Narrow Results

Addressing the weakened Hilbert's 16th problem or the Hilbert–Arnold problem, this paper gives an upper bound B(n) ≤ 7n + 5 for the number of zeros of the Abelian integrals for a class of Liénard systems. We proved the main result using the Picard–Fuchs equations and the algebraic structure of the integrals.

In this paper, we study the number of limit cycles of a kind of polynomial Liénard system with a nilpotent cusp and obtain some new results on the lower bound of the maximal number of limit cycles for this kind of systems.

In this paper, we mainly discuss Hopf bifurcation for planar nonsmooth general systems and Liénard systems with foci of parabolic–parabolic (PP) or focus–parabolic (FP) type. For the bifurcation near a focus, when the focus is kept fixed under perturbations we prove that there are at most k limit cycles which can be produced from an elementary weak focus of order 2k + 2 (resp.k + 1)(k ≥ 1) if the focus is of PP (resp. FP) type, and we present the conditions to ensure these upper bounds are achievable. For the bifurcation near a center, the Hopf cyclicicy is studied for these systems. Some interesting applications are presented.

As we know, Liénard system is an important model of nonlinear oscillators, which has been widely studied. In this paper, we study the Hopf bifurcation of an analytic Liénard system by perturbing a nilpotent center. We develop an efficient method to compute the coefficients b_l appearing in the expansion of the first order Melnikov function by finding a set of equivalent quantities B_2l+1 which are able to calculate directly and can be used to study the number of small-amplitude limit cycles of the system. As an application, we investigate some polynomial Liénard systems, obtaining a lower bound of the maximal number of limit cycles near a nilpotent center.

We study a polynomial Liénard system depending on three parameters a, b, c and exhibiting the following properties: (i) The origin is the unique equilibrium for all parameters. (ii) If a crosses zero, then the origin changes its stability, and Andronov–Hopf bifurcation arises. We consider a as control parameter and investigate the dependence of Andronov–Hopf bifurcation on the "unfolding" parameters b and c. We establish and describe analytically the existence of surfaces and curves located near the origin in the parameter space connected with the existence of small-amplitude limit cycles of multiplicity two and three (existence of degenerate Andronov–Hopf bifurcation).

In this paper, we study the number of limit cycles of some polynomial Liénard systems. Using the methods of the Melnikov functions and Hopf, homoclinic and heteroclinic bifurcation theory, we prove that H(2, 5) ≥ 4, H(6, 5) ≥ 7, H(10, 5) ≥ 11.

In this article, we study the limit cycle bifurcation of a Liénard system of type (5,4) with a heteroclinic loop passing through a hyperbolic saddle and a nilpotent saddle. We study the least upper bound of the number of limit cycles bifurcated from the periodic annulus inside the heteroclinic loop by a new algebraic criterion. We also prove at least three limit cycles will bifurcate and six kinds of different distributions of these limit cycles are given. The methods we use and the results we obtain are new.

In this paper, we consider a class of piecewise smooth Liénard systems. We classify the unperturbed system into three types and study the bifurcation of limit cycles under perturbations. By studying the expansions of the first order Melnikov function, we give some new results on the number of limit cycles in homoclinic bifurcations.

In this paper, we first present some general theorems on bifurcation of limit cycles in near-Hamiltonian systems with a nilpotent saddle or a nilpotent cusp. Then we apply the theorems to study the number of limit cycles for some polynomial Liénard systems with a nilpotent saddle or a nilpotent cusp, and obtain some new estimations on the number of limit cycles of these systems.

We will consider two special families of polynomial perturbations of the linear center. For the resulting perturbed systems, which are generalized Liénard systems, we provide the exact upper bound for the number of limit cycles that bifurcate from the periodic orbits of the linear center.

In this work, we study the Abelian integral $I(h)$ corresponding to the following Liénard system,

This paper is concerned with the bifurcation problem of limit cycles by perturbing a piecewise Hamiltonian system with a double homoclinic loop. First, the derivative of the first Melnikov function is provided. Then, we use it, together with the analytic method, to derive the asymptotic expansion of the first Melnikov function near the loop. Meanwhile, we present the first coefficients in the expansion, which can be applied to study the limit cycle bifurcation near the loop. We give sufficient conditions for this system to have $14$ limit cycles in the neighborhood of the loop. As an application, a piecewise polynomial Liénard system is investigated, finding six limit cycles with the help of the obtained method.

In this paper, we investigate the existence problem of periodic orbits for a planar Liénard system, whose solution mappings are interrupted by abrupt changes of state. We first present the geometrical properties of solutions for the planar Liénard system with state impulses, then by using Bendixson theorem of impulsive differential equations and successor function method, several new criteria on the closed orbits and discontinuous periodic orbits are established in the impulsive Liénard system.

We report the existence of bursting oscillations and mixed-mode oscillations in a Liénard system when it is driven externally by a sinusoidal force. The bursting oscillations transit from a periodic phase to spiking trains through chaotic windows, as the control parameter is varied. The mixed-mode oscillations appear via an alternate sequence of periodic and chaotic states, as well as Farey sequences. The primary and their associated secondary mixed-mode oscillations are detected for the suitable choices of system parameters. Additionally, the system is also found to possess multistability nature. Our investigations involve numerical simulations as well as real time hardware experiments using a simple analog electronic circuit. The experimental observations are in conformation with numerical results.

In this paper, we consider a class of Liénard systems, described by $ẍ+f(x)ẋ+g(x)=0$, with $Z_2$ symmetry. Particular attention is given to the existence of small-amplitude limit cycles around fine foci when $g(x)$ is an odd polynomial function and $f(x)$ is an even function. Using the methods of normal form theory, we found some new and better lower bounds of the maximal number of small-amplitude limit cycles in these systems. Moreover, a complete classification of the center conditions is obtained for such systems.

This paper deals with planar piecewise linear slow–fast Liénard differential systems with three zones separated by two vertical lines. We show the existence and uniqueness of canard limit cycles for systems with a unique singular point located in the middle zone.

In this paper, we consider some piecewise smooth Liénard systems and study the Hopf bifurcation of limit cycles from the origin after a perturbation. We define the cyclicity $N(k,l,m,n)$ of the piecewise smooth Liénard systems at the origin and denote $N_{1,1}(m,n)$ for fixing $k=l=1$. Then we obtain $N_{1,1}(1,n)=n,$$n\ge 1$; $N_{1,1}(m,1)=m,$$m\ge 1$ and $N_{1,1}(m,n)=m+n-2,$$m=n\ge 2$.

In this paper, we investigate the dynamical behavior of the planar polynomial system which describes the evolution of the isotropic star. It is shown that the system has no global $C^{\infty }$ and no global analytic first integrals. It has an invariant algebraic curve with algebraic multiplicity $1$ and an exponential factor that comes from the multiplicity of the infinite invariant straight line. It is proved that the system can be changed into the Liénard system. By using a dominant balance analysis, we prove the system has a general solution which eventuates to a finite-time singularity. Finally, we prove the trajectories of the vector field associated with the planar system of the isotropic star do not create a trajectory manifold which means there is no pseudo-Riemannian metric $g$ in the sense of trajectory metric such that the trajectories of the isotropic star system be geodesic.