0093-6413/96 $12.00 + .00
r,n s o o ~ l a(s~')aoaas-a
HOMOCLIN1C BIFURCATIONS IN SELF-EXCITED OSCILLATORS
M. Belhaq and A. Fahsi
Laboratory of Mechanics, Faculty of Sciences Ain Chok, BP. 5366, M~iarif, Casablanca, Morocco
(Received 2 May 1996; accepted for print 25 May 1996)
Introduction
This paper is concerned with saddle-loop bifurcations of self-excited oscillations described by a single-degree of freedom equation
x + o~ 2 x = af(x, x, I-t) (1)
in which e is a small perturbation parameter and I.t is a two-parameter family. The function f is supposed to be sufficiently smooth with respect to their arguments. Such equations are encountered in many engineering problems; see for instance [ ! ].
For e = 0, equation (I) reduces to a single harmonic oscillator with a natural frequency to.
Assume that for e positive and sufficiently small, the perturbed system (1) may have self- excited oscillations (or periodic solutions) emanating from homoclinic or heteroclinic bifurcations.
To determine the homoclinic bifurcation in periodically tbrced oscillators one usually applies the so-called Melnikov's integral. This technique uses the intersection phenomenon of stable and unstable manifolds coming from the hyperbolic fixed point of the associated Poincare map [2], [3]. Moreover, this technique can also be applied to planar autonomous problems by constructing a certain Poincare-map related to the periodic solution [4].
In this paper we suggest an other analytical criterion to predict homoclinic (or heteroclinic) bithrcations, resulting from the vanishing or emergence of periodic solutions in the system ( I ).
This approach consists of attacking directly the period of the approximate periodic solution.
The natural condition to be satisfied at the saddle-loop connexion is given by T goes to infinity. In other words, a homoclinic bifurcation criterion can be given by vanishing the frequency f2 of the periodic solution. To illustrate this approach we study two specific cases: a quadratic nonlinear oscillator and a generalized Van Der Pol oscillator. The results o f this work are compared with both Melnikov's method and numerical calculations.
3 8 1
Criterion of homoclinic bifurcation
Assume that for e is positive but small, isolated periodic solution may survive in the system (I). Suppose that the saddle point of (I) may be either inside or outside this solution (see fig.
1). We shall illustrate the method providing the prediction of saddle-loop connexions emanating from the emergence or vanishing of the self-excited periodic solution in the simple case for which the saddle is outside the periodic orbit (fig. la). The case for which the saddle point is inside the periodic solution (fig. l.b) can be considered and studied similarly.
Fig. 1: The saddle point outside the periodic solution (a) or inside the periodic solution (b) Let x(t, a(td-t)) the general solution of (1) in which a(t,l.t) denotes the amplitude of the solution. Using the generalized averaging method [5], this solution can be sought in the form
x(t) = acos~lJ + eUl(a,~) + ~2 U2(a, tl0 + ... (2)
where each Ui(a,~lJ) is 27t-periodic in ~11. The amplitude a(t,!tt) and the phase ~(t,l.t) are assumed to vary with time according to
~ = + • d_E = = + +
eml(a) D(a) co ~Bl(a) (3)
dt '
where the functions Ai(a) and Bi(a) are determined by the condition of vanishing of secular terms in the correction functions Ui(a,~10.
Define ~(t,'g(I.t)) as the periodic solution of period T(ff(t.t) ) verifying the stationary condition d~'(I.t)/dt = 0 such as "~(la) is the amplitude of the solution and D("ff(I.t) ) = 2n/T(~(I.t) ) is the corresponding frequency.
