Bifurcations of limit cycles in a cubic system

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Abstract

This paper is concerned with the number of limit cycles in a cubic system. Eleven limit cycles are found and two different distributions are given by using the methods of bifurcation theory and qualitative analysis.

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 systemẋ=y(1+x2+cy2),ẏ=x(1−ax2−y2)with 0>c>−1>a, ac>1.

The system (2.1) has seven finite singular points O(0,0), S10,1c, S20,−1c, A1c+1ac−1,a+11−ac, A2c+1ac−1,a+11−ac, A3c+1ac−1,−a+11−ac, A4c+1ac−1,−a+11−ac, 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 formH(x,y)=12(y2−x2)+14(ax4+cy4+2x2y2)=h.From (2.2) we have H(O)=0, H(Ai)=

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), 0,±1c are always the saddles of system (1.1) for all ε>0 small. Then (1.1) becomesẋ=y(1+x2+cy2)+εP(x,y),ẏ=x(1−ax2−y2)+εQ(x,y),whereP(x,y)=a10x+a30x3+a21x2y+a12xy2+a01y+ca01y3,Q(x,y)=b10x+b01y+b21x2y+b12xy2+b30x3+cb01y3.Since 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 L={(x,y)|H(x,y)=−14c,y⩾0}, Li=L|(−1)ix⩽0, i=1, 2, L3={(

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