Bifurcations of limit cycles in a cubic system
Section snippets
Introduction and main results
Part of the well-known Hilbert’s 16th problem is to consider the existence of maximal number of limit cycles for a general planar polynomial system. In general, this is a very difficult question and it has been studied by many mathematicians (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], for example). From [14] we know that there exists a quadratic system having four limit cycles. It was proved in [1] that any quadratic system has at most three limit cycles with
The phase portraits of the unperturbed system
In this section we study the phase portraits of the unperturbed systemwith 0>c>−1>a, ac>1.
The system (2.1) has seven finite singular points O(0,0), , , , , , , and no infinite singular points. O, S1, S2 are saddle points and Ai, i=1, 2, 3, 4 are centers of system (2.1). Eq. (2.1) has a first integral of the formFrom (2.2) we have H(O)=0,
The proof of the main result
Consider system (1.1) with 0>c>−1>a, ac>1. Without loss of generality, we may suppose the points (0,0), are always the saddles of system (1.1) for all ε>0 small. Then (1.1) becomeswhereSince the system (3.1) is symmetric in point O(0,0), then we only consider the half-plane of y⩾0 in the following. Let , Li=L|(−1)ix⩽0, i=1, 2, L3={(
References (14)
- et al.
Bifurcation set and distribution of limit cycles for a class of cubic Hamiltonian system with higher-order perturbed terms
Chaos, Solitons & Fractals
(2000) - et al.
On the stability of double homoclinic and heteroclinic cycles
Nonlinear Anal.
(2003) - et al.
Detection function method and its application to a perturbed quintic Hamiltonian system
Chaos, Solitons & Fractals
(2002) - et al.
Bifurcation set and distribution of limit cycles for a class of seven-order Hamiltonian system with fifteen-order perturbed terms
Chaos, Solitons & Fractals
(2002) - Bautin NN. On the number of limit cycles which appear with the variation of coefficients from an equilibrium position...
Cyclicity of planar homoclinic loops and quadratic integrable systems
Sci. China Ser. A
(1997)- et al.
On the number of limit cycles in double homoclinic bifurcations
Sci. China Ser. A
(2000)
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