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SUPPRESSION OF CHAOS IN A NONLINEAR OSCILLATOR WITH PARAMETRIC AND EXTERNAL EXCITATION
MOHAMED BELHAQ and MOHAMED HOUSSNI Laboratory of Mechanics, Faculty of Sciences mn Chock,
BP 5366, M&f, Casablanca, Morocco
Key words and phraws: Weakly nonlinear system, averaging method, bifurcation, quasi-periodic solution, Melnikov technique, transversal homoclinic orbit, chaotic dynamic, parametric resonance, suppression of chaos.
1. INTRODUCTION
This paper is concerned with the study of the dynamic response of one-degree-of-freedom system with quadratic and cubic nonlinearities subjected to combined parametric and external excitation, which can model the one mode vibration of a suspended elastic cable driven by a quasi-periodic forcing. The equation governing this system can be written in the form
i + ai + o,z(l+ hcos( vr>)x+/3x~ + 5.9 = ycos(wt), (1.1)
where damping a, nonlinearities fi and 5, parametric excitation amplitude h, frequency v and external excitation amplitude y are small. Assume that the frequency w is close to cc,,. An overdot denotes a differentiation with respect to time t. From physical point of view the quadratic term may be due to the curvature of the cable, whereas the cubic term may be due to the symmetric material nonlinearity. The parametric term may be due to a harmonic axial load.
An equation of type (1.1) including only a cubic nonlinearity (i.e., p = 0) were studied by Yagasalci et al. [l]. He used the averaging method via the Van der Pol transformation to approximate quasi- periodic orbits. Benedittini et al. [2] investigated a similar system in the special case h = 0, corresponding to a system forced by external excitation. They used the multiple scales method to study the weakly nonlinear dynamic of the system. They also developped numerical calculations to investigate chaotic motions.
The purpose of this paper is to study the quasi-periodic and chaotic motions of the general system (1.1) close to strong resonances. The main object here is to show how to bring the system from a chaotic regime to a regular one. In other words, the problem of suppression of chaos is analyzed by introducing a resonant parametric perturbation in the cubic nonlinearity of the system (1.1).
2. ASYMPTOTIC METHOD
In order to construct an approximation of the quasi-periodic solutions of (1.1) close to resonance
points of order q we impose: ID,’ = (pw/q>’ + 6, where 6 is a detening parameter. Let introduce two small parameters E and p, such that andlet a=@=pc’&, y=cy, v=&C, h=pi=pe’i, P=E~,
v = EC , { = ~‘4 and 6 = c26. Therefore, the system (1.1) becomes
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i+($o)2x=E[-gx’+jcos(wl)i+E’[ -6x -/L&i -&pm / q)2 cog vqx - <x3
1
+E4 -6h -I cos( i’z)x 1
This system (2.1) contains a “fast” time t and a “slow” time r = &t that are assumed to be independent.
Using the Bogolioubov-Mitropolsky technique [3], adapted to quasi-periodic oscillations problems, the third order approximation of quasi-periodic solutions of (2.1) is sought in the form
(2.2)
! ~=~~(u,e,~~)+~~A~(a,8,i’z)+~~A&,e,G~),
(2.3)
where the unknown coefficients 4, B, ( 27r-periodic in both 8 and vt) are determinated by the condition of vanishing of secular terms in the correction functions U,, U,( 27r-periodic in 0, wt and vt); p, q are relatively prime. The amplitude a and the phase 8 are assumed to vary with time according to (2.3). Using the same process than [4], we obtain the following approximate second order solution
da 4
-=--
dt 2PW I jSp,pFt_tll(u, Wsin(qO/p)]
+&*[sg(q- ~)G,_,,f~~,_,(u, Wsin(qO/p)+G~~(a, CT)] ’ (2.4)
de .+&~~W, %cos(qe/p)]
-=-_
dt +E2[sg(q - P)~,_,,,F~\,_, (u, ik)c0s(qe/f7) + FE;tu, CT)] ’
and F’2_’ _ (a CT) = PW - 1) q2P7
4P.1 ’
(I-(qlp)“)&F
The sg(.) is the sign function and ~5,,~ is the kronecker symbol. This expansion to second order allows one to study simultaneously the two resonances cases p = q = 1 and p = 1;q = 2 of the system (1.1).
Indeed, c?,,~ is different from zero for these values of p and q . Let us consider the fundamental resonance case: p = 1;q = 1. Hence, we obtain from(2.4) the following nonautonomous (amplitude- phase) system
~=y&7sin(@-p(&.J), dt 2w
%L+A,J- y
dt 2w 4w ‘zz7
(2.5)
lop2
where a2 =2J and A, =35-g. Note that the periodic solutions of (2.5) correspond to the quasi- periodic solutions of (1.1). In the case p = 0, one obtain the unperturbed system of (2.5) in the form
(2.6)
Q!J d6
The fixed points of the modulation equations (2.6), corresponding to - = - = 0, are the roots of the dt dt
cubic equation
AJ3+BJ2+CJ+D=0, (2.7)
with A=l, B=4$, C=(FJ and D=-2(:1. L&ng J=R- 5 B m equation (2.7), one obtains . R3+pR+q=0,
which gives real solutions when one of these following conditions are satisfied (see Fig. 1).
