Almost Invariant Sets in Chua�s Circuit

Michael Dellnitz�and Oliver Junge

Mathematisches Institut

Universit�at Bayreuth

D������ Bayreuth

http���www�uni�bayreuth�de�departments�math��mdellnitz

http���www�uni�bayreuth�de�departments�math��ojunge

March �� ���

Abstract

Recently multilevel subdivision techniques have been introduced in the

numerical investigation of complicated dynamical behavior� We illustrate the

applicability and e�ciency of these methods by a detailed numerical study

of Chua�s circuit� In particular we will show that there exist two regions in

phase space which are almost invariant in the sense that typical trajectories

stay inside each of these sets on average for quite a long time�

� Introduction

Suppose that we want to study the complicated dynamical behavior of a givendi�erential equation Then � due to the nature of chaos � it could be misleading toextract information from long term computations of single trajectories Rather itwould be desirable

�a� to approximate directly the topological structure of the underlying invariantset and then

�b� to derive statistical properties of the dynamical behavior on this set

For instance one object which is of interest in this context is the related �natural�invariant measure �an SBR�measure say�� on the one hand the support of such ameasure provides the topological information and on the other hand its range yields

�Research of the authors is partly supported by the Deutsche Forschungsgemeinschaft underGrant De ��������

the statistics how frequently certain parts of phase space are visited on average bytypical trajectories

However the invariant measure on its own does not provide all the informationon the dynamics which could be of interest For an illustration of this fact considertwo parts in phase space which both have measure �� Then the chances are eventhat typical trajectories are observed in each of these regions On the other handnothing can be said about transitions between these two sets For instance it maybe the case that trajectories are moving rapidly back and forth between them or incontrast to this they typically stay in each of them for quite a long period of timebefore moving to the other one This observation leads naturally to the questionwhether there exist almost invariant sets that is regions in phase space wheretypical trajectories stay �on average� for quite a long period of time The knowledgeabout the existence of almost invariant sets combined with the information on thenatural invariant measure provides an

approximation of the essential dynamics

on invariant sets � this is precisely what we aim for Recently subdivision techniques combined with the approximation of the Fro�

benius�Perron operator have been developed for the numerical extraction of theessential dynamics �see �Dellnitz � Hohmann ���� �Dellnitz � Junge ���� orfor an alternative approach using index theory �Eidenschink ����� In this articlewe employ these techniques in order to study in detail the dynamical behavior ofChua�s circuit

�x � ��y �m�x��

�m�x

���y � x� y � z�z � ��y�

� �

This system of di�erential equations describes the behavior of an autonomous elec�trical circuit Concerning the technical background as well as a detailed study ofthis model by classical methods the reader is referred to e g �Huang et al� ���� Throughout our considerations we have �xed the values of the parameters as follows

� � �� � � ��� m� � ���� and m� � ����

In our discussions we will particularly focus on the situation where the underlyingsystem has a Z��symmetry since also Chua�s circuit is Z��symmetric In general theaction of Z� on R

n can be represented by fIn� �g with an orthogonal matrix �satisfying �� � In �In the identity on Rn� In Chua�s circuit � � �I� since the righthand side in � � is an odd function in �x� y� z� Correspondingly �x�t�� y�t�� z�t��is a solution of � � if and only if ��x�t���y�t���z�t�� is a solution

The bottom line of this article is to approximate the essential dynamics of Chua�scircuit In particular we compute invariant measures on the closure of two two�dimensional unstable manifolds and show numerically that there exist two almostinvariant sets in Chua�s circuit which are symmetrically related

All the algorithms described in the following are integrated into the C�� code

GAIO �Global Analysis of Invariant Objects�

A link to a detailed description of GAIO can be found on the homepages of theauthors

�

� Construction of a Box Covering

The basic technique underlying the approximation of invariant measures and almostinvariant sets is the computation of transition probabilities for boxes covering theinteresting dynamics in phase space Hence we begin by describing the numericalmultilevel methods we have used to construct such a box covering We remark thatin principle also the classical cell mapping techniques could have been used �e g �Kreuzer ���� �Hsu ����� However our approach has the advantage that thecomputational e�ort primarily depends on the complexity of the dynamical behaviorrather than on the dimension of phase space

A Subdivision Algorithm

The purpose is to approximate invariant sets of discrete dynamical systems of theform

xj�� � f�xj�� j � �� � � � � �

where f is a continuous mapping on Rn The central object which is approximatedby the subdivision algorithm developed by Dellnitz � Hohmann ����� is the so�called relative global attractor

