ABDELHAK FAHSI
Faculty of Sciences and Technique of Mohammedia, BP 146, Mohammedia, Morocco (Received: 18 January 1998; accepted: 20 October 1998)
Abstract. An analytical method to predict the homoclinic bifurcation in a planar autonomous self-excited weakly nonlinear oscillator is presented. The method is mainly based on the collision between the periodic orbit undergo- ing the homoclinic bifurcation and the saddle fixed point. To illustrate the analytical predictive criteria, two typical examples are investigated. The results obtained in this work are then compared to Melnikov’s technique and to a previous criterion based on the vanishing of the frequency. Numerical simulations are also provided.
Keywords: Periodic orbit, planar autonomous systems, homoclinic bifurcations, multiple scales technique, criteria.
1. Introduction
Homoclinic and heteroclinic orbits are of great interest from the application point of view.
For instance, in reaction-diffusion problems, they form profiles of traveling wave solutions.
Their existence can cause complicated dynamics in three-dimensional systems [1, 2]. In a static-dynamic analogy, a homoclinic orbit corresponds to a spatially localized post-buckling state [3]. Recently, the investigation of homoclinic bifurcation has received much attention from both the analytical and numerical points of view [4–6]. A classical tool for predicting bifurcations of homoclinic orbits is Melnikov’s technique. This approach is mainly based on the vanishing of the distance between manifolds of the perturbed system. Recently, another analytical method to predict the homoclinic bifurcations for planar autonomous self-excited systems was presented in [7]. This approach consists in attacking directly the period of the periodic solution. More precisely, the condition considered at such bifurcations is that the limit of the period goes to infinity or the vanishing of the frequency of the periodic solution.
In [8], a semi-analytical and numerical process was developed to determine the separatrices and the limit cycles of strongly nonlinear oscillators. Conditions under which a limit cycle is created or destroyed were derived.
In this paper, we present a new analytical method to predict homoclinic bifurcation in self- excited planar autonomous systems. The method we propose is mainly based on the collision (at the bifurcation) between the two objects involved in the bifurcation. In planar autonomous self-excited systems, a collision occurs between the periodic orbit and the saddle point. At the bifurcation, the periodic orbit disappears or gives rise to other attractors. In order to show the
Figure 1. Homoclinic bifurcation according to criterion (3).
validity of the proposed method, two typical examples have been treated. Comparisons with numerical simulations and previous methods [7, 9, 10] are reported.
2. Criterion
Consider a system of the form
˙
x =y+εf1(x, y, µ, t), y˙ =αx+εf2(x, y, µ, t), (1) wheref1and f2are supposed to be sufficiently smooth with respect to their arguments and periodic int,α is a constant, andµ =(µ1, µ2, . . .)are a family of parameters. Equation (1) can model the dynamic behavior of many dynamical systems [11, 16].
Assume that in the case where the functions f1andf2 aret-independent, andε 6= 0 but small, system (1) undergoes a homoclinic (or a heteroclinic) bifurcation to a saddleS(as, bs)at µ=µc, whereµcis the critical parameter value corresponding to the homoclinic bifurcation.
Suppose that this homoclinic bifurcation is produced by the disappearance (or appearance) of a periodic orbit(C). LetXA(xA(t, a(µ)), yA(t, a(µ)))be an approximation of the bifurcating periodic solution with the amplitude a(µ)and the frequency (a(µ)) = 2π/(T (a(µ))) =
µ. The criterion for predicting the homoclinic bifurcations given in [7], consists in comput- ing explicitly, using a perturbation method [12, 13], an approximation of the frequencyµ. In the limitµ→µc, the periodic orbit approaches the saddleSand so the periodTµtends to infinity at the bifurcation. Therefore, the natural condition considered was given by
µ=0. (2)
In this paper, we consider another strategy. Instead of using the period (or the frequency) of the periodic orbit, we directly attack the periodic solution itself and the saddle S(as, bs)as well. Denote byX¯A(x¯A(a(µ)),y¯A(a(µ)))the value of the periodic solution located, for some value oft, on an axis(1)connecting Sto the focus F (see Figure 1). In the limit µ →µc, the periodic orbit passes closer and closer to the saddle, that isX¯A → S(as, bs), and at the bifurcation, aliasµ=µc, the condition to be satisfied can be given by
X¯A(x¯A,y¯A)=S(as, bs). (3)
xA(t)=acos(φ)+εU1(a, φ), (5)
yA(t)= −aωsin(φ)+εV1(a, φ), (6)
where the amplitudeaand the phaseφmodulations are solutions of the system da
dt =εµ˜1a 2 −ε2
µ˜3a3
8ω2 (µ˜2+ ˜µ4ω2)
, (7)
dφ
dt =ω−ε2
"
5µ˜22
12ω3 +5µ˜4µ˜2
12ω +µ˜42
ω 6 + µ˜32
24ω
! a2
#
, (8)
and
U1(a, φ)= a2 2
˜ µ4+ µ˜2
ω2
+a2 6
˜ µ4−µ˜2
ω2
cos(2φ)−µ˜3a2
6ω sin(2φ), (9)
V1(a, φ) = −µ˜4ω2− ˜µ2
3ω a2sin(2φ)+µ˜1a
2 cos(φ)−µ˜3
3 a2cos(2φ). (10)
Using criterion (3) to the first-order approximation, aliasxA(t)=acos(φ)andyA(t)= ˙xA(t)
= −aφ˙sin(φ), we obtain
¯
xA =acos(φ)= ω2 µ2
, (11)
¯
yA = −aωsin(φ)=0. (12)
Resolving these last equations leads to the relation a = ω2
µ2
. (13)
Taking into account the expression of the amplitude given by considering the stationary solu- tion of Equation (7), we find
a = s
4µ1ω2
µ3(µ2+µ4ω2). (14)
Figure 2. Comparisons of Homoclinic bifurcation curves of Equation (4),+ + +criterion (2),− − −criterion (3) to first order,∗ ∗ ∗criterion (3) to second order, Melnikov’s method.
