We consider the class of polynomial ial differential equation x =pn (x, y) +p n+m (x, y)+pn+2m (x, y), y=Qn (x, y)+Qn+m (x, y)+Q n+2m (x, y). For m, n ≥1 where Pi and Qi are homogeneous polynomials of degree i. Inside this class of polynomial differential equation we consider a subclass of Darboux integrable systems. Moreover, under additional conditions we proved such Darboux integrable systems can have at most 1 Limit cycle.      

The paper is concerned with the bifurcation of Limit cycles in general quadratic perturbations of a quadratic reversible and non-Hamiltonian system, whose period annulus is bounded by an elliptic separatrix related to a singularity at infinity in the Poincare disk. Attention goes to the number of Limit cycles produced by the period annulus under perturbations. By using the appropriate Picard-Fuchs equations and studying the geometric properties of two planar curves, we prove that the maximal number of Limit cycles bifurcating from the period annulus under small quadratic perturbations is two.

This paper considers the problem of stable Limit cycles generating in a class of uncertain nonlinear systems which leads to stable oscillations in the system’s output. This is a wanted behavior in many practical engineering problems. For this purpose, first the equation of the desirable Limit cycle is achieved according to shape, amplitude and frequency of the required output oscillations. Then, the nonlinear control law is designed such that the phase portrait of the closed-loop system includes this stable Limit cycle. The design of controller is based on the Lyapunov stability theorem which is suitable for stability analysis of the positive Limit sets (the stable Limit cycle is a positive Limit set for the nonlinear dynamicl system). The proposed robust controller consists of two parts: nominal control law and additional term which guarantees the robust performance and vanishing the effect of uncertain terms. Finally, to show the applicability of the proposed method, an inertia pendulum system (with parametric uncertainties in its dynamical equations) is considered and the robust output oscillations are achieved by creating the desirable Limit cycle in the close-loop system.

Lyapunov's theorem is the basic criteria to establish the stability properties of the nonlinear dynamical systems. In this method, it is a necessity to find the positive definite functions with negative definite or negative semi-definite derivative. These functions that named Lyapunov functions, form the core of this criterion. The existence of the Lyapunov functions for asymptotically stable equilibrium points is guaranteed by converse Lyapunov theorems. On the other hand, for the cases where the equilibrium point is stable in the sense of Lyapunov, converse Lyapunov theorems only ensure non-smooth Lyapunov functions. In this paper, it is proved that there exist some autonomous nonlinear systems with stable equilibrium points that despite stability don’t admit convex Lyapunov functions. In addition, it is also shown that there exist some nonlinear systems that despite the fact that they are stable at the origin, but do not admit smooth Lyapunov functions in the form of V(x) or V(t,x) even locally. Finally, a class of non-autonomous dynamical systems with uniform stable equilibrium points, is introduced. It is also proven that this class do not admit any continuous Lyapunov functions in the form of V(x) to establish stability.