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In this paper, we prove the existence of twelve small (local) limit cycles in a planar system with third-degree polynomial functions. The best result so far in literature for a cubic order planar system is eleven limit cycles. The system considered in this paper has a saddle point at the origin and two focus points which are symmetric about the origin. This(More)
The focus of the paper is mainly on the existence of limit cycles of a planar system with third-degree polynomial functions. A previously developed perturbation technique for computing normal forms of differential equations is employed to calculate the focus values of the system near equilibrium points. Detailed studies have been provided for a number of(More)
In this paper, the existence of 12 small limit cycles is proved for cubic order Z 2-equivariant vector fields, which bifur-cate from fine focus points. This is a new result in the study of the second part of the 16th Hilbert problem. The system under consideration has a saddle point, or a node, or a focus point (including center) at the origin, and two weak(More)
This paper investigates periodic bifurcation solutions of a mechanical system which involves a van der Pol type damping and a hysteretic damper representing restoring force. This system has recently been studied based on the singularity theory for bifurcations of smooth functions. However, the results do not actually take into account the property of(More)
Angiopoietin-like protein 1 (ANGPTL1) is a potent regulator of angiogenesis. Growing evidence suggests that ANGPTL family proteins not only target endothelial cells but also affect tumor cell behavior. In a screen of 102 patients with lung cancer, we found that ANGPTL1 expression was inversely correlated with invasion, lymph node metastasis, and poor(More)
In this paper, the concept of globally exponentially attractive set is proposed and used to consider the ultimate bounds of the family of Lorenz systems with varying parameters. Explicit estimations of the ultimate bounds are derived. The results presented in this paper contain all the existing results as special cases. In particular, the critical cases, b(More)
Keywords: Hilbert's 16th problem Limit cycle Bifurcation Focus value Normal form Maple a b s t r a c t This paper is concerned with the practical complexity of the symbolic computation of limit cycles associated with Hilbert's 16th problem. In particular, in determining the number of small-amplitude limit cycles of a non-linear dynamical system, one often(More)
Keywords: General Lorenz family Globally exponentially attractive set Positive invariant set Generalized Lyapunov function Ultimate boundedness of chaotic system a b s t r a c t In this paper, we give a constructive proof on the existence of globally exponentially attractive set and positive invariant set of general Lorenz family, which contains four(More)
Liénard systems and their generalized forms are classical and important models of nonlinear oscillators, and have been widely studied by mathematicians and scientists. The main problem considered is the maximal number of limit cycles that the system can have. In this paper, two types of symmetric polynomial Liénard systems are investigated and the maximal(More)