Volume 124, number 3 PHYSICS LETTERS A 21 September 1987

C H A O S IN T H E Q U A S I P E R I O D I C A L L Y F O R C E D D U F F I N G O S C I L L A T O R ~

Stephen W I G G I N S Applied Mathematics 217-50, California Institute of Technology, Pasadena, CA 91125, USA

Received 29 May 1987; accepted for publication 9 July 1987 Communicated by D.D. Holm

We study chaotic dynamics of the quasiperiodically forced Duffing oscillator. We find that the mechanism for chaos is trans- verse homoclinic tori. Utilizing a generalization of a global perturbation technique of Melnikov we are able to give a criterion for the existence of chaos and we demonstrate the effect of the number of forcing frequencies on the region of chaos in parameter space. Our results give insight into the recent experimental results on the quasiperiodically forced Duffing oscillator obtained by Moon and Holmes.

In recent years the possibi l i ty of chaot ic dynamics for per iodical ly forced single degree of f reedom non- l inear oscil lators has become a well-established fact. The analysis of such systems is faci l i ta ted by study- ing an associated two-dimensional Poincar6 map. The Poincar6 map reveals a rich underlying global struc- ture as well as fractal proper t ies which might go undetec ted by s tandard methods of analysis for such systems. In many systems (e.g. the periodically foced Duffing osci l lator [ 1 ] and the per iodical ly forced Josephson osci l lator [ 2,3 ]) the underlying structure of the Poincar6 map which is responsible for the cha- otic dynamics is the t ransverse intersect ion of the stable and unstable manifolds of a hyperbol ic peri- odic orbit. Such intersect ions yield t ransverse homo- clinic orbi ts resulting in chaot ic dynamics of the Smale horseshoe type [4] . For a large class of per iodical ly forced single degree of f reedom systems Melnikov [5] has developed a computable mea- surement of the dis tance between the stable and unstable manifolds of hyperbol ic per iodic orbi ts which can be used to establish the presence of trans- verse homoclinic orbits in specific systems and hence, can also provide a cr i ter ion for the existence of cha- otic dynamics in specific systems.

Although there has been substant ial progress made

Work partially supported by the US Office of Naval Research, contract N00014-85-K-0205.

in the study of the nature of chaotic dynamics in per iodical ly forced single degree of f reedom nonlin- ear systems, little effort has been put into the study of quas iper iodical ly force systems although recently the existence of chaotic dynamics for systems undergoing quasiper iodic exci tat ion has been estab- lished through exper iment [6] and numerica l sim- ulat ion [7,8]. This lack of progress can probably be t raced to the fact that there is no natural reduct ion to a two-dimensional Poincar6 map for quasiper iod- ically forced systems. In this let ter we wish to report new results ~ which describe a mechanism for cha- otic dynamics in quasiper iodical ly forced systems; in par t icular we will consider the quasiper iodical ly forced Duffing oscillator, describing the geometrical structure in the phase space responsible for chaotic dynamics and giving the region in pa ramete r space where chaotic dynamics may occur. Our results will explain the recent exper imenta l observat ions of Moon and Holmes [ 6]. Our character izat ion of the chaot ic dynamics arises from a general izat ion of the

,t A generalization of the Smale-Birkhoffhomoclinic to the case of orbits homoclinic to normally hyperbolic invariant tori as well as a generalization of Melnikov's method to such systems was given by Wiggins [9,10]. Simultaneously and indepen- dently Meyer and Sell [ 11 ] obtained similar results for almost periodically forced nonlinear oscillators. Additionally, Scheurle [ 12 ] utilized a Melnikov type technique to obtain random-like solutions to an almost periodically forced oscillator.

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Volume 124, number 3 PHYSICS LETTERS A 21 September 1987

Smale-Birkhoff homoclinic theorem [ 4 ] to the case of orbits homoclinic to normally hyperbolic invar- iant tori. We have also generalized a global pertur- bation technique originally due to Melnikov that allows us to predict the regions in parameter space where chaotic dynamics may occur.

Consider the following equation

3) = ½ x( 1 - x 2) + e (fcos 01 + fcos 02 - 7Y),

02 =co2 , (1)

where e is presumed small, f a n d y are positive, and 0)1 and co2 are positive real numbers. Eq. (1) is the quasiperiodically forced Duffing oscillator having the four-dimensional phase space given by R 2 × S ~ × S 1. We can reduce the study of (1) to the study of an associated three-dimensional Poincar6 map obtained by defining a three-dimensional cross-section to the four-dimensional phase space by fixing the phase of one of the angular variables and allowing the remaining three variables that start on the cross-sec- tion to evolve in time under the action of the flow generated by (1) until they return to the cross-sec- tion. The return occurs after one period of the angu- lar variable whose phase was fixed in order to define the cross-section. To be more precise, the cross-sec- tion, E, is given by

Z ={(X, y, 0l, 0 2 ) e R 2 ×S 1 ×S 1102 =020} ,

where, for definiteness, we fix the phase of 02 and the Poincar6 map P~: Z _,5: is defined as