A<OorA=OorA>O, 6A,<O.
(2.8)
(2.9)
The quantity A=(f)‘+(t) is the discriminant of (2.8), p = : - -$ 2 B3 0
BC D and q=- - -2+A.
27 A 3A
In Fig. 1, we illustrate the bifurcation curves of quasi-periodic solutions of (1.1) for 4 = 2, /3 = 1, and w, = 1. Note that three regions corresponding to different frequency can also be distinguished. The Hamiltonian energy related to the system (2.6) is given by
(2.10)
We now describe this integrable structure, which will be used in the following sections. Here we consider the case 6A, < 0 and A < 0. Then there exists two centers (J, 0) = (j,, 0, ),(j,, 6,) and a hyperbolic saddle (jr, 0,) such that 0 < j, < j, < j,. The level set given by
H( J, 0) = H( j, , e2) = H,, (2.11)
is composed of two homoclinic orbits, r+, r_, and the fixed point (j,, 0,). The homoclimc orbits, (J&X %(t>), are given by
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(2.12)
where
I Tk 2c,c_
$0) =
(c, -c_)cosh(ct)+(c+ +c_) ifA, >O
+ 2c+c_
(c, -c_)cosh(ct)+(c+ +c_) if& <O’
c=@&z, c,=2(t+a)andk=-2$-j2.
3. CHAOTIC MOTIONS
There exists an analytical tool to detect the existence of transverse homoclinic points using Melnikov Method [5]. 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 Poincare map. For a detailed description of this method and a large bibliography see [6]. Applying the Melnikov method to the reduced system (2.9, in which u plays the role of the perturbation parameter, one obtains the following Melnikov functions for I-*
M,(?,) =t<
[-&dmsin(tI+(r))[ecos( v(fffg))) -_
\[
6 A,
- z+&*(t)- ycos(6*(r)) a--I 2w 2.!*(t) (-q(r))
dt. (3.1)
Substituting (2.12) into (3.1) and evaluating the integral by the method of residues, we have (in the case A, CO)
4ni;wv,sinh 4vo~lpm”)
M+(ro) = - i 1
/Ao/sinh(~)
sirr(vro)+i$(3&p-8S(n-),)).
M_(f,)=-
(3.2a)
(3.2b)
c++c_
where @o=arccos - , v,=2wvandp=~.Letusdefine [ c, -c_ 1
It follows from the Melnikov theory that if
~>Q(b,v,,S,y), or ~~R(&vo,6,y),
(3,3a)
(3.3b)
(3.4a,b) then the reduced system (2.5) has transverse homoclinic orbits resulting in chaotic dynamics. In Fig.2, we show the homoclinic bifurcation curves hco/a=&(A,,v,,&y) with 5=2, p=l, w=2, w, =l and y = 1.15 whereas in Fig.3, we illustrate the combined effect of the nonlinearities and the effect of the external excitation on the dimensions of the chaotic regions in the parameter space of the system as well.
4. SUPPRESSION OF CHAOS
In order to analyze the suppression of chaos in the original system (l.l), we introduce a resonant parametric perturbation. Assume that this perturbation is introduced in the cubic nonlinearity of the system as
~+~+W,~(~+~C~~(V~))~+~X~+~(~+~COS(~~)X~)=~COS(W~). (4.1)
Let 17 = prj and Q = &fi such that the parameters h and 517 are evaluating at the same order in both smallness parameter E and p. Using the same process as before, we obtain
I
$=$T7sin(@-~(&J), - A,~~~+--&J_----?1-
2w&7 cos(fq+p i-
3fi5
~cos(v~)+40.1cos(s&) . 1
(4.2)
\2
For rj = 0 we recover the unperturbed oscillator (1.1) of (4.1); the corresponding Melnikov distance for I+ (in the case A, >O) is given by
Ao(to)=W+Oo)=-
(4.3)
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A(h)=A&,)+~ 35’~[-~J2j;if)sinj8,(r))](J+(1)cos(l(l+r,)))dt.
-m (4.4)
Denoting by @(t,) the integral in the right-hand side of (4.4), which can be performed by the method of residues for I+ to yield (case A, > 0)
-
With obvious notation, we rewrite the equation (4.4) in the following form
A(t,)=A(v,)sin(vt,)+B(rR,)sin(S&,)+C. (4.6)
sin( Qt, ). (4.5)
For understanding the discussion below, we have plotted the functions A( vO), B( no) and C in Fig.4b.