Hence, using Equation (13), the homoclinic bifurcation curve is then approximated, at the first order, by
µ1= µ3(µ2+µ4ω2)ω2
4µ22 . (15)
At the second order, criterion (3) is written as acos(φ)+εU1(a, φ)= ω2
µ2
, −aωsin(φ)+εV1(a, φ)=0, (16)
whereU1(a, φ)and V1(a, φ) are given by Equations (9) and (10). Resolution of this system using the package Bifpack [5] leads to the second-order approximation of the homoclinic bifurcation curve. On the other hand, using the approach given in [7], based on the vanishing of the frequency in Equation (8), we obtain in the particular case ofµ4 =0
µ1= 6µ3µ2ω2
10µ22+µ23ω2. (17)
Applying the Melnikov technique to Equation (4) (for details, see [14]) yields µ1= µ3ω2
7µ2
. (18)
In Figure 2, we have plotted the approximate homoclinic bifurcation curves in theωversus µ1 parameter plane, for the fixed values µ4 = 0, µ2 = 0.4 andµ3 = 0.2. The crossed line represents criterion (2) given by Equation (17), the dashed line represents the approximation obtained by using criterion (3) to first order given by Equation (15), the star line denotes the criterion (3) up to second order given by Equation (16), and the solide line represents the Melnikov approximation given by Equation (18).
contained in this example, it is convenient to apply the multiple scales method [13, 17] in order to perform an approximation of the periodic orbit.
Introducing the variable changeX =x− √µ2, settingσ =µ1−µ2and lettingX =εu, y =εvandσ =ε2σ, where˜ εis introduced as a tool to indicate the smallness of the nonlinear terms and the parameterσ, the system (19) becomes
˙ u = v,
˙
v = (−2µ2+ε2σ )u˜ +εσ˜√
µ2−ε√
µ2(3u2+2uv)−ε2(u3+u2v). (20) Taking into account the above change of coordinates, the saddle is now located at(−√µ2,0).
Using the multiple scales method, the approximate periodic solution of system (20) up to second order and the modulation equations of amplitude and phase are given, respectively, by
uA(t)=acos(φ)+O(ε) . . . , (21)
vA(t)= −ap
2µ2sin(φ)+O(ε) . . . , (22)
and da
dt = −σ 2a+a3
4 , dφ
dt = p
2µ2+ σ
√2µ2
− 9+µ2
6√ 2µ2
a2. (23)
The stationary solution of Equation (23) gives the amplitude of the periodic solution
a2=2σ. (24)
Hence, at the first order of the approximation, criterion (3) is written as acos(φ)= −√
µ2, −ap
2µ2sin(φ)=0.
Resolving this last system leads to the approximate homoclinic bifurcation curve µ2= 2
3µ1. (25)
Using the Melnikov technique, the homoclinic bifurcation curve of Equation (19) is approxi- mated by (for details, see [10])
µ2= 4
5µ1. (26)
Figure 3. Comparisons of Homoclinic bifurcation curve of Equation (19),+ + +criterion (2),− − −criterion (3) to first order, Melnikov’s method,. . .Numerical results.
On the other hand, using the approach given by criterion (2), based on the vanishing of the frequency of the periodic solution, we obtain
µ1=µ2+ 6µ2
6+µ2
. (27)
We compare the results in Figure 3, where we have plotted the approximate homoclinic bifurcation curves in the µ2 versus µ1 parameter plane. The crossed line corresponds to criterion (2), Equation (27), the dashed line illustrates criterion (3), Equation (25), the solid line relates to Melnikov’s technique, Equation (26) and the dotted line corresponds to the numerical simulation using the package Phaser [18].
