Chaos\ Solitons + Fractals Vol[ 8\ No[ 7\ pp[ 0168Ð0176\ 0887Þ 0887 Elsevier Science Ltd[ All rights reserved\ Pergamon Printed in Great Britain

9859Ð9668:87 ,08[99¦9[99

PII] S9859!9668"87#99951!8

Boundary of Two Frequency Behaviour in a System

of Three Weakly Coupled Electronic Oscillators

PETER ASHWIN$

Department of Mathematics and Statistics\ University of Surrey\ Guildford\ GU1 4XH U[K[

"Accepted 13 February 0887#

Abstract*We review and report observations of complex dynamics associated with loss of stability of twofrequency quasiperiodicity at a saddle!node bifurcation of invariant two!tori[ In particular\ for a systemof three weakly coupled electronic oscillators we _nd {bubbles| in parameter space interrupting a curve onwhich there is a saddle!node bifurcation of two!tori[ These are comparable to Chenciner|s bubbles foundnear a codimension two bifurcation of a planar map[ Þ 0887 Elsevier Science Ltd[ All rights reserved[

0[ INTRODUCTION

In this paper we consider an electronic experiment with three coupled oscillators in which onecan observe two frequency quasiperiodicity\ saddle node bifurcations of this and bubble!likebehaviour[ This system provides an example of loss of stability of two frequency quasiperiodicitywhere the dynamics is easier to follow than that for the degenerate Hopf example of Chenciner[This is due to the topology of the torus^ there is reinjection of trajectories to the region of interestin contrast to the Chenciner|s example where much of the dynamics of interest is unstable[ Section1 sets a theoretical background in terms of torus maps\ while Section 2 details the experimentalresults[ The system itself is not described in detail because this has been done in another placeð4Ł[ Finally\ in Section 3 we conclude the paper with a discussion of the observations[

0[0[ Back`round

Although two!frequency behaviour is not persistant in a generic dissipative dynamical systemssense\ it has a very real physical relevance when one samples points in parameter space at random[Notably\ KAM type results such as ð10\ 01Ł indicate that there can be two!frequency behaviouron positive measure Cantor sets in parameter space[ This means that given a system with stabletwo!frequency behaviour\ there can be an open region in parameter space that does not showthis behaviour^ however there may be a large measure set that does[

If one tries to describe the breakup of quasiperiodic behaviour via bifurcation theory\ thepresence of Cantor sets in parameter space causes extra problems that\ to the authors knowledge\have not been fully resolved[ See for example the idea of quasi!genericity of ð06Ł for localbifurcation to tori[ Typically progress can be made only on restricting to the subset of parameterspace that display a certain winding!number[ For systems with extra structure\ for example

$Author for correspondence[ E!mail] p[ashwinÝsurrey[ac[uk

0168

0179 P[ ASHWIN

quasiperiodically forced systems\ there can be connected paths of quasiperiodic attractors\ as inthe work on strange non!chaotic attractors\ e[g[ ð03Ł[

The questions of stability and bifurcation of quasiperiodic behaviour have been consideredfrom a mathematical viewpoint by Broer et al[ ð01Ł[ They put forward de_nitions for bothconcepts in terms of viewing the system as being close to a normal form[ Using a KAM result ofPo�schel ð10Ł they show persistence of hyperbolic quasiperiodicity on a positive measure Cantorset in parameter space\ and bifurcation happening on a boundary of such a set[ Their approachgeneralises theory of Chenciner ð02Ł and Iooss and Los ð06\ 08Ł who consider local bifurcationscreating quasiperiodic behaviour[ In summary\ they show that quasiperiodicity with any numberof frequencies is a persistant phenomenon in that it occurs typically for a positive measure set ofpoints in parameter space[

These studies show that bifurcations of quasiperiodic behaviour can typically occur on setsthat are graphs over a positive measure Cantor set on a submanifold[ By de_nition a Cantor sethas an in_nite set of {holes|\ however\ and the existence of these holes can give rise to verycomplicated behaviour involving resonance\ periodic and homoclinic orbits and chaos[ Chencinerhas named these holes {bubbles| "bulles#\ and has done much to understand what can genericallyhappen within them "for a review of this\ see Arrowsmith and Place ð3Ł or Arnold ð1Ł#[

Closely related to the question of bifurcations of quasiperiodic orbits is the so!called break!upof tori[ Even in smooth dynamical systems\ one cannot assume that invariant tori on which thereis quaisperiodic ~ow are smoothly embedded manifolds[ Typically in fact they only have _nitesmoothness and varying a parameter may bring this smoothness right down to a point where itdisappears[ A thorough study of this e}ect was made by Aronson et al[ ð2Ł for a particularmapping of the plane[

Systems that have n frequencies in some limit have been observed by many authors to providerich _elds for investigating quasiperiodicity with k frequencies for any k³n\ in particular\ weaklycoupled oscillator systems which consist of dynamical systems with a coupling parameter o andtuning parameters vi "i�0\[[[\n# such that at o�9 the system splits into product of n hyperboliclimit cycles each with frequency vi[ It is simple to see that for o�9\ there is a full measure set oftuning parameters at which the system displays n!quasiperiodicity[

