Existence Conditions of Thirteen Limit Cycles in a cubic System

As we know, the second part of the Hilbert problem is to find the maximal number and relative locations of limit cycles of polynomial systems of degree n. Let H(n) denote this number, which is called the Hilbert number. Then the problem of finding H(n) is divided into two parts: find an upper and lower bounds of it. For the upper bound there are important works of Écalle [1990] and IIyashenko and Yakovenko [1991]. However, if H(n) < ∞ holds or not is still open, even for the case n = 2. On the other hand, many works have been done on the lower bound, especially for quadratic and cubic systems. See [Li, 2003] for a detailed introduction to recent advancement of the problem. For example, Bautin [1952] proved H(2) ≥ 3 by studying Hopf bifurcation. Chen and Wang [1979] and Shi [1980] separately proved H(2) ≥ 4. Li and Huang [1987] first found a cubic system having 11 limit cycles, giving H(3) ≥ 11. Li and Liu [1991], and Liu et al. [2003] respectively found more cubic systems having 11 limit cycles with the same distribution. Later, Han et al. [2004] and Han et al. [2004, 2005] used the method of stability-changing of a homoclinic loop to give more cubic systems having 11 limit cycles with two different distributions. Then Zhang et al. [2005] studied an asymmetric cubic system and found three different distributions of 11 limit cycles. Yu and Han [2004, 2005a, 2005b] proved further H(3) ≥ 12 by studying Hopf bifurcation in some centrally symmetric cubic systems. Recently, Liu and Li [2008] obtained a sufficient condition for