Repeated resonances to chaos in forced oscillators

Pergamon

Medmnics Research Communications, Vol. 21, No. 5, pp. 415-422, 1994 Copyright © 1994 Elsevier Science Lid Printed in the USA. All right~ reserved 0093-6413/94 $6.00 + .00

0093-6413(94)00002-6

R E P E A T E D R E S O N A N C E S T O C H A O S IN F O R C E D O S C I L L A T O R S

M. B e l h a q a n d A. F a h s i

Laboratory of Mechanic, Faculty of Sciences Ain Chock, BP 5366, Maiarif, Casablanca, Morocco.

(Received 6 January 1993; accepted for print 26 May 1994)

I n t r o d u c t i o n

It is important to predict the occurence of homoclinic tangencies in forced dissipative nonlinear oscillators. It can be shown for these systems that, as a parameter is varied ( i.e. the amplitude of external forcing), repeated resonances of successively higher periods occur. It is known that the homoclinic bifurcation is the limit of these subharmonic bifurcation sequences.

There exists an analytical tool to detect the existence of transverse homoclinic points using Melnikov's method [1]. This method consists in evaluating the distance, measured along the homoclinic loop, between the stable and unstable manifolds coming from the hyperbolic fixed point of the associated Poincar6 map. For a detailed description of this method and a large bibliography see [2].

In this brief paper we suggest an analytical approach which allows one to approximate simultaneously, the subharmonic solutions of order q, their bifurcation curves as well as the homoclinic bifurcation curves. Indeed, the main object here is to show that from approximations of q-subharmonic solutions, the analytical conditions for the successive subharmonic bifurcations lead to the condition for homoclinic bifurcation as q --~ oo. We use here the Smale-Birkhoffs result which stipulates that every transverse homoclinic point is the limit of periodic orbits. For more details see [3], p. 276-289.

We illustrate the above idea by the following example concerning a class of quadratic nonlinear single degree of freedom systems submitted to parametric excitation

~-Ot~+O~2o(1 + h c o s o ~ t ) x = CX2-~X~+'[~ 2, ₍₁₎

415

where ~, 13, coo, h, y and c are real constants, co a real positive. We define the parameter space as ~t = (o~, coO)"

A s y m p t o t i c f o r m u l a t i o n

Let us consider a nonlinear single degree of freedom system submitted to periodic forcing

+ co2 x = E fl (x, x, It, cot) + e2f2 (x, ~, It, cot) + E 3 f3 (x, ~, It, cot), (2)

where COO is the natural frequency and co is the frequency of the excitation. Each function fi is sufficiently smooth in its arguments and in addition 2x-periodic in cot; the parameter e is assumed to be small and It = (It1, It2) ~ IR2 • We seek the approximation up to second order of q-subharmonic solutions of (2) in the form of power series in ~ as

x = a cos ~ + EU1(a, ~/, cot) + e2U2(a, ~g, cot); ~ = pcot/q + 0, (3)

d_&a = e Al(a, 0, g, q)+ E 2 A2(a, 0, It, q), ~0t0 = E Bl(a, 0, It, q)+ e 2 B2(a, 0, It, q), (4) dt

where the unknown coefficients A i, B i (2x-periodic in 0) are determineted by the condition of vanishing of secular terms in the correction functions UI,U 2 (2x-periodic in both 0 and cot); p, q are relatively prime. The amplitude a and the phase 0 are assumed to vary with time according to (4).

For c = 0, (2) reduces to a harmonic oscillator and therefore the unperturbed phase plane is filled with periodic orbits of period T = 2x/co0.

We assume that for E > 0 and sufficiently small and for some conditions on the parameters, the associated Poincar6 map of (2) possesses perturbed orbits and transverse homoclinic orbits;

i.e. the stable and unstable manifolds o f the hyperbolic fixed point of the associated Poincar6 map intersect transversally.

As in ref. [4], the subharmonic solutions of order q and their bifurcation curves N~, N~ can be

obtained from (3) and (4). In fact, the steady-state periodic solutions o f (4) are given by two

CHAOS IN FORCED OSCILLATORS 417

algebraic equations for the unknown a and 0: da/dt = 0, d0/dt = 0. Eliminating 0 in this system yields an equation of the form G(a, ~t, q) = 0 which can have real solutions if the conditions

gl(I.tl, q) - ~t2 -< g2(l.tl, q) (5)

are satisfied. Thus the subharmonic bifurcation curves N 1, Nq 2 are d e f i n e d by

~t2 = gl(~tl, q) and 1~2 = g20tt, q) and the homoclinic bifurcation curves H l, H 2 are obtained by considering the limit of gl(~q, q) and g2(~tl, q) as q ~ **. We therefore obtain

lira NJq = HJ ; j = 1,2.