On increasing the coupling parameter o from zero\ the measure of v such that the systemdisplays n!periodicity gradually decreases while positive measure regions in parameter space candevelop where the system displays k!quasiperiodicity for any k³n\ periodicity or chaos[ Fig[ 0"cf[ Battelino\ ð09Ł# shows schematically how these measures can vary on varying o[ Baesens et al[ð09Ł investigate a particular mapping of the two!torus "which can be thought of as a Poincaresection for a system of three weakly coupled oscillators# and _nd this sort of behaviour common[The question of the scaling behaviour of the measure of k!quasiperiodicity with o has been tackledby Galkin ð04Ł who has shown that if one restricts to the so!called Mathieu families "where thenonlinearity is a polynomial in trigonometric functions# the measure can scale as O"ok#\ k×0\ butfor generic nonlinearities it will scale O"o#[

Although a numerical experiment by Battelino et al[ ð00Ł suggested that large measure regionsof chaos are usually associated with the breakup of the invariant torus rather than being chaoticmotion on the invariant torus\ there is evidence that there can be chaotic behaviour that remainson the invariant torus^ for example Ashwin and Swift ð6Ł have found examples for electronicexperiments on coupled oscillator systems as have ð8Ł in torus mappings[

1[ THREE OSCILLATORS

Consider a system of three oscillators governed by an ODE of the form

0170Boundary of two frequency behaviour in a system

Fig[ 0[ Schematic diagram showing the measure of three!\ two! and one!frequency quasiperiodicity and chaos as couplingstrength between oscillators is varied[

u¾ i � Vi¦ofi"u#[ "0#

for i�0\1\2\ where Vi and o are taken to be "real# parameters\ and ui is taken modulo 0[We examine the dynamics near a rational line in the V!parameters[ By rede_ning u2 if necessary\

taking an appropriate combination we can assume that V0 and V1 are order one and V2 � ov[Thus we can write the original system in the form

u¾0 � V0¦of0"u#

u¾1 � V1¦of1"u#

u¾2 � o"v¦f2"u##[ "1#

If V0 and V1 are rationally independent then we can apply the method of averaging "Lochak andMeunier ð19Ł# to gain the approximate equation

u¾0 � V0

u¾1 � V1

u¾2 � o"v¦F"u2##[ "2#

which is O"o# close to the evolution of "1# on a timescale of order O"0:o# and the average of f2 is

F"u2#� gu0\u1$T1

f2"u# du0 du1[

Thus we can approximate _nite time dynamics by a product between a linear ~ow on a two!torusand a ~ow in the u2!direction[ For example\ suppose that F"u2#� sin"u2#^ then the averagedsystem will display saddle!nodes of invariant tori at v� 20[

0171 P[ ASHWIN

1[0[ Persistence of QP1

Applying results of Broer et al[ ð01Ł means that for a given small enough o\ the quasiperiodicbehaviour of the averaged system is persistent in the following sense] there exists a Cantor setPo$R2 with non!zero measure\ such that if "V0\V1\v#$Po then there exists an invariant torus withtwo!frequency behaviour at this parameter value[

One can characterise the two!frequency quasiperiodic orbits that persist in this system byorbits such that u2 remains bounded and the rotation number de_ned by

"r0\r1#� limt:�

0t"u0"t#\u1"t##[

Note that for o small\ "r0\r1# is close to "V0\V1#[ The Cantor set on which the two frequencyquasiperiodicity is persistant is characterised by rotation vectors that satisfy a Diophantinecondition\ i[e[ ri such that there exist C×9 and q×9 with

=k0r0−k1r1=×C=k=−0−q

for all "k0\k1#$Z1−"9\9#[Note that the set of parameters giving a particular rotation number will be a subset with

codimension one in the parameter space[ This\ on varying two parameters\ a particular QP1 willpersist on a line in parameter space[

1[1[ Saddle!node of QP1

At the ends of such a line of existence of QP1 with a given Diophantine rotation number oneof a number of bifurcations can happen[ We concentrate on the case of a saddle!node of invariantcircles\ as studied in Chenciner|s example[ It can be shown that due to the Diophantine condition\such a saddle!node is essentially a well!de_ned object[ However\ this only happens on a Cantorset in phase space[

1[2[ Bubbles

There are an in_nite number of {holes| contained within the Cantor set where there is noinvariant two!torus^ these correspond to the existence of resonances in the averaged equations[As discussed by Chenciner ð02Ł and at length by Baesens et al[ ð8Ł for the case of three oscillators\one can approach the generic dynamics by performing averaging over one angle variable "alter!natively\ by looking at a suitable iterate of a Poincare map and noting that this is near identity#[Having gained a ~ow approximation on a two!torus\ one can study bifurcation behaviour in thissetting before adding a small generic perturbation breaking the averaging approximation[

In this respect\ the statement in Baesens et al[ ð8Ł that they did not found evidence of Chencinerbubble e}ects is misleading^ in fact the resonance regions they discuss correspond to them forhomotopically nontrivial invariant circles[ They did not see any bubble e}ects for the homo!topically trivial invariant curve saddle!nodes\ however[

