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NEW ANALYTICAL TECHNIQUE FOR PREDICTING HOMOCLINIC BIFURCATIONS IN AUTONOMOUS DYNAMICAL SYSTEMS
M. Belhaq
Laboratory of Mechanics, Faculty of Sciences ALn Chock, BP 5366, M~arif, Casablanca,
Morocco
e-mail: sc.bm@casanet.net.ma
(Received 20 May 1997; accepted for print 3 December 1997)
I n t r o d u c t i o n
Homoclinic and heteroclinic orbits are of great interest from applied point of view. For instance, in reaction-diffusion problems, they form the profiles of traveling wave solutions [ 1]. The existence of homoclinic orbits can also cause complicated dynamics in the three- dimensional systems, as in the Shilnikov example [2]. The classical approach to predict the bifurcations of these homoclinic orbits in planar autonomous (or driven) systems, deals with the Andronov-Melnikov method [3], [4], or uses its variants [6]. This technique consists in evaluating the distance between the stable and unstable manifolds to the hyperbolic fixed point, associated with the Poincar6-map; see for instance [5]. In the planar autonomous systems, the correspondence between a periodic orbit and a fixed point of the Poincar6-map for manifolds using the Melnikov function, leads to a criterion of homoclinic bifurcation [6]. This approach is mainly based on information concerning the distance between manifolds.
Recently, another analytical approach to predict the homoclinic bifurcations for self-excited autonomous systems was presented in [7]. This approach consists of attacking directly the period T of the bifurcating periodic orbit. More precisely, the condition we considered at such bifurcations is established by considering the limit of the period T goes to infinity, or the vanishing of the related frequency 12
The purpose of this paper is to present a new method to approximate this homoclinic bifurcation in self-excited autonomous systems. The technique involves principally the periodic orbit and the hyperbolic point (saddle) as well. Note that the period (or the frequency) is not needed in this approach. This method is based on the collision between the periodic orbit and the saddle point. Therefore, it can be applied only in the cases for which the homoclinic bifurcation lead to the birth or the death of a periodic orbit (a limit cycle).
We will apply this new criterion to three examples undergoing these global bifurcations.
Comparisons of this criterion to numerical results and previous methods ([4], [5], [6] and [7]) are reported.
4 9
50 M. B E L H A Q
C r i t e r i o n
Consider the planar differential system in the Iorm
Jc = .~y + 6f~ ( x , y , p ) , (1)
~" = - Z ~ x + 6f :(x, y,p),
where ZI and Z 2 a r e positive, f~ and j2 are supposed to be sufficiently smooth with respect to their arguments, and 6, ~t = (/11, P2) are parameters (6 is a small dimensionless perturbation parameter). This equation (1) can model the dynamic behavior of many mechanical systems. We assume that the system (1) undergoes a homoclinic or a heteroclinic bifurcation to a saddle S (x~, ys) at p = pc, where pc is the critical parameter values corresponding to the homoclinic bifurcation. In addition, we consider only the cases where this bifurcation results from the vanishing or emergence of a periodic orbit.
For simplicity, we consider the particular case of (1) corresponding to f~ (x, y,/l) = 0 and
• ~1 = 1. Note that the general case fl ix, y,p ) ~: 0 may be investigated similarly.
Let (x(t, a(t,p)), y = .f) be the general solution of (1) in which a(t,p) denotes the related amplitude. Using a perturbation method, for instance the generalized averaging one [8], this solution can be sought using the ansatz
x(t) = acos Ig + 6U 1 (a, N) + 62U2 (a, I,g)+ .... (2)
d a -~ --_
- - = e_A l(a) + e'A2(a)+ .... d ~ _ eB, (a) + £ 2 B z ( a ) + .... (3)
dt dt
where each U~(a, ~ ) in (2) is 2~-periodic in ~ The amplitude a(t,#) and the phase ~ ( t , p ) are assumed to vary with time according to (3). The functions Ai(a) and B~(a) are determined by vanishing the secular terms in the correction functions U~(a, ~t). Note however that tbr some particular systems, for which the ansatz (2)-(3) does not work, as for the cases fl (x, y , p ) ¢ 0, it is convenient to apply the multiple scales method [ 11 ].
Let us define XA(t,a(p)) as the approximation of the periodic solution, in which a ( p ) denotes the amplitude, and f~(a(p)) = ~ 2 ' ~ = f ~ the related frequency.
