Related papers: On the number of hyperelliptic limit cycles of Li\…

Lienard systems are very important mathematical models describing oscillatory processes arising in applied sciences. In this paper, we study polynomial Lienard systems of arbitrary degree on the plane, and develop a new method to obtain a…

In this paper, we study the maximum number of limit cycles for the piecewise smooth system of differential equations $\dot{x}=y, \ \dot{y}=-x-\varepsilon \cdot (f(x)\cdot y +{\rm sgn}(y)\cdot g(x))$. Using the averaging method, we were able…

We will consider two special families of polynomial perturbations of the linear center. For the resulting perturbed systems, which are generalized Li\'enard systems, we provide the exact upper bound for the number of limit cycles that…

We focus on the second part of Hilbert's 16th problem and provide an upper bound on the number of limit cycles that a polynomial, differential, planar system may have, depending exclusively on the degree $n$ of the system. Such a bound…

This paper is concerned with the limit cycles for planar semi-quasi-homogeneous polynomial systems. We give some explicit criteria for the nonexistence and existence of periodic orbits. Let $N=N(p,q,m,n)$ be the maximum number of limit…

In this paper, we study the problem of limit cycle bifurcation in two piecewise polynomial systems of Li\'enard type with multiple parameters. Based on the developed Melnikov function theory, we obtain the maximum number of limit cycles of…

An upper bound for the number of Hamiltonian cycles of symmetric diagraphs is established first in this paper, which is tighter than the famous Minc's bound and the Br$\acute{e}$gman's bound. A transformation on graphs is proposed, so that…

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…

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…

For a given natural number $n$, the second part of Hilbert's 16th Problem asks whether there exists a finite upper bound for the maximum number of limit cycles that planar polynomial vector fields of degree $n$ can have. This maximum number…

This paper studies the number of limit cycles that may bifurcate from an equilibrium of an autonomous system of differential equations. The system in question is assumed to be of dimension $n$, have a zero-Hopf equilibrium at the origin,…

We give an explicit upper bound for a number of limit cycles of the Li\'enard equation $\dot{x}=y-F(x)$, $\dot{y}=-x$ of even degree in the case its unique singular point $(0,0)$ is a focus.

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.

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…

We Prove That The Uniform Upper Bound for the Number Of Limit Cycles Of The Lienard Equation of Degree 4 Can be equals to 2. Further We Suggest to Embedding Planar Lienard Equations In Higher Dimension and Present question of completly…

Let $\mathcal{H}(n)$ be the maximum number of limit cycles that a planar polynomial vector field of degree $n$ can have. In this paper we prove that $\mathcal{H}(n)$ is realizable by structurally stable vector fields with only hyperbolic…

This paper presents new results on the limit cycles of a Li\'enard system with symmetry allowing for discontinuity. Our results generalize and improve the results in [33,34]. The results in [34] are only valid for the smooth system. We…

We study the number of limit cycles that a planar polynomial vector field can have as a function of its number $m$ of monomials. We prove that the number of limit cycles increases at least quadratically with $m$ and we provide good lower…

We study limit cycles in piecewise complex systems with switching manifold $\mathbb{S}^1$. Using M\"obius transformations we establish an equivalence between circular and straight-line discontinuities that preserves periods, stability, and…

The second part of Hilbert's 16th problem concerns determining the maximum number $H(m)$ of limit cycles that a planar polynomial vector field of degree $m$ can exhibit. A natural extension to the three-dimensional space is to study the…