Related papers: Limit Cycles of a Quadratic System with Two Parall…
In this paper, by using Picard-Fuchs equations and Chebyshev criterion, we study the bifurcate of limit cycles for quadratic Hamilton system $S^{(2)}$ and $S^{(3)}$: $\dot{x}= y+2axy+by^2$, $\dot{y}=-x+x^2-ay^2$ with $a\in(-\frac{1}{2},1)$,…
This paper deals with the problem of limit cycle bifurcations for piecewise smooth integrable differential systems with four zones. When the unperturbed system has a family of periodic orbits, the first order Melnikov function is derived…
We solve the center-focus problem in a class of piecewise quadratic polynomial differential systems with an invariant straight line. The separation curve is also a straight line which is not invariant. We provide families having at the…
In recent decades, piecewise linear differential systems have attracted considerable attention due to their ability to describe a wide range of phenomena. A central problem, as in the theory of general planar differential systems, is to…
In this paper, we study the number of limit cycles that can bifurcate from a periodic annulus in discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines, such that the…
In this paper we consider the limit cycles of the planar system $$\frac{d}{dt}(x,y)=\mathbf X_n+\mathbf X_m, $$ where $\mathbf X_n$ and $\mathbf X_m$ are quasi-homogeneous vector fields of degree $n$ and $m$ respectively. We prove that…
The already proved Lum-Chua's conjecture says that a continuous planar piecewise linear differential system with two zones separated by a straight line has at most one limit cycle. In this paper, we provide a new proof by using a novel…
This article deals with the study of the number of limit cycles surrounding a critical point of a quadratic planar vector field, which, in normal form, can be written as $x'= a_1 x-y-a_3x^2+(2 a_2+a_5)xy + a_6 y^2$, $y'= x+a_1 y + a_2x^2+(2…
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…
By introducing a max-plus dynamical system having limit cycles, we discuss their periodicity, especially the number of discrete states in them. We also find that quasi-periodic cycles exist depending on the bifurcation parameter in the…
In this paper, we consider the realization of configuration of limit cycles of piecewise linear systems on the plane. We show that any configuration of Jordan curves can be realized by a discontinuous piecewise linear system with two zones…
We investigate the maximum number of limit cycles bifurcating from the period annulus of a family of cubic polynomial differential centers when it is perturbed inside the class of all cubic piecewise smooth polynomials. The family…
We prove that coexistence of two limit cycles with disjoint interior can not occur in certain quadratic systems
The main purpose of this article is to study from the geometric point of view the problem of limit cycles bifurcation of perturbed completely integrable systems.
For planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non existence of periodic orbits not contained in this given…
This paper studies the family of piecewise linear differential systems in the plane with two pieces separated by a cubic curve. By analyzing the obtained first order Melnikov function, we give an upper bound of the number of limit cycles…
This paper studies the family of piecewise linear differential systems in the plane with two pieces separated by a switching curve $y=x^{m}$, where $m>1$ is an arbitrary positive. By analysing the first order Melnikov function, we give an…
In this paper we are concerned with determining lower bounds of the number of limit cycles for piecewise polynomial holomorphic systems with a straight line of discontinuity. We approach this problem with different points of view: study of…
We apply the averaging theory of high order for computing the limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. These discontinuous piecewise differential systems are formed by two…
We prove that the cyclicity of a quadratic slow-fast integrable system of Darboux type with a double heteroclinic loop is finite and uniformly bounded.