Related papers: Small limit cycles bifurcating in pendulum systems…
We consider perturbed pendulum-like equations on the cylinder of the form $ \ddot x+\sin(x)= \varepsilon \sum_{s=0}^{m}{Q_{n,s} (x)\, \dot x^{s}}$ where $Q_{n,s}$ are trigonometric polynomials of degree $n$, and study the number of limit…
In this paper, the general planar piecewise smooth Hamiltonian system with period annulus around the center at the origin is considered. We obtain the expressions for the first order and the second order Melnikov functions of it's general…
This paper deals with the problem of limit cycle bifurcations for a piecewise near-Hamilton system with four regions separated by algebraic curves $y=\pm x^2$. By analyzing the obtained first order Melnikov function, we give an upper bound…
In this paper, we study limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a homoclinic loop around the origin. By using the…
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 $a\in(-\frac{1}{2},1)$,…
In this paper, we study the bifurcate of limit cycles for Bogdanov-Takens system($\dot{x}=y$, $\dot{y}=-x+x^{2}$) under perturbations of piecewise smooth polynomials of degree $2$ and $n$ respectively. We bound the number of zeros of first…
In the present paper, we study the number of zeros of the first order Melnikov function for piecewise smooth polynomial differential system, to estimate the number of limit cycles bifurcated from the period annulus of quadratic isochronous…
This paper contains two parts. In the first part, we shall study the Abelian integrals for Zoladek's example [13], in which it is claimed the existence integrals of 11 small-amplitude limit cycles around a singular point in a particular…
In this paper, we deal with limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order 2 by studying expansions of the first order Melnikov functions near the loop and coefficients in these expansions. More…
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…
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 consider three examples of weekly perturbed centers which do not have {\it geometrical equivalence}: a linear center, a degenerate center and a non-hamiltonian center. In each case the number and amplitude of the limit cycles emerging…
In this paper, we present a method of higher-order analysis on bifurcation of small limit cycles around an elementary center of integrable systems under perturbations. This method is equivalent to higher-order Melinikov function approach…
In this paper, we study limit cycle bifurcations for a class of general near-Hamiltonian systems near a heteroclinic loop with an elementary saddle and a nilpotent saddle. Firstly, we consider the behaviors of the unperturbed system,…
In this paper, we study the number of limit cycles that can bifurcating from a periodic annulus in discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines. We prove…
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…
In this paper, we perturb the global center of the planar polynomial vector fields $\mathcal{X}(x,y)=(-y(x^2+a^2),x(x^2+a^2))$ ($a\neq0$) inside cubic piecewise smooth polynomials with switching line $y=0$. By using average function of…
In this paper, we consider the bifurcation of limit cycles for generic L-V system ($\dot{x}=y+x^2-y^2\pm\frac{4}{\sqrt{3}}xy,~\dot{y}=-x+2xy$) and B-T system ($\dot{x}=y,~\dot{y}=-x+x^2$) under perturbations of piecewise smooth polynomials…
Limit cycles of planar polynomial vector fields have been an active area of research for decades; the interest in periodic-orbit related dynamics comes from Hilbert's 16th problem and the fact that oscillatory states are often found in…
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…