2[ ELECTRONIC EXPERIMENTS

Electronic oscillator systems are amenable systems for experiments on coupled oscillatorsowing to their easily controllable parameters and rapid dynamics^ see for example the work of

0172Boundary of two frequency behaviour in a system

Linsay and Cumming ð07Ł on mode locking in a three oscillator system[ In this section we reportobservation from such a system of three electronic oscillators[ The circuit for each oscillator isidentical save a resistance Ri "i�0[\[[\ 2# which sets the frequency of the uncoupled oscillator[We _xed the value of R0 and varied R1 and R2 via non!dimensional parameters r1 and r2 "arbitraryunits#[ The oscillators used are of van der Pol type with two degrees of freedom and are detailedin ð5\ 6Ł[ The coupling strength is chosen to be weak but non!zero and is _xed[

To visualize the behavior of the system\ the four voltage outputs "y0\ y1\ y2# are fed throughthe {torus unfolder| described in ð5Ł[ This gives a voltage proportional to the phase of the ithoscillator when the 2rd oscillator has zero phase to give a Poincare� section of the dynamics[ Thesamples taken were between 4999 and 019999 points[

The Fig[ 1 shows a scan through parameter space with the regions of existence and non!existence of a particular winding type of invariant circle marked[ The boundary between the twocorresponds to a saddle!code bifurcation of invariant tori\ and one can observe the presence of{bubbles| in this curve[ These are resonances of the invariant torus^ the period in the Poincare�section is marked for each resonance[ Observe that there is a Farey sequency of periods\ i[e[between a resonance of order p and one of order q there is one of order p¦q[ Observe also thatthe high order resonances exist for very small regions in parameter space^ Fig[ 2 shows how the

Fig[ 1[ The location of the stable invariant curve in the parameter plane "r1\r2#[ Observe the interruption of the curve bya number of resonance regions[ These resonances are labelled by the period of periodic orbit in the Poincare� section[

0173 P[ ASHWIN

Fig[ 2[ Upper and lower estimates of the width of the resonance region as a function of period of resonance\ measuredfrom Figure 1[ Observe the fast convergence to zero consistent with the observation that the measure of the resonances

is less than full[

width of the strips vary with order^ this is consistent with there remaining a positive measurewidth of strip for which there are no resonances[

Fig[ 3 show some typical examples of Poincare� sections of stable dynamics "after transientshave been allowed to die away# from this system[ Fig[ 3"a# shows the stable invariant curve nearloss of stability\ close to the order 5 resonance[ Fig[ 3"b# is nearby in parameter space and displaysa passage through the {ghost| of the previously stable invariant curve[ This is a candidate fortoroidal chaotic behaviour[ "c# shows mode!locking to a periodic orbit with period 00 sitting onthe invariant curve[ "d# shows breakup of the invariant curve and loss of stability of the period00 orbit[ The folding evident in the trajectory indicates that it is not an invariant curve but anexample of annular chaos[ "e# shows a period 10 resonance while _nally "f# shows a three torusthat has been created by a saddle!node bifurcation of invariant two!tori[ This is typical awayfrom resonances[ Note the in Fig[ 3 "c# and "e# circles have been added to emphasise the locationof the periodic points[ There are a variety of generic bifurcations that occur on the boundariesof such behaviour^ many of these are studied in detail in ð8Ł[

3[ CONCLUSIONS

We have shown that a number of theoretical predictions^ the persistence of QP1 and theinterruption of a curve of saddle!nodes of invariant circles by {bubbles| of complex behaviourcan be observed in a physical system of coupled electronic oscillators[ It is comparatively easy toverify the existence of bubbles of resonant behaviour near saddle!node bifurcations of invarianttori that have non!trivial homotopy on the 2!torus because just after saddle node\ the globaldynamics on the torus can be such that trajectories are reinjected to a neighbourhood of thesaddle node[ Thus one can easily _nd chaotic behaviour of intermittent type[ Chenciner hasconjectured the existence of Birkho} attractors that remain in a neighbourhood of the saddle!node\ although these would only exist for very small regions in this system and we have not beenable to _nd them in practise[

0174Boundary of two frequency behaviour in a system

Fig[ 3[ Samples from the electronic circuit projected onto the unfolded two!torus[ These are Poincare� sections taken atthe zero!crossings of the third oscillator[ "a# shows the invariant torus "an invariant curve in the section# of type "0\−0#at "r1\r2# � "480\624#[ "b# shows toroidal chaos at "r1\r2# � "480\625#[ "c# shows a period 00 orbit at "r1\r2# � "399\325#[ "d#shows annular chaos at "r1\r2# � "287\359#[ "e# shows a period 10 orbit at "r1\r2# � "201\220#[ "f# shows a three!torus at

"r1\r2# � "385[4\487#[ In "c# and "e# circles have been added to show the location of the periodic orbits[

0175 P[ ASHWIN

Fig[ 3*continued[

Acknowled`ement*The author acknowledges the support of the Nu.eld foundation through a {Newly appointed sciencelecturer| award[

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0176Boundary of two frequency behaviour in a system

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