Z2 Z2 x Z2 28 672 S 4 D4 S4 D4 Z 2 × Z 2 Z 2 × Z 2 S 4 Z 2 x Z 2

29696 S 4 O4 S4 Z~ N 4 O 4 84 Z~

30720 S4 S4 $4 Z~ $4 $4 $4 1 31744 84 D 4 $4 84 54 Z2 × 22 84 84

There are several transitions in symmetry apparent during this scan. The first row refers to the zero threshold chosen for that method. Note in particular the sample starting at 30 720 is predicted to have $4 symmetry. This is an anomalous reading due to the detective not being a one-parameter detective (see text).

F r o m our exper ience of many measu red examples

we summar i se be low some of the advantages and

d isadvantages of the var ious methods . For an ideal

detect ive, one requires that the answer converges

quickly and re l iab ly to the correct symmet ry group

for t h e vast ma jo r i ty of cases. However , g iven a

real sys tem that does not have a per fec t symmetry ,

there wi l l a lways be the poss ib i l i ty of states that do

not fit into the c lass i f icat ion by symmet ry and one

should a lways be aware o f this. Note, however, that

the c lass i f icat ion U n k n o w n in Table 6 usual ly corre-

sponds to an incorrect choice of the zero threshold

rather than the averaged poin t be ing in any other way

' d i s a l l owed ' .

P. Ashwin, J. Tomes/Physica D 100 (1997) 71~84 83

Table 6 continued n ~,E,p z~E,s ~I,p EI,s TE,P TE,s TI,p TI,S

32768 $4 D4 S4 S4 84 Z2 × Z2 S4 $4 33792 $4 D4 $4 1 $4 Z 2 x Z 2 S 4 1

34 816 S 4 84 S4 $4 $4 Z2 x Z2 $4 S4 35 840 $4 84 84 D3 S4 84 84 84

36 864 S4 $4 $4 $4 $4 $4 $4 $4 37888 $4 $4 $4 Unk. $4 Unk. $4 1

38912 S 4 Unk. S 4 Z~ 84 22 x 22 S 4 22 x 22

39936 $4 Z2 x Z 2 $4 1 $4 Z2 x Z2 $4 Z~ 40960 S 4 D 4 84 Z~ 84 22 × 22 S 4 1

41984 S 4 Unk. D 4 Z~ D 4 Z~ Z 2 × Z 2 1

43008 D4 Z~ D4 1 Z~ Z~ 1 1 44032 Z~ 1 Z4 1 1 1 1 1

45056 24 24 24 Z~ 1 1 1 1 46080 Z 4 Z4 Z 4 Z~ 1 1 1 1 47104 Z4 Z4 Z4 1 1 1 1 1 48128 Z4 Z4 Z 4 Z 4 1 1 1 1 49152 Z4 Z4 Z4 1 1 1 1 1 50176 Z4 Z4 Z4 Z~ 1 1 1 1 51200 Z4 Z4 Z~ Z4 1 1 1 1 52224 Z4 Z4 Z4 Z~ 1 1 1 1 53248 Z4 Z4 Z4 Z4 1 1 1 1 54272 Z4 Z4 Z4 Z~ 1 1 1 1 55296 Z4 Z4 Z4 Z~ 1 1 1 1 56320 Z~ Z4 Z4 Z~ 1 1 1 1 57344 Z~ Z4 Z4 1 1 1 1 1 58368 Z~ D4 Z 4 Z 4 1 1 1 1 59 392 Z~ 04 Z 4 Z~ 1 1 1 1 60416 Z4 D4 Z4 Z4 1 1 1 1 61440 Z4 D4 Z4 Z4 1 1 1 1 62464 Z4 Z4 Z4 Z4 1 1 1 1 63488 24 24 D 4 Z 4 1 1 1 1 64512 Unk. Z 4 D4 Z4 1 1 1 1

It is of course a difficult p rob lem to correctly choose

a zero threshold and thereby judge whether an attrac-

tor has a part icular symmetry or not. A test of ' robust-

ness ' of a detective is how tolerant it is to this choice.

We have demonstra ted for this system that the smooth

detective cp s is better and it is easy to see why this is the

case, name ly a po lynomia l detective has (necessari ly)

a high-order zero near the origin (all mixed derivatives

up to and inc luding fifth order are zero) whereas the

smooth detective has third-order nonzero derivatives at

the origin. Thus the po lynomia l method assigns much

higher impor tance to points far away from the origin

whereas the smooth detective is ' fairer ' .

We have exper imented with a ' fundamenta l doma in '

detective of the form

f f ( x l , x2, x3, x4)

: H ( x 2 - x l ) H ( x 3 - x 2 ) H ( x 4 - x3),

where H is a uni t step funct ion (i.e. not even cont inu-

ous). Even this piecewise constant funct ion gives good

predict ions for attractors which are (a) connected and


(b) have T(A) = 1, trivial point symmetry. However

when this is not the case they can be unreliable. Note

that f f corresponds to a characteristic function sup-

ported on a fundamental domain for the group action.

This study also furnishes an example of how the de-

tectives examined can fail for a one-parameter system.

One of the major problems with carrying out such an

investigation as this is that it is easy to become buried

in the amount of information generated; for this rea-

son we have concentrated on finding just the conju-

gacy class of the subgroups involved rather than the

specific subgroup.

Acknowledgements

We thank Matt Nicol and Ian Stewart for interest-

ing conversations with regard to this work, and partic-

ularly Viktor Tchistiakov for sending us a preprint of

[9]. The research of PA was partially supported by the

EPSRC at the Mathematics Institute of the University

of Warwick while both authors acknowledge the sup-

port of an Anglo-German academic research collabo-

ration funded by the British Council and the Deuscher

Academischer Austauschdienst.

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