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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)
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)
In this paper, based on a generalized Lyapunov function, a simple proof is given to improve the estimation of globally attractive and positive invariant set of the Lorenz system. In particular, a new estimation is derived for the variable x. On the globally attractive set, the Lorenz system satisfies Lipschitz condition, which is very useful in the study of(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 absolute stability theory and methodology for nonlinear control systems are employed to study the well-known Chua's circuit. New results are obtained for the globally exponent synchronization of two Chua's circuits. The explicit formulas can be easily applied in practice. With the aid of constructing Lyapunov functions, sufficient(More)
A computationally efficient method is proposed for computing the simplest normal forms of vector fields. A simple, explicit recursive formula is obtained for general differential equations. The most important feature of the approach is to obtain the " simplest " formula which reduces the computation demand to minimum. At each order of the normal form(More)
In this paper, we consider the weakened Hilbert's 16th problem for symmetric planar perturbed polynomial Hamiltonian systems. In particular, a perturbed Hamiltonian polynomial vector field of degree 9 is studied, and an example of Z 10-equivariant planar perturbed Hamiltonian systems is constructed. With maximal number of closed orbits, it gives rise to(More)
This paper investigates the bifurcations due to time delay in the feedback control system with excitation. Based on an self-sustained oscillator, the delayed velocity feedback control system is proposed. For the case without excitation, the stability of the trivial equilibrium is discussed and the condition under which the equilibrium loses its stability is(More)
A general explicit formula is derived for controlling bifurcations using nonlinear state feedback. This method does not increase the dimension of the system, and can be used to either delay (or eliminate) existing bifurcations or change the stability of bifurcation solutions. The method is then employed for Hopf bifurcation control. The Lorenz equation and(More)