q ~ o o (6)

S u b h a r m o n i c a n d h o m o c l i n i c b i f u r c a t i o n s

We now investigate the oscillator (1). Let us consider that the parameter e is such that 0 < e << 1. In order to construct the q- subharmonic solutions of (1) close to resonance points of order q we require the condition (CN): t ~ = (pto/q)2 + 5, where 6 is a small real p a r a m e t e r . W e i n t r o d u c e a n e w set o f p a r a m e t e r s as f o l l o w s :

o~ = e2a, ~ = e26_ h = eh, c =e ~, 13 = e ~ , y = ey. Therefore, the system (1) becomes

pto 2 x xcos

~ + ( P ~ ) 2 x = ~ c ' x 2 - ~ x ~ + ~ 2 - h ( r ~ - ) x c o s t o t ] + e ~ - " ~ + ~ ] + e ~ - h " ~ tot]. (7)

In previous works (ref. [5, 6]) we considered the cases of resonance p = 1; q = 1, 2, 3 and 4 and predicted the regions in the parameter space where these subharmonics exist. These theoretical results were tested by numerical integrations and very good agreement was found. In the present paper we will consider the resonances of order q. To simplify the calculation we choose q # 2p ( this leads to A 1 =0, B 1 = 0).

Then, the approximation up to second order of the subharmonic solutions of order q is given by

x(t) ~/+ e[((q) 2 2__ + ~)a 2 ((q)2 ._~ - 2 q ~a 2

= a cos m 2 2 - m 2 - 7) 6a~cos 2~ - ~ - ~ sin 2~

(8)

~ q ~'a ((q_p,~. qa)];

p2 h'a cos ((p+q)~-- ~ 0 ) - p2 ~ = qP-~t + 0,

+ q 2(2p+q) q 2(~-p-q)cos ,-p- ~, j

dt 2--~ ( q 4rap dO _ e 2 [ q ( ~ a - h2m2p4

d-T - L ~ 2q2(4p2. q2)

(9)

6°)2I) 2 ~ ] 1

1~_~ 5c2q2 ~ 23_~P_ ~

a - ( + + + ) 2 a3 )

o

According to the condition (CN) the system (9) can be written as follows

da = E2 _ _ _ ~c

d t 8c02 a3 + O(e3) '

dO __e2 8 q2 ~'2m~ _ ( + 5~2 + ~ a2 ) +O(e3)"

d-T 2 - ~ - 4p2_ q2 4m2 12m 3 ~ + )

(lO)

The stationary regimes of (10) corresponding to the subharmonic solutions of (7), are the roots of the algebraic system

_ _ _

O~a_ [3c a 3 = 0 '

2 8o~

q2 h2to~ _ ( ~ + 2 5c 2 "yc 2 ~ m 2 a2 4p2 _ q2 2~2 " 12 60)~ + ~- + ~ J = O.

(11)

In consequence, it follows that on one hand x = 0, y = 0, corresponding to the solution a = 0 which is stable for ~ < 0 and unstable for et > 0. On the line ot = 0 the Naimar-Sacker [8]

bifurcation from this stable fixed point (0, 0) occurs, yielding an unstable fixed point and a

stable closed orbit [5]. On the other hand one gets

CHAOS IN FORCED OSCILLATORS 419

a = ~c

(12a)

132 ~ 4°~t~2

o~ 0 = ( q~__~)2 + q2 h2°~ + 5~ 2 + ~ (12b)

4p2_ q---5 20~2 ( ~ + 60~ T + )

Hence, eqs (12a,b) approximate the amplitude of subharmonics and their bifurcation curves Nq (tangent to Nq 1, Nq 2 at resonance points Pp/q). These are illustrated in fig. 1 for p = 1 and some q values. The approximation of the homoclinic bifurcation curve H (tangent to H l, H 2) is obtained by considering the limit as q ~ ** in (12b) (fig. 1). The locus Nq intersects theNaimar-Sacker bifurcation line at resonance points Pp/q corresponding in the parameter space to

2(4p2 _ q2) . (13)

tO2 = (-~)2 2(4p2 _ q2) _ p2h2

Furthermore, the sequences of subharmonic bifurcations given by (12b) accumulate on homoclinic bifurcation curves as q ~ -0 ( p is small with respect to q ). As a result one obtains

ct = ( 1 + ). (14)

10c 2 + 1~24 + 6"yc~ + 8 ' ~ 4

By using the Melnikov technique [7], the condition for which the Melnikov function has a quadratic zero is

1 13t~ + 5 h x o 2 ( 0 2 - 1). (15)

Note that there is no "t in the expression obtained by the Melnikov technique. This above relation determines the homoclinic bifurcation curves H 1, H 2 for sufficiently small perturbation.