Recall that the criterion to predict the homoclinic bifurcations presented in [7], consists of computing explicitly, using (2)-(3), an approximation of the period T u (or its frequency
~2u). In the limit p ---) Pc, the periodic orbit passes closer and closer to the saddle S and so the period T u grows to infinity at the bifurcation. Therefore, the natural condition which was considered at this bifurcation, was merely given by T. ~ ~ , or
f~u = O. (4)
We will now present the new technique. Instead of using the period or the frequency, we attack directly the approximation of the periodic solution itself and the saddle S as well.
Denote by X a ( a ~ ) ) = (xA(a(la)),ya(a(~))) the value of the periodic solution calculated, for certain value of ~ , on an axis (A) connecting the saddle S to the focus F , see figure 1. In the limit /~--->/l c, the periodic orbit approaches closer and closer to the saddle (~'a ---) S). At the homoclinic bifurcation, alias/a = / ~ c , the condition to be satisfied is that the saddle point S and the point ~'a, both located on (A), collide. This is given by
Xa (xa,ya) = S(x,,ys), (5)
(A)
Fig. 1: Homoclinic bifurcation according to the criterion (5)
S: saddle, F: unstable focus, (A): axis, Xa: value of the periodic solution on (A)
E x a m p l e s and R e s u l t s
In this section, we apply the new approach (5) to three cases. As a first example, w e consider the following oscillator with the general quadratic nonlinearities
3c=y,
3) = -092x + e(&y + ~x 2 - / ) x y + ~y2), (6) where a = e&,c = e~,/3 = e/~, and 7 = e~' (or, c, /3, and 7 a r e the original parameters supposed to be small); the parameter ¢o denotes the natural frequency of the system. Using the ansatz above (2), (3), the approximation up to second order of the general solution, is given by
x a (t) = a cos v/+ eU 1 (a, I/t), (7)
where
52 M. B E L H A Q
J d a _ = 0 ) - • 2 I ( 6.a f/~aS ,~ 5~.~ + 5 9 ? + ) 2 0 ) + + ] ) a2 .
[ a t L 120)" 120) 6
(8)
Using the criterion (5) to the first order approximation, alias x A ( t ) = a cos I/t and YA ( t ) = k A (t) = - - a ~ s i n V, we obtain
0)2
acos ty = - - , -af~sin ~ = 0. (9)
C
This is because S = ~'" ( 7 , 0 ) and ~'z = ((1COS ~ , - a f 2 s i n t/t). Resolving these last equations for ~ = 0 , we obtain the relation
0)2
(1 = - - . (10)
(,
/~24cg~...~2 ([7]), the homoclinic Taking into account the expression of the amplitude a = ~/~{~T~+r~2)
bifurcation curve is approximated to the first order in tem~ of the original parameters of the system by
fl(c+ 1,0)2)0) -`
tx - (lla)
4C 2
Let us consider now the approximation of the homoclinic bifurcation to second order. We can postulate that the homoclinic bifurcation is imminent when the velocity of the periodic orbit is equal to zero. This corresponds to the maximum of the amplitude of the orbit, that is
0) 2
Xma x = - - . At the second order, the expression (9) becomes C
(12 c" (12 C ~t7 2
a c o s v + T ( 1 , + - ~ ) + - - ~ ( 1 , - ~ - ) c o s 2 I/t - 2-09 sin2 ~ = --,0}2 C
a 2 c . fla 2
- a s i n ¢ + - ~ - ( y - ~ - ) s m 2 ¢ - 3co c o s 2 ¢ = 0.
( l l b )
In the particular case 7 = O, the above system is written as
a 2 c a 2 c fla2sin21g 0)2
a c o s ~ + 2 0)2 6 0) 2c0S21ff 60) c a" C
- a s i n V + ~ - ~ T s i n 2 ~ t - c o s 2 v =0.
( l l c )
The resolution of the system (1 lc) leads to the second order approximation of the homoclinic bifurcation.
