Abstract:
In this paper, we study a Lienard system of the form dot{x}=y-F(x), dot{y}=-x, where F(x) is an odd polynomial. We introduce a method that gives a sequence of algebraic approximations to the equation of each limit cycle of the system. This sequence seems to converge to the exact equation of each limit cycle. We obtain also a sequence of polynomials R_n(x) whose roots of odd multiplicity are related to the number and location of the limit cycles of the system.

Abstract:
We present a simpler proof of the existence of an exact number of one or more limit cycles to the Lienard system $\dot{x}=y-F(x) $, $\dot {y}=-g(xt)$, under weaker conditions on the odd functions $F(x) $ and $g(x) $ as compared to those available in literature. We also give improved estimates of amplitudes of the limit cycle of the Van Der Pol equation for various values of the nonlinearity parameter. Moreover, the amplitude is shown to be independent of the asymptotic nature of $F$ as $|x| \to\infty$.

Abstract:
Lienard systems of the form $\ddot{x}+\epsilon f(x)\dot{x}+x=0$, with f(x) an even function, are studied in the strongly nonlinear regime ($\epsilon\to\infty$). A method for obtaining the number, amplitude and loci of the limit cycles of these equations is derived. The accuracy of this method is checked in several examples. Lins-Melo-Pugh conjecture for the polynomial case is true in this regime.

Abstract:
The main objective of this paper is to study the number of limit cycles in a family of polynomial systems. Using bifurcation methods, we obtain the maximal number of limit cycles in global bifurcation.

Abstract:
In this paper, we consider two dimensional autonomous system of the form:( )( ),,x P x yy Q x y= ..= ..&&(A)in which P and Q are polynomials in x and y. We write the system A in the form of( ) ( )( ) ( )2 32 3, ,, ,x x y p x y p x yy x y q x y q x yll= + + += - + + +&&(B)Where 2 p , 2 q and 3 p , 3 q are homogeneous quadratic and cubic polynomials in x and y. The question of interest is the maximum possible number of limit cycles (a limit cycle is an isolated closed orbit) of such systems which can bifurcate out of the origin in terms of the degree of P and Q. It is second part of known Hilbert’s sixteenth problem. Research on Hilbert’s sixteenth problem in general usually proceeds but the investigation of particular classes of polynomial system. In this paper, in particular it is given that up to six limit cycles can bifurcate from fine focus of some examples of cubic system (B). Also we have given one example of quadratic systemwith at most one limit cycle.

Abstract:
In this paper, we study the bifurcation of limit cycles in Lienard systems of the form dot(x)=y-F(x), dot(y)=-x, where F(x) is an odd polynomial that contains, in general, several free parameters. By using a method introduced in a previous paper, we obtain a sequence of algebraic approximations to the bifurcation sets, in the parameter space. Each algebraic approximation represents an exact lower bound to the bifurcation set. This sequence seems to converge to the exact bifurcation set of the system. The method is non perturbative. It is not necessary to have a small or a large parameter in order to obtain these results.

Abstract:
We give an effective method for controlling the maximum number of limit cycles of some planar polynomial systems. It is based on a suitable choice of a Dulac function and the application of the well-known Bendixson-Dulac Criterion for multiple connected regions. The key point is a new approach to control the sign of the functions involved in the criterion. The method is applied to several examples.

Abstract:
Lienard systems of the form $\ddot{x}+\epsilon f(x)\dot{x}+x=0$, with f(x) an even continous function, are considered. The bifurcation curves of limit cycles are calculated exactly in the weak ($\epsilon\to 0$) and in the strongly ($\epsilon\to\infty$) nonlinear regime in some examples. The number of limit cycles does not increase when $\epsilon$ increases from zero to infinity in all the cases analyzed.

Abstract:
In recent papers we have introduced a method for the study of limit cycles of the Lienard system: dot{x}=y-F(x), dot{y}=-x, where F(x) is an odd polynomial. The method gives a sequence of polynomials R_n(x), whose roots are related to the number and location of the limit cycles, and a sequence of algebraic approximations to the bifurcation set of the system. In this paper, we present a variant of the method that gives very important qualitative and quantitative improvements.

Abstract:
A theorem on the existence of exactly $N$ limit cycles around a critical point for the Lienard system $\ddot{x}+f(x) \dot{x}+g(x) =0$ is proved. An alogrithm on the determination of a desired number of limit cycles for this system has been considered which might become relevant for a Lienard system with incomplete data.