AQ ��j��

f j�Q�� �� �

where Q � Rn is a compact subset Roughly speaking the set AQ should be viewed

as the union of unstable manifolds of invariant objects inside Q In particular AQ

may contain subsets of Q which cannot be approximated by direct simulation The subdivision algorithm for the approximation of AQ generates a sequence

B��B��B�� � � � of �nite collections of boxes with the property that for all integers kthe set

Qk ��B�Bk

B

is a covering of the relative global attractor under consideration Moreover thesequence of coverings is constructed in such a way that the diameter of the boxes

diam�Bk� � maxB�Bk

diam�B�

converges to zero for k �� Given an initial collection B� one inductively obtains Bk from Bk�� for k �

� �� � � � in two steps

�i� Subdivision� Construct a new collection �Bk such that�B� �Bk

B ��

B�Bk��

B and diam� �Bk� � � diam�Bk���

for some � � � �

�

�ii� Selection� De�ne the new collection Bk by

Bk �nB � �Bk � f���B� � �B �� for some �B � �Bk

o�

The following proposition establishes a general convergence property of this al�gorithm

Proposition ��� Let AQ be the global attractor relative to the compact set Q� andlet B� be a �nite collection of closed subsets with Q� � Q� Then

limk��

h �AQ� Qk� � ��

where we denote by h�B�C� the usual Hausdor distance between two compact subsets B�C � R

n�

For a proof as well as details concerning the implementation of the subdivisionalgorithm we refer to �Dellnitz � Hohmann ����

The Approximation of Invariant Manifolds by Continuation

The subdivision algorithm can be used as a basis to construct coverings of invariantmanifolds of hyperbolic objects In our computations we have used this methodand we now brie�y describe a simpli�ed version of it For a detailed discussion thereader is referred to �Dellnitz � Hohmann ����

Let p be a hyperbolic �xed point and denote by W u�p� its unstable manifold As in the subdivision algorithm we �x a �large� compact set Q � R

n containingp in which we want to approximate part of W u�p� The idea of the method is asfollows� �rst we apply the subdivision algorithm to a �small� neighborhood C of pwhich leads to a box covering of the local unstable manifold W u

loc�p� �Observe thatW u

loc�p� is the relative global attractor AC if C is small enough � Then this coveringis extended to the global part of W u�p� by an action of f on the covering

To be more precise let Bk be a covering of the relative global attractor

AC � W uloc�p� � C

obtained by the subdivision algorithm applied to C after k steps For simplicityassume that the boxes in Bk are generalized rectangles of identical size and denoteby

P a partition of Q

into a �nite collection of disjoint generalized rectangles of this type For the compu�tation of unstable manifolds we now proceed as follows� de�ne a sequence C�� C�� � � �of subsets Cj � P by

�i� Initialization�C� � Bk�

�

�ii� Continuation� For j � �� � �� � � � de�ne

Cj�� � fB � P � B � f�B�� �� for some B� � Cjg

until a j is reached such that Cj�� � Cj

Intuitively it is clear that the algorithm as constructed generates an approxi�mation of part of the unstable manifold W u�p� Indeed if we set W� � W u

loc�p��Cand de�ne inductively for j � �� � �� � � �

Wj�� � f�Wj� �Q�

then one can show the following convergence result�

Proposition ��� �Dellnitz � Hohmann� ��� For every j the set Cj is acovering of Wj� Moreover� for �xed j� Cj converges to Wj in Hausdor distance ifthe number k of subdivision steps in the initialization goes to in�nity�

Aspects Concerning Z��Symmetry

We now outline how to reduce the numerical e�ort if the underlying dynamicalsystem is Z��symmetric As in the introduction we assume that Z� is given byfI� �g where � is an orthogonal matrix with �� � I �Recall that Chua�s circuit isZ��symmetric where the nontrivial element in Z� is given by � � �I� �

Suppose that the relative global attractor AQ is invariant under the action of �that is

�AQ � AQ�

Then both the subdivision� and the selection step in the subdivision algorithm canbe simpli�ed in an obvious way For this one has to work with box collections Bkwhich are ��symmetric

B � Bk � �B � Bk� �� ��

Indeed in this case one can reduce the numerical e�ort as well as the memoryrequirements by a factor of two since in the subdivision algorithm one just needs totake half the number of boxes into account

Essentially the same argument can be applied to the continuation method forthe approximation of invariant manifolds� one just needs to consider a symmetricQ �i e �Q � Q� and has to �symmetrize the Cj�s in each continuation step