Using the approach (4) given in [7], the homoclinic bifurcation curve in the parameters space is approximated (to first order) by
6/3(c + 1,co~)o~ 2
a = (12)
10c 2 + lOycco ~ + 4y2co 4 +fl2co 2"
Furthermore, by applying the Melnikov technique, the homoclinic bifurcation curve corresponding to the condition for which the Melnikov function has a quadratic zero is given by [9]
c~ = flc°---~2 (13)
7c
To compare these last results, we let y = O, c = 0 . 4 a n d f l = 0.2. W e can point out that the results given by (1 la), (1 lc) and (12), given respectively by the criterion (5) and (4), are "t- dependent, whereas the expression (13) given by Melnikov method is not. The figure 2 illustrates the comparison of these results in the co versus a parameter plane. The circles represent the approach (12) following [7], the squares " a " represent the new technique of this work to first order (1 la), the crosses " x " represent the results of this new technique to second order abtained by resolving the system (11c), and finally the dots" ° "
are related to the Melnikov technique (13). In this figure, one can see the good agreement between the new criterion to second order" x " and the Melnikov method" • ". Note that it is possible to perform higher order approximations (by performing the functions UE(a , gt), U3(a , v/) . . . . ) of the periodic solution (7)-(8), using symbolic calculations. This can improve the precision of the predictive criterion of both the new approach and of the one given in [7]. However, this point is not the main object of this paper.
0,3
0,2 -
o,1
0,0
v
o
' I "
0,1
o
0 •
0 • X
o ~ x
o I 1 m
I I I I I I
0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Fig. 2. Comparisons of homoclinic bifurcation curves of (6), o : The criterion [7], a : This work - first order, x : This work - second order, • : Melnikov method,
54 M. B E L H A Q
The second example is devoted to the generalized Van der Pol oscillator in the form
.i = y (14)
;, = - x + e ( - ~ , x 2 - ~ x ' + ( & - ~ x : )y),
where, as above, we set e~ = )~,e/5 =/~,e& = a and e/~ =/3 (2,;.t,a,/3, are the original parameters). In the conservative case ( a = 0, /3 = 0), the equation (14) contains, at least, one critical point (0,0), which is always a center, and at most, three critical points (equilibrium solutions): (0,0) and (x+,0), (x_,0)determined by
x+ = ~-~(-)~ + ' ~ f - 1 4U). (15)
The pitchfork bifurcation of the origin occurs on the curve 22 - 4/1 = 0 (see figure 3, solid line). The Hopf line for the center C,, ( x = O, y = 0) is given by a = 0, whereas the Hopfline for the center C_ (x = x_, y = 0), in the case 2 ) 0 , ~ ) 0 and A ~- - 4~)0, is given by
a -fix_ 2 = 0. The saddle point is located at S ( x = x+, y = 0).
Let us focus attention on the periodic solution following the Hopf bifurcation around C o.
Using the same process as above, the approximation up to second order of the general solution, in term of the original parameters of the system (14), is given by
Xa(t) = aCOS I//+ Ul(a, ty), (16)
d a _ a a Ba3 + A a S , dllt = C + Da2 _ Ea4 '
d t 2 d t
a 2 3/~ + O~/3 522 7/32 + 15/./2 w h e r e A = / 3 / ' t B = ~ + 3 a / ' t C = I - - - D - - - and E -
32 8 16 8 8 12 256
Similarly, we will use only the first order approximation of the solution. The expression of the function U 1 (a, gt) in (16) is not necessary here. The periodic solution must verify the
d a _ _
stationary condition "z" - 0 to give
. = s
aZl,2 2A 2A (17)
in which, the root a 2 corresponds to the amplitude of the stable serf-excited periodic orbit.
Applying the criterion (5), and taking into account the coordinate of the saddle S(x+,O) given in (15), and (17), the approximation of the homoclinic bifurcation satisfies (in the case £ ) 0 ) the relation
( - 2 + ~fAS- 4/.t): = ~ - ~ ( B - ' ~ - - 2 c ~ ) . (18)
Similarly, in the case 2 ( 0 , we have
(-Z - 4 g 2 - 4p)~ = ~ ( B - ~]B 2 - 2~xA). (19)
Using the criterion (4), the homoclinic bifurcation curve is approximated by the equation (see [71)
e.E EB B ~ / B ' - 2 t z A ) = 0 .
C + _--7 + ( D - n ) ( ~-~ (20)
2 A A 2A 2A
The figure 3 illustrates the comparison of these results. The drawn line represents the result of the criterion (4) (equation (20)), the square-solid line is obtained following [5] using the Melnikov technique, and the dotted one represents the new criterion to first order given by (18) for ~)0, and by (19) for ~,(0.