Figure 2 shows the comparison of these curves in Ix-space obtained by both methods for co =

2, h = .1, 13 = .2, "t = 0 and c = 1. The solid line represents the curve obtained by the

technique developed here (14) and the dotted one represents the curve given by the Melnikov

analysis (15).

co 0 / N4

Subharmonic ~ - N 5 Naimar- Sacker

0.60

PII4

P115 P116

- - ~bifurca~on c ~ N6

/ Homoclinic bifurcalion

CO 0

.35

-0.1

I

~ H

0 0;

Fig. 1 Fig. 2

Fig. 1. Subharmonic bifurcation curves Nq for some q and homoclinic bifurcation curve H as q ---> oo of the system (1). P1/q indicates the resonance point of order q (p=l).

Fig.2. Comparison of the theoretical method performed here (solid line) and Melnikov technique (dotted line) for homoclinic bifurcation curves for the system (1).

If we consider the special case h = 0 (i.e. the perturbation of the integrable Hamiltonian system:

= y, ~, = cx 2 - c0~x is a constant vector field), the Melnikov function is not time-dependent and therefore the global saddle-loop bifurcation occurs on the curve

13co

0t = - - (16)

7c

obtained from (15). Furthermore, the method developed in the present paper gives, from (14)

313°)20 (17)

O ~ - -

1 0 c '

where, for 13, 7 and co o small, the term 132o)20 + 6ycco~ + 8 ~ o ~ is neglected in (14). This is a

CHAOS IN FORCED OSCILLATORS 421

type of standard bifurcation where a periodic orbit, bifurcating from the stable fixed point as we cross the Hopf bifurcation, grows to touch a saddle point creating the homoclinic orbit through the saddle point. The amplitude of this periodic orbit is approximated by (12a) and its analytical expression is obtained from (8) as follows

x ( t ) = a c o s ~ g + e ( - ~ + y ) a 2 - o ~ 2 (--c---y)-~-cos2~g-CO2o a 2 sin2 ; V = o o t + e . (18)

C o n c l u s i o n

In this paper we have suggested a theoretical approach to investigate simultaneously the subharmonic solutions of order q, their bifurcation curves as well as their bifurcation sequences to chaos as q ~ 0o in the forced nonlinear oscillators.

In Melnikov's technique the unperturbed system considered is the one having saddle-loop whereas in our approach the unperturbed system is the harmonic oscillator. This choice results in computational simplicity and furthermore the generating functions are simple trigonometric functions instead of elliptic functions. Note that no restriction has been put on p; the special case q # 2p has only been assumed to facilitate the calculations. Note that the parameter y (of the term ~ 2 ) does not appear in the Melnikov formulation.

When q becomes large in the q-subharmonic solutions (8), we can state that the subharmonic orbits lead to an approximation of homoclinic orbits.

In the special case h = 0, the method performed here allowed us again to construct simultaneously, the amplitude of the bifurcating periodic orbit, its explicit expression as well as its saddle-loop bifurcation curve.

For small parameter values a good agreement is found between our theoretical prediction for homoclinic bifurcation and the standard Melnikov analysis.

A similar investigation of third-order q-subharmonic solutions should be useful to refine the

approximation of homoclinic bifurcation curves by using the limit as q --~ **.

R e f e r e n c e s

[ 1 ] V. K. MELNIKOV, On the stability of the center for time periodic perturbations, Trans.

Moscou Math. Soc. 12, 1963, p. 1-57.

[2] J. GUCKENHEIMER and P. J. HOLMES, Nonlinear oscillations, dynamical systems and bifurcation of vector fields, Appl. Math. Sciences 42, 1983.

[3] I. GUMOWSKI and C. MIRA, Dynamique chaotique, transition ordre-dEsordre, Eds.

Cepadues, 1980.

[4] M. BELHAQ, Etude asymptotique au voisinage d'une resonance 4 d'un oscillateur param~trique ~ non linEaritEs quadratiques, C. R. Acad. Sc. Paris, 314, SErie II, 1992, p.859-864.

[5] M. BELHAQ, Numerical study for parametric excitation of differential equation near a 4- resonance, Mech. Res. Comm. Vol. 17(4), 1990, p. 199-206.

[6] M.BELHAQ and A. FAHSI, Subharmonic vibrations close to degenerate PoincarE-Hopf bifurcations, Mech. Res. Comm. Vol. 20(4), 1993, p. 335-341.

[7] M. BELHAQ, R. L. CLERC and C. HARTMANN, Bifurcations homocliniques pour des Equations de LiEnard forcEes pEriodiquement, J. MEc. Th. Appl. Vol. 6(6), 1987, p. 865- 877.

[8] D.K. ARROWSMITH, C.M. PLACE, An introduction to dynamical systems, Cambridg

University press, Chap. 5, 1990.