(x(0) , y(0) , 01(0) ~010)

For e = 0 (1) is a completely integrable hamilton- ian system having a two-dimenional normally hyper- bolic invariant torus given by

{(x, y, 01,02)sR 2 x S 1 ×S J I x - - y = 0 ,

01, 0 2 e [ 0 , 2n]} ,

Y ~ 0 1 x

Fig. 1. HomoclJnic geometry of the phase space of Po, cut away half view.

with trajectories on the torus given by

(x( t), y( t), 01(0, 02(0)

= (0, 0, o)1 t+01o, 0 9 2 / ' + ' 0 2 o ) •

The torus has a symmetric pair of coincident stable and unstable manifolds with trajectories in the respective branches given by

(x( t), y( t), 01(0, 02(0)

= ( + x / 2 sech ~ 2 2, -T sech ~ 2 tanh ~ 2 2 ,

a)~ t +010, 002t +020 ) .

Utilizing this information we can obtain a com- plete picture of the global integrable dynamics of the perturbed Poincar6 map, Po. In particular, Po has a one-dimensional normally hyperbolic invariant toms, To, that has a symmetric pair of two-dimensional stable and unstable manifolds, WS(To) and WU(To), that are coincident, see fig. 1. For E =0 there is no chaotic dynamics since Po is integrable.

For ~ ~ 0 and small the perturbed Poincar6 map, Po, still possesses an invariant one-dimensional nor- mally hyperbolic invariant torus [ 13,14 ], T~, having two-dimensional stable and unstable manifolds, WS(T~) and W"(T~), which may now intersect trans- versely yielding transverse homoclinic orbits to T~. The behavior of W~(T~) and WU(T~) can be deter- mined by computing the Melnikov function for quasiperiodically forced systems. The Melnikov function is the O(e) term in a power series expan-

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Transverse Homoclinic Tori

T,

21 September 1987

f Double Poincar6 Section

PHYSICS LETTERS A

x

Fig. 2. Transverse homoclinic torus for P,.

sion m ~ for the distance between WS(T,) and WU(T,). Thus if the Melnikov function has simple zeros then WS(T,) and WU(T,)intersect transversely. In this case the Melnikov function for each respec- tive branch of the stable and unstable manifolds is given by

M+-(Olo, 020)

= -- -~ X/2y + 2~f~, sech(~(n,/x/2) sin 0 ~o

± 2rcfo)2 sech(rco92/xf2 ) sin 020. (2)

Upon examination of the Melnikov function we see that for

f > ~/27{3n [o9, sech(~a) , /x~)

" ~ 0 ) 2 sech(Ko92/,J2)]} l , (3)

WS(T,) and WU(T,) intersect transversely in a set of points that is topologically a one-dimensional torus which we call a transverse homoclinic torus, see fig. 2. Arguing similarly to the case of transverse homo- clinic points to periodic orbits we see that there must be a countable infinity of such transverse homoclinic tori yielding a geometric structure for P, as shown in fig. 3. Now the Smale-Birkhoff homoclinic theorem does not apply in this situation: however we have proven a generalization of this theorem that tells us that in a neighbourhood of a transverse homoclinic torus there exists an invariant Cantor set of tori. Denoting this Cantor set of tori by A, we show that P, restricted to A is topologically conjugate to a full shift on N symbols. Thus the dynamics normal to the

x

Fig. 3. Homoclinic torus tangle for P~ and the double Poincar6 section.

tori in A is "Smale-horseshoe-like" and chaotic. This structure forms the backbone of the strange attractor experimentally observed by Moon and Holmes for this system. They studied the structure of the strange attractor by utilizing a technique due to Lorenz [ 15 ] that involves constructing a double Poincar6 section or Lorenz cross-section by fixing the phase of one of the angular variables and a small window about a fixed phase of the remaining angular variable. The map of this "section of a section" into itself revealed a fractal nature of the strange attractor similar to that found in the usual Duffing-Holmes strange attractor [ 16] which was not apparent in the three-dimen- sional Poincar6 map. Our results give much insight into the nature of this phenomenon. In fig. 3 it is clear that the intersection of WS(T~) and WU(T~) with the double Poincar6 sections yields a geometric struc- ture quite similar to the homoclinic tangle that occurs in the periodically forced Duffing equation that is responsible for the fractal structure of the Duff- ing-Holmes strange attractor.

Next we want to consider the effect on the region where transverse homoclinic tori exist caused by adding additional forcing fucntions to (1), i.e. we consider

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Volume 124, number 3 PHYSICS LETTERS A 21 September 1987

af f=mlY f = m 2 7 "

f= tony

~y

Fig. 4. Regions of chaos in f-7 space as a function of the number of forcing frequencies.