Box Coverings for Chua�s Circuit

In Chua�s circuit � � we have approximated the union of the two two�dimensionalunstable manifolds of the steady state solutions �������� ��������� inside the boxQ � ���� �� ������ ���� ����� ��� Note that Q � �Q and therefore we can usethe underlying symmetry of Chua�s circuit We have computed the time�one�mapf numerically using a fourth order Runge�Kutta method with constant step lengthh � �� Four di�erent box coverings are shown in Figures and � Already after

�

−10−5

05

10

−2

−1

0

1

2

−20

−10

0

10

20

xy

z

a � subdivisions� �� boxes

−10−5

05

10

−2

−1

0

1

2

−20

−10

0

10

20

xy

z

b � subdivisions� ��� boxes

−10−5

05

10

−2

−1

0

1

2

−20

−10

0

10

20

xy

z

c � subdivisions� ���� boxes

Figure � Three coverings of an attracting set A for Chua�s circuit The set Ais the closure of the union of the two�dimensional unstable manifolds of the twononsymmetric steady states

� steps in the subdivision algorithm the shape of the invariant set given by theclosure of the unstable manifolds becomes apparent

�

Figure �� Covering of an attracting set for Chua�s circuit ��� subdivisions ������boxes�

� Approximation of the Essential Dynamics

In the previous section we have seen how to construct a box covering of the relevantdynamical behavior in phase space In other words we have the topological informa�tion needed and it remains to approximate the dynamical behavior on this object This is done by

�a� an approximation of the natural invariant measure supported on the invariantset �e g an SBRmeasure if it would exist� and

�b� an identi�cation of subsets of the relative global attractor which are almostinvariant

The main purpose of this article is to apply the related numerical techniques toChua�s circuit and hence we will brie�y recall them from �Dellnitz � Junge ����

Discretization of the Frobenius�Perron Operator

The crucial observation is that the calculation of invariant measures can be viewedas a �xed point problem Let M be the set of probability measures on R

n with

�

respect to the Borel �algebra Then � M is invariant if and only if it is a �xedpoint of the FrobeniusPerron operator P �M�M

�P��B� � �f���B�� for all measurable B � Rn � �� �

To discretize the operator P � M � M we replace M by a �nite dimensionalset Mk� let Bi � Bk i � � � � � � N denote the boxes in the covering obtained afterk steps in the subdivision algorithm We choose Mk to be the set of �discreteprobability measures on Bk that is

Mk �

�u � Bk � �����

����NXi��

u�Bi� �

��

Then the discretized Frobenius�Perron operator Pk �Mk �Mk is given by

v � Pku� v�Bi� �NXj��

m�f���Bi� � Bj�

m�Bj�u�Bj�� i � � � � � � N� �� ��

where m denotes Lebesgue measure Now a �xed point u � Pku of Pk provides anapproximation to an invariant measure of f Observe that Pk is represented by astochastic matrix

Remark ��� For the mathematically precise statement on the convergence of thismethod one would have to introduce the concept of small random perturbations The reason is that this allows one to use a result of Kifer ����� on the convergenceof invariant measures in the perturbed systems to the SBR�measure The reader isreferred to �Dellnitz � Junge ���� for the rigorous mathematical treatment �Seealso �Kifer ���� �

Aspects Concerning Z��Symmetry

If the box covering is chosen in a symmetric way �see �� ��� then the numericale�ort for the computation of the discretized Frobenius�Perron operator in �� �� canbe reduced by a factor of two Indeed since � is orthogonal

m�f���Bi� � Bj�

m�Bj��

m�f����Bi� � �Bj�

m��Bj�for all i� j �� ��

and therefore just half of the entries of Pk have to be computed This also reducesthe memory requirements � observe that typically the matrix Pk is extremely sparse

By �� �� one can easily verify that the discretized Frobenius�Perron operatorcommutes with the action of � that is

�Pk � Pk�� �� ��

�Here we have abused notation and denoted by � also the matrix which is permutingthe indices of symmetrically placed boxes � It follows that an eigenvector v of Pk

corresponding to a simple eigenvalue is either

�

symmetric ��v � v� or antisymmetric ��v � �v�

To see this use �� �� to obtain

Pkv � �v �� Pk��v� � �Pkv � ��v � ���v��

and therefore �v must be a real multiple of v Finally note that � is orthogonal

The Natural Invariant Measure � for Chua�s Circuit

Suppose that there exists a unique natural invariant measure with support onthe closure of the two two�dimensional unstable manifolds considered in Section � For the approximation of we have to compute the eigenvector u of the discretizedFrobenius�Perron operator corresponding to the eigenvalue By the uniqueness ofthe invariant measure the support of has to be symmetric with respect to � � �I� Hence we can use the Z��symmetry in the computations as indicated above Inparticular we know that u has to be symmetric since its entries are nonnegative bystandard results on eigenvectors of stochastic matrices