0,6 0,4"
0,2 - 0,0"
-0,2 -
-0,4 -2
. ..1 / 7 /
-1 0 1
Fig. 3. Comparisons of homoclinic bifurcation curves of ( 1 4 ) , a = f l = 1
__: Melnikov method . . . . : The criterion (4) by [7] ... : The criterion (5) of this work The last example of this work is concerned with the universal unfolding Bogdanov-Takens system
)¢=y,
(21) 2? =/-/l +/-t2Y + x2 + xy.
The fixed points of (21) are given by
(x±, y) = (+ -a~l,0). (22)
For illustrating the criterion (5) to predict the homoclinic bifurcation in this system, it is more convenient to apply the multiple scales method [11]. Performing this perturbation technique, the approximation to second order of the periodic orbit may be sought in the form
x~(t,e) = Xo(ro, r , ) + ax,(To, r , ) + e~XAro, T~)+ ....
ya(t,e) = yo(To,TO+ ey~(To,T~)+ e2y2(To,T~)+ .... (23)
56 M. BELHAQ
where the different time scales T = e"t with e is a small parameters. By translating the fixed point to the origin, substituting (23) into (21), and equating coefficients of like power of e , we obtain systems at different orders of e. Without detailling the calculations, the approximation to second order of the periodic orbit is given by
where
XA (t) = aCOS( ~) 'Ue, 2 yz(t) = - a 2 ~ 2 siIl(q) ).
a = -4(/./~ +/~1);q9 = ( ~ + 0).
We immediately see that the criterion (5) written explicitly as a cos(~0)- ~-~ = - - x / ~ , - a 2~/~2 sin(~0) = 0, leads to the approximate homoclinic bifurcation curve
The Melnikov technique gives (see [61).
Some comparisons are illustrated in the table below.
(24)
(25)
(26)
(27)
(28)
N u m e r i c a l M e l n i k o v - eq. (28) This w o r k - criterion
calculatiorm (5), eq. (27)
~ 2
13 1 = - 1 0 0 9 7.1 7.3
I~ 1 = -1 O. 74 O. 71 O. 73
Finally, note that other bifurcations which occur in the system (21), namely Hopf and saddle node bifurcation, can be also approximated using only the approximation of periodic orbit and the coordinates of the saddle. Indeed, since at the Hopf bifurcation the amplitude of the orbit vanishes, we obtain from (25), the Hopf bifurcation curve
~j = -/-/~, (29)
corresponding to a = 0 in (25). This same result has been obtained in [6] by studying the linearized vector field at the fixed points (+ -x/-~,0). On the other hand, to predict the saddle node bifurcation, at which these fixed points meet to disappear, one only has to equal their coordinates given by (22). We obtain, as found in [6] using a classical study, the curve
//1 = 0. (30)
C o n c l u s i o n
To investigate the homoclinic bifurcations, one usually applies the standard Andronov- Melnikov method, or one of its variants. This technique especially based on the calculation of the distance between the stable and unstable manifolds (separatrices) to the hyperbolic fixed point. Recently, another alternative to predict these global bifurcations in the self- excited autonomous systems was established in [7]. This alternative considers the limit of the periodic orbit tends to infinity at the homoclinic connection. Therefore, the condition to be satisfied was merely given by the vanishing of the frequency of this orbit. In this paper, we have focused attention on the periodic orbit itself and the saddle as well. After all, the main object involved in this bifurcation is precisely the periodic orbit. We have then presented a new analytical criterion based on the analysis of the periodic orbit moving toward the saddle point when approaching the homoclinic bifurcation. In the limit, the periodic orbit and the saddle collide. Note that these criteria evolve quite simple expressions including trigonometric functions, and then specific higher order approximations can be performed by carrying out a direct symbolic algebraic manipulation. In contrast to the previous method [7], the comparisons shown here, reveal the good prediction given by the new technique of this work. This technique can also be applied to investigate the saddle node bifurcations for fixed points, as shown in the Bogdanov-Takens example, and the Hopf bifurcation as well. Furthermore, a generalization of the technique (4) to predict homoclinic bifurcations in periodically driven oscillators was proposed in [10].
Note that the homoclinic bifurcations in the three-dimensional systems can also be investigated applying the techniques of this work. The challenge here is to construct a good analytical approximation of the periodic solution of the three-dimensional systems. The collision between the periodic orbit and the fixed point should give an analytical prediction of the homoclinic bifurcation in these systems. This stimulating direction is our future main purpose.
A c k n o w l e d g m e n t
The author would like to thank the unknown reviewer for his numerous and relevant remarks that have been helpful in improving the presentation of this paper.
58 M. BELHAQ R e f e r e n c e s
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