3)= ½x(1 - x 2) + e0Ccos 01 + f c o s 02

+... + f c o s 0n - ? y ) ,

01=091, ..., 0 . -----O.).. (2)

We reduce the study of (4) to the study of an asso- ciated ( n - 1 )-dimensional Poincar6 map having a ( n - 2)-dimensional normally hyperbolic invariant manifold with (n -1 ) -d imens iona l stable and unstable manifolds. Intersection of the stable and unstable manifolds is determined by calculating the Melnikov function. In fig. 4 the lines f = m ly, f = may, and f = m.y represent lines above which transverse homoclinic tori occur for the Duffing oscillator forced at 1, 2, and n frequencies, respectively; ml, m2, and m. are obtained from the Melnikov function and are given by

mn =xf2{3n [091 sech(n09,/x/2)

+o92 sech(n092/x/-2) +...

+to,, sech(n09,,/x/~)] } -1

ml =x/-2 [3n091 sech(nO) l/x//2)] - ' ,

m2 =x~{3n[09 , sech0t 09 , /x~)

+092 sech(n092/xf2)]} - j ,

Thus we see that the effect of increasing the number of forcing frequencies is to increase the area in parameter space where chaotic behavior can occur

and hence to increase the likelihood of chaotic dynamics.

Finally we want to comment on the generality and possible implications of our results. In this letter we have described a structure that is common to many quasiperiodically forced oscillators which is respon- sible for chaotic dynamics, namely transverse homo- clinic tori. The dynamics in the angular direction is trivial, it is just the flow given by 01 (t) = co i t + 01 o, 02(t) =092t+020, ..., On(t)=09,t+0no. However, the theorem which we have proven is much more gen- eral in that the result applies to multi-degree of free- dom systems with coupling amongst the frequencies. In particular, we state the result below.

Theorem [9]. Let fi R ~ X R m x T I ~ R ~ X R m X T I be a C r diffeomorphism having an /-dimensional normally hyperbolic invariant torus, V, possessing an ( n + 1 )-dimensional stable manifold, WS(V), and an (m +l)-dimensional "transverse homoclinic torus". Thus f contains an invariant Cantor set of tori, A. Moreover, flA is topologically conjugate to the shift map acting on a space of bi-infinite sequences of N symbols. Additionally, we have a generalization of Melnikov's global perturbation technique to multi-degree of freedom systems [ 1 0 ] that can be used to detect such dynamics. This brings to light the possibility of even more new dynamical phenomena occurring in such systems. In particular, it is well known that the flow on invariant tori may exhibit entrainment and detrainment as parameters are varied. It appears reasonable that the chaotic Cantor set of tori near the transverse homoclinic tori described in our theorem should also exhibit some type of entrainment or detrainment as parameters are varied if there is coupling amongst the frequen- cies. Hence the possible strange attractor associated with such homoclinic behavior may also exhibit some form of entrainment or detrainment leading to an "Arnold tongue" type of phenomenon for strange attractors in multi-frequency systems. We remark that in order for this form of chaotic dynamics to occur in a differential equation there must be at least two angular variables in order to get a nontrivial invar- iant torus, plus at least two more dimensions in order for the torus to have stable and unstable manifolds which interact. Thus quasiperiodically forced single degree of freedom systems or two degree of freedom

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Volume 124, number 3 PHYSICS LETTERS A 21 September 1987

d i s s i p a t i v e s y s t e m s are t he " s i m p l e s t " t ype o f sys-

t e m s w h i c h m i g h t e x h i b i t t h i s b e h a v i o r .

References

[ 1 ] P.J. Holmes, Philos. Trans. R. Soc. A 292 (1979) 419. [2] Z.D. Genehev, Z.G. Ivanov and B.N. Todorov, IEEE Trans.

Circ. Sys. CAS-30 (1983) 633. [ 3 ] F.M.A. Salam and S.S. Sastry, 1EEE Trans. Circ. Syst. CAS-

32 (1985) 784. [4] J. Guckenheimer and P.J. Holmes, Nonlinear oscillations,

dynamical systems, and bifurcations of vector fields (Springer, Berlin, 1983).

[5] V.K. Melnikov, Trans. Moscow Math. Soc. 12 (1963) 1. [6] F.C. Moon and W.T. Holmes, Phys. Lett. A 111 (1985) 157. [ 7 ] A. Bondeson, E. Ott and T.M. Antonsen Jr., Phys. Rev. Lett.

55 (1985) 2103. [ 8 ] Y. Pomeau, B. Dorizzi and B. Grammaticos, Phys. Rev. Lett.

56 (186) 681. [ 9 ] S.Wiggins, Caltech preprint (1986).

[ 10] S.Wiggins, Caltech preprint (1986). [ 11 ] K. Meyer and G. Sell, Univ. of Cincinnati preprint ( 1986 ). [ 12] J. Scheurle, Z. Angew. Math. Phys. 37 (1986) 12. [ 13 ] M.W. Hirsch, C.C. Pugh and M. Shub, Invariant manifolds

(Springer Berlin, 1977). [ 14] N. Fenichel, Indiana Univ. Math. J. 21 (1971) 193. [ 15] E.N. Lorenz, Physica D 13 (1984) 90. [16] F.C. Moon, Phys. Rev. Lett. 53 (1983) 962.

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