For the visualization we have used two di�erent techniques which have beendeveloped in �Dellnitz et al� ���� In Figure � we cut the invariant set by a planeand show the invariant measure along the intersection for six di�erent cuts Here alsothe complicated topological structure of the unstable manifolds becomes apparent The invariant measure on the entire invariant set is illustrated in Figure �� yellowcolored boxes indicate high density and hence typical trajectories can frequently beobserved in these regions On the other hand typical trajectories will hardly visitthe regions which are colored red and probably never enter the green areas

Remark ��� An animation with a �moving cutting plane can be found on thehomepages of the authors

�

a

b

c

d

e

f

Figure �� Cutting planes illustrating the structure of the attracting set in Chua�scircuit as well as the natural invariant measure ��� subdivisions ������ boxes� Inthe cutting planes the density ranges from green �low density� �� red �� orange�� yellow �high density�

�

Figure �� The natural invariant measure in Chua�s circuit ��� subdivisions ������boxes! coloring scheme as in Figure ��

Almost Invariant Sets in the Presence of Z��Symmetry

The natural invariant measure provides the information where on average typicaltrajectories will be observed in the given system However the invariant measuredoes not provide all the information about the actual dynamical behavior Forinstance if there are two parts in state space which both have measure �� then thechances are even that trajectories are observed in one of those regions But it is notclear whether trajectories are moving quickly back and forth between these sets orstay in each of them for quite a long time before moving to the other one In thelatter case the components should be viewed as being almost invariant� let � � Mbe a probability measure We say that the set B is almost invariant with respectto � if

��f���B� � B�

��B�� �

Thus can be viewed as the ��probability that points in B are mapped into Bunder f In particular if B is an invariant set that is f���B� � B then � Thebottom line of this section is to relate the size of to the size of certain eigenvaluesof the Frobenius�Perron operator P

Suppose that � �� is a real eigenvalue of P with corresponding real valued

eigenmeasure � � MC that isP� � ���

where � is scaled so that j�j � M �We denote by MC the set of bounded complexvalued measures �

Remark ��� Obviously � itself cannot be a probability measure � in fact it isnot di"cult to see that ��R�� � � However if the eigenvalue � is close to one itis reasonable to assume that the probability measure j�j is close to the invariantmeasure of the system To see this consider the �unperturbed case where � � is a double eigenvalue Then there are two disjoint invariant sets A� A� and onecan choose independent eigenmeasures and � such that j�j � In particular forthis choice one has ��B�� � �B�� and ��B�� � ��B�� for all Bj � Aj

Recall that Chua�s circuit is ��symmetric where � � �I� It is easy to see thatthe Frobenius�Perron operator P commutes with � where the action of � on ameasure � is given by

������D� � ���D� for all measurable D � R�

Since by assumption � is a simple eigenvalue it follows as in the �nite dimensionalcase that the eigenmeasure � is either

symmetric ���� � �� or anti�symmetric ���� � ���

In any case the probability measure j�j is symmetric From now on we assume that �is anti�symmetric since this is the situation we are concerned with in Chua�s circuit Then the related attracting invariant set A inside Q � ���� �� ������ ���� ����� ��� can be decomposed as

A � B � �B� where ��B� � �

�and ���B� � ��

�

To see this recall that ��A� � � �Remark � �� In the following proposition we relate the size of the eigenvalue � to the proba�

bility occurring in the de�nition of an almost invariant set A proof in the moregeneral context of small random perturbations can be found in �Dellnitz � Junge����

Proposition ��� The set �B is almost invariant with respect to j�j if and onlyif B is almost invariant� Moreover is given by

���

�� �� ��

Proof� Let B be �almost invariant with respect to j�j Then by de�nition

j�j�f���B� � B� � j�j�B� �

��

Using the invariance of A we obtain

�

�� ���B� � ��f���B� � �B� � ��f����B� � �B��

�

Since � � j�j on B it follows that

��f���B� � �B� � ���f����B� � �B��

�� ��f���B� � B��

�

�

�� � ��

Finally the fact that � is an eigenvalue with eigenmeasure � implies that

�

�� ���B� � ��f���B�� � ��f���B� � B� � ��f���B� � �B� �

��� � ��

Now �� �� immediately follows

Two Almost Invariant Sets in Chua�s Circuit

We now apply the results of the previous paragraph and compute eigenvectors corre�sponding to real eigenvalues of the discretized Frobenius�Perron operator Pk whichare close to one