Assume that the system (1.1) is in a chaotic motion. Hence, the Melnikov distance A,, changes sign at some t,, , i.e., \A( v,)l -ICI = d > 0. Therefore, if B(&) < d, it is easy to see that from (4.6), the situation remains unchanged (i.e., for some t,, A(&) will change sign). On the other hand, if the frequency of the resonant parametric perturbation Q is in resonance with the driving frequency v, a necessary and sufficient condition [7] for A(&) to be positive for all lo is given by
(B(Q,)l>d, (4.7)
In Fig.4a, c, d, and according to the condition (4.7), we illustrate the regions in which chaos behavior of the system (1.1) can be suppressed. The effect of the cubic nonlinearity and of the external excitation parameter on the parameter regions of suppression of chaos are also shown.
5. CONCLUSIONS
In this work, the generalized averaging method was performed to construct a second order expansion of the response of a weakly nonlinear single-degree-of-freedom system subjected to combined parametric and external excitation. The methods elaborated generally to study the dynamics close to resonance points, consist in constructing a particular expansion for each resonance case. In contrast, we develop here a method allowing one to construct an asymptotic expansion giving several resonances simultaneously (in instance, we obtain here a formulation giving the two resonance cases p = I; q = 1 andp=l;q=2).
On the other hand, the Melnikov technique was applied to the reduced (averaged) system to predict the regions in the parameter space in which chaotic orbits may exist in the original system. First, we show the influence of the nonlinearities on the dimensions of chaotic regions. By introducing a nonlinear parametric perturbation in the system, the suppression of the chaotic dynamic was analyzed. This complex dynamic can become regular provided that the amplitude parameter of the perturbation is
larger than some critical value and the frequency of the perturbation is in resonance with tthe frequency of the forcing. This fact was confirmed by an analytical estimation. Finally, the possiblity to control the dimmension of regular regions, acting on some physical parameter of the oscillator (1. I), was shown.
ACKNOWLEDGMENTS
The first author (M.B) is grateful to the “Ministere des Affaires Etranghes et de la Cooperation” of Morocco and to the PNUD for supporting his travel to Athens to present this work.
REFERENCES
1. YAGAZAKI K., SAKATA M. & KIMURA K., Dynamics of a weakly nonlinear system subjected to combined parametric and external excitation, J. of Applied Mech. 57, 209-217 (1990).
2. BENEDETI’INI F. & REGA G., Nonlinear dynamics of an elastic cable under planar excitation, Int. J. Nonlinear Mech.
22(6), 497-509 (1987).
3. BOGOLIOUBOV N. Br MITROPOLSKY I., Les m&odes asymptotiques en theotie des oscillations non lin6aires. Gauthier- Villards, Paris (1962).
4. BELHAQ M. & FAHSI A., Subharmonic vibrations close to degenerate Poincare-Hopf bifurcations, Mech. Res. Corn.
20(4), 335-341 (1993).
5. MELNIKOV K., On the stability of the center for time periodic perturbations, ‘Dans. Moscou Math. Sot. 12, l-57 (1963).
6. GUCKENHEIMER J. & HOLMES P. J., Nonlinear oscillations dynamical systems and bifurcation of vector fields, in Applied Mathematical Sciences 42 (Edited by F. John, J. E. Marsden and L. Sirovicb), pp. 184-204. Springer-Verlag. New York (1983).
7. LIMA R. & PETTINI M., Suppression of chaos by resonant parametric perturbations, Physical Review A. 41(2), 726.733 (1990).
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Fig.1 Bifurcation curves of quasi-periodic soILlions in the -y-w plane; w,=l. C=2. p=l.
w=l.45 r.J= 1.05 0=0.80
Fig 3 Moterlal non-linecrit~es cna pororrictcrs effect on Ihe cI:‘oIIc
Homocllnic bifurcation curves (case &CO);
P=l. y=1.15, 0=2. 0.=1.
- =o.s
__^_
---.. i
=l.O
=1.5 . . . =2.0
~ y=o.5
---- yz1.0 . . y=l.5
w=1.04. y=1.5. w.=2. j?=I, a=0.05. v.=l, h=O.&
Functions A(v,,) and B(f).) in Eq. (26) for 0~3, 7~0.1, 7=1.5, 0,=2, p=l, 1x=0.005, .4=0.5,h=O.O88.
---- w=l.OS 3 (d) ’ *- 7=0.5
: o=O.BO : :... ’ -___ y=,.o
8. : : . . y=l.S
: 6 .
ll
y=l.S, w,=2. p=1. a=o.o5, o=l.l. C&=2, f9=1. a=o.os
t=O.5, u.= 1.5. h=O 8. [=0.5, v.= 1.5. h=0.8
Fig.4 Material non-lineorities ond external excitation parameters effect on the suppression of chaos regions.