Figure 4 shows phase portraits corresponding to the homoclinic bifurcation of Equa- tion (19) forµ1 =0.50 and three values ofµ2. It is seen that asµ2decreases, the two limit cycles move towards the saddle point and collide with it at the homoclinic bifurcation. This leads to the disappearance of the two limit cycles and the appearance of a new orbit.
4. Conclusion
In a recent work [7], a technique to predict homoclinic bifurcations in planar self-excited autonomous oscillators was established. This technique mainly considers the limit of the periodic orbit tending to infinity or the vanishing of the frequency at the homoclinic con- nection. In this paper, we have focused attention on the periodic orbit itself and the saddle as well. After all, the principal object involved in this bifurcation is precisely the periodic orbit.
We have presented an analytical criterion based on the analysis of the periodic orbit moving toward the saddle point when approaching the homoclinic bifurcation. At the bifurcation, the condition to be satisfied was merely the collision between the periodic orbit and the saddle.
The comparisons shown here reveal the good agreement between the criterion of this work and the Melnikov technique or numerical simulations. The criterion developed here gives much
Figure 4. Phase portraits of Equation (19) forµ1=0.5 and respectivelyµ2=0.42,µ2=0.4034 andµ2=0.40.
better approximations than the previous approach [7] based on the vanishing of the frequency of the periodic orbit. Of course, the prediction of the homoclinic bifurcation can be improved by performing high-order approximations.
Furthermore, a generalization of criterion (3) to predict homoclinic bifurcations in pe- riodically driven oscillators was proposed in [15]. Note that the homoclinic bifurcations in 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 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.
Acknowledgment
The first author (M. B.) thanks the Morrocan-American Commission for Cultural and Educational Exchanges for its support.
References
1. Shil’nikov, L. P., ‘On a new type of bifurcation of multi-dimensional dynamical systems’, Soviet Mathemat- ics Doklady 10, 1969, 1368–1371.
2. Parker, T. S. and Chua, L. O., Practical Numerical Algorithms for Chaotic Systems, Springer-Verlag, New York, 1989.
3. Thompson, J. M. T. and van der Heijden, G. H. M., ‘Homoclinic orbits, spatial chaos and localized buckling’, in Proceedings of IUTAM Symposium 1997, Application of Nonlinear and Chaotic Dynamics in Mechanics, F. C. Moon (ed.), Kluwer, Dordrecht, to appear.
4. Nayfeh, A. H. and Balachandran, B., Applied Nonlinear Dynamics, Wiley, New York, 1995.
5. Seydel, R., Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, Springer-Verlag, New York, 1994.
6. Fiedler, B., ‘Global pathfollowing of homoclinic orbits in two-parameter flows’, Report No. 4, Institut für Angewandte Analysis und Stochastik, Berlin, 1992.
7. Belhaq, M. and Fahsi, A., ‘Homoclinic bifurcations in self-excited oscillators’, Mechanics Research Communications 23(4), 1996, 381–386.
8. Xu, Z., Chen, H. S. Y., and Chung, K. W., ‘Separatrices and limit cycles of strongly nonlinear oscillators by the perturbation-incremental method’, Nonlinear Dynamics 11, 1996, 213–233.
9. Doelman, A. and Verhulst, F., ‘Bifurcations of strongly nonlinear self-excited oscillations’, Mathematical Methods in the Applied Sciences 17, 1994, 189–207.
10. Guckenheimer, J. and Holmes, P. J., Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Applied Mathematical Sciences, Vol. 42, Springer-Verlag, New York, 1983.
11. Moon, F. C., Chaotic Vibrations: An Introduction for Applied Scientists and Engineers, Wiley, New York, 1987.
12. Bogolioubov, N. and Mitropolsky, I., Les méthodes asymptotiques en théorie des oscillations non linéaires, Gauthier-Villars, Paris, 1962.
13. Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.
14. Belhaq, M., Clerc, R. L., and Hartmann, C., ‘Bifurcations homocliniques pour des équations de Liénard forcées périodiquement’, Journal de Mécanique Théorique et Appliquée 6(6), 1987, 865–877.
15. Belhaq, M. and Fahsi, A., ‘Repeated resonances to chaos in forced oscillators’, Mechanics Research Communications 21(5), 1994, 415–422.
16. Thomson, J. M. T. and Stewart, H. B., Nonlinear Dynamics and Chaos. Geometrical Methods for Engineers and Scientists, Wiley, New York, 1986.
17. Nayfeh, A. H., Perturbation Methods, Wiley, New York, 1973.
18. Hale, J. and Koçak, H., Dynamics and Bifurcations, Springer-Verlag, New York, 1991.