As a parameter is varied, the amplitude may grow to approach the separatrix while the period
of the solution increases to infinity. At the saddle-loop connexion, the period becomes infinity
and therefore, it is natural to postulate the condition that the limit T(~(I.t) ) goes to infinity as
a criterion o f this bifurcation. This limit corresponds indeed to the vanishing o f the frequency f~(g(~t) ) and consequently the condition to be satisfied at this bifurcation is merely given by
f~f~(~t) ) = 0 (4)
A u u l i c a t i o n
Consider the quadratic nonlinear single degree o f freedom oscillator
+ 0)2 x = e ( otx + cx 2 - 13xx + ~ ) ( 5 )
The approximation up to second order o f the general solution is given by
a 2 13 a2 ~ a 2
c ) -ff cos2~v - sin2~/ + ( ~ / + ) ~ - ( 6 ) x ( t ) = a c o s ~ + ( y - 0) 2 60)
da eta ~a 3 2
- ~ - - 80)2 (c + y0) )
dw 5c2 + 57c +Y 20 , 1 [32 , 2 (7) dt = 0) - ( 1 - ~ 3 120) 6 ~- 240) ) a
The periodic solution can be obtained by the stationary condition da/dt = 0. In consequence, we obtain either a = 0, corresponding to the trivial solution x(t) = 0, which is stable for ct < 0 and unstable for ct > 0 or
~' = ~ 4°t'0)2
13(c + ~0)2) (8)
corresponding to the amplitude o f the unique self-excited periodic solution which has the frequency
5c 2 57c + ~ 132 ).if2 (9)
~(~ ) = 0)" ( l - ~ 3 + 120) 6 + 240)
Finally, by using the criteruim (4), the homoclinic bifurcation curve is approximated by
(x = 3 [3(c+'/0)2)0) 2 (10)
10c 2 + 107c0) 2 + 4y20) 4 + 1320) 2
Using the Melnikov's technique, the condition for which the Melnikov's function has a quadratic zero is
13t02
ct = 7c (11)
Figure-2 shows the comparison o f results obtained by both methods. The solid line represents the approach o f this work (10) and the dotted one represents the Melnikov's technique (11).
03
0 . 0 0 I I i
,I
' lOt
- 0 . 2 0 - 0 . 1 0 0.00 O. I 0 0.20
Fig. 2: The saddle-loop bifurcation curve o f quadratic nonlinear oscillator
As second example, we consider the generalized Van Der Pol oscillator
x + x = t ~ ( - ~ , x 2 -j.tx 3 +(ct-13 x2)x ) (12)
First, consider the conservative system (or = 0, 13 = 0). Then, the equation may admit at least e one critical point x = 0 and at most three critical points : x = 0, x = x_ and x = x+ where
1 ( _ ~ . ± . ~ 41.t
×± - - ~ - ) ( ] 3 )
We consider the case X > 0, I-t > 0 and ~2 _ 4~t ~ 0 for which the phase plan is described in figure 3. Hence we have two centers C o (x = 0, y = 0 ) , C 1 (x = x_, y = 0) and one saddle at S (x = x + , y = 0). Besides, we have two homoclinic orbits through the saddle S.
The Hopf bifurcation condition near C 0 is given by et = 0 and the Hopf bifurcation near the center CI occurs at a - 13~ = 0.
Figure 3: Phase plan in the case: X > O, I-t > O, X 2 - 4g > 0
For the homoclinic bifurcation around C 0 , the approximation up to second order o f the
general solution is given by
x(t) = acos~ + Ul(a,w) •
da eta d_~ = C + Da 2 - Ea 4
dt - 2 "Ba3 + A a S dt
(14)
_ ~_t ~ + 3oq.t ct 2 _ 3~t + ot~ 5K 2 _ 7]32 + 15bt 2 where A - 3 2 " B = 16 ; C = I - - - ~ - ' D - 8 - 1"--2- " E - 256 The periodic solution must verify the stationary condition da/dt = 0. One gets
~2 B q B 2 - 2otA
a l - 2 - 2 A + 2A (15)
in which the root a2 corresponds to the stable self-excited periodic solution o f frequency
_2 ~4
f ~ ( ' f f 2 ) = C + D a 2 - E a 2 (16)
In consequence, using the criterion (4) the homoclinic bifurcation curve is approximated by the condition
orE EB B q B 2 - 2otA
C+2A + (D --A--) (~-X 2A )=0 (17)
For the homoclinic bifurcation around C I , the equation (12) become, alter translating C 1 to the origin
+ ¢00 y = e (or y - 3qry - 13y 2 y -Ky2 _ i.ty3 ) 2 (18)
where y = x - x _ • co 0 = l + 2 ~ . x _ +3~tx " ~ . = k + 3 ~ t x _ • o t = a - 1 3 x ! " y=213x_
The general solution o f ( 1 8 ) is approximated by
x(t) = acos~l/+ Ul(a,~10 " (19)
da _~.a _ ~ a 3 + , ~ a S . ~ : C + l ~ a 2 E a 4 (14
dt 2 dt -
where + ~ 2 3p + o~3 5~, 2
~, = 13j.t . ~ = ~ 3~.t _ yK . ~ = % ) -8¢°o " ~ = 3
32 , 2, '
y2 . ~ = 7132 + 151a2 240) o 256¢0 o 256~t3 "
Similarly, applying tile criterion (4), the homoclinic bifurcation curve is approximated by the
condition
C+-=+(D-
2A ~ - ) ( ~ - 2A- )--0 (20)
In the table below we compare these results with previous works [4], for ot = [3 = l.
I