Again we compute box�coverings Bk for k � �� �� �� � by the methods de�scribed in Section � In order to obtain reliable numerical estimates we use time�T �maps with T � �� ��� and ��� In the following table we show the approximations�� to the eigenvalue of the corresponding discretized Frobenius�Perron operatorPT�k as well as the second largest real eigenvalue �� Observe that we need to

consider �����T� for a comparison of the di�erent results

Table � Largest real eigenvalues of PT�k

k T � �� T � ��� T � ���

�� �� �� ��

�

� �� ��

�

�

� � �� � ��� � �� � ��� � ��� � ���� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ���� � ��� � ��� � ��� � ���

Under the hypothesis that �� is quite accurate if �� � the numbers in Table indicate that PT�k has a real eigenvalue � which is close to ���� for T � �� Asmentioned before the corresponding eigenmeasure � is antisymmetric and thereforean application of Proposition � � leads to � ����� for a set B with ��B� � �

�

Thus � ����� may be interpreted as the j�j�probability for the system to stay inB for �� seconds if the trajectory starts in B at a randomly chosen point

In Figure � we visualize approximations of the almost invariant sets B and �B More precisely we show in red resp green the positive resp negative componentsof eigenvectors vT�k corresponding to the eigenvalue �� of PT�k for T � ��� and

�

−10−5

05

10

−2−1

01

2

−20

−15

−10

−5

0

5

10

15

20

xy

z

a � subdivisions

−10−5

05

10

−2−1

01

2

−20

−15

−10

−5

0

5

10

15

20

xy

z

b � subdivisions

−50

510

−2

−1

0

1

2−20

−15

−10

−5

0

5

10

15

xy

z

c � subdivisions

Figure �� Box coverings of two almost invariant sets in Chua�s circuit

k � �� �� � Since we want to avoid inaccuracies induced by the numerical ap�proximation we show in each sub�gure only those boxes Bj in the covering withjvT�k�Bj�j � ���

Finally in Table � we compute approximations of the probability to stay withinthe almost invariant set B for a certain period of time by a computation of powersof

�

Table �� Approximations of the probability to stay inside B for T seconds

T � �� T � T � � T � � T � ��

� ����� �� � ����� �� � ����� ��� � ����� ��� � �����

Obviously the numbers in Table � are dubious for small T The reason is thattrajectories do not enter the set B at random locations and therefore the interpre�tation of fails in this situation On the other hand due to the chaotic behaviorof the system the numbers are reasonable for larger T In fact Table � indicatesthat the event for the system to stay in B for more than �� seconds is very rareand this can indeed be con�rmed by a direct simulation of Chua�s circuit �see Fig�ure ��� roughly the sign of x�t� indicates whether the system is inside B or �B andtherefore the system stays just once inside one of these components for more than�� seconds �namely for t around ����

600 620 640 660 680 700 720 740 760 780 800−10

0

10

200 220 240 260 280 300 320 340 360 380 400−10

0

100 20 40 60 80 100 120 140 160 180 200

−10

0

10

800 820 840 860 880 900 920 940 960 980 1000−10

0

10

t

400 420 440 460 480 500 520 540 560 580 600−10

0

10

x

Figure �� A simulation of Chua�s circuit� x versus time t in seconds

Acknowledgement� For Figures ��� we have used the visualization platformGRAPE which has been developed at the SFB ��� at the University of Bonn andat the Institute for Applied Mathematics at the University of Freiburg

�

References

Dellnitz M � Hohmann A ����� �The computation of unstable manifolds usingsubdivision and continuation in Nonlinear Dynamical Systems and Chaos� eds Broer H W van Gils S A Hoveijn I � Takens F �Birkh�auser PNLDE ��� pp �������

Dellnitz M � Hohmann A ����� �A subdivision algorithm for the computation ofunstable manifolds and global attractors Numerische Mathematik �� ������

Dellnitz M Hohmann A Junge O � Rumpf M ����� �Exploring invariant setsand invariant measures To appear in CHAOS

Dellnitz M � Junge O ����� �On the approximation of complicated dynamicalbehavior Submitted

Eidenschink M ����� Exploring Global Dynamics� A Numerical Algorithm Based onthe Conley Index Theory �PhD Thesis Georgia Institute of Technology�

Hsu H ����� �Global analysis by cell mapping Int� J� Bif� Chaos � ������

Huang A Pivka L Wu C W � Franz M ����� �Chua�s equation with cubicnonlinearity Int� J� Bif� Chaos �

Kifer Yu ����� �General random perturbations of hyperbolic and expanding trans�formations J� Analyse Math� � ���

Kifer Yu ����� �Computations in dynamical systems via random perturbations Preprint

Kreuzer E �����Numerische Untersuchung nichtlinearer dynamischer Systeme �Sprin�ger�

