Lienard Equation Research Articles

Stability to vector Liénard equation with constant deviating argument

In this paper, we consider the vector Lienard equation with the constant deviating argument, τ>0, $$X''(t) + F\bigl(X(t),X'(t) \bigr)X'(t) + H\bigl(X(t - \tau)\bigr) = P(t) $$ in two cases: (i) P(.)≡0, (ii) P(.)≠0. Based on the Lyapunov–Krasovskii functional approach, the asymptotic stability of the zero solution and the boundedness of all solutions are discussed for these cases. We give an example to illustrate the theoretical analysis made in this work and to show the effectiveness of the method utilized here.

Lazer-Leach Type Conditions on Periodic Solutions of Liénard Equation with a Deviating Argument at Resonance

We study the existence of periodic solutions of Liénard equation with a deviating argumentx′′+f(x)x'+n2x+g(x(t-τ))=p(t),wheref,g,p:R→Rare continuous andpis2π-periodic,0≤τ<2πis a constant, andnis a positive integer. Assume that the limitslimx→±∞g(x)=g(±∞)andlimx→±∞F(x)=F(±∞)exist and are finite, whereF(x)=∫0x‍f(u)du. We prove that the given equation has at least one2π-periodic solution provided that one of the following conditions holds:2cos(nτ)[g(+∞)-g(-∞)]≠∫02π‍p(t)sin(θ+nt)dt, for allθ∈[0,2π],2ncos(nτ)[F(+∞)-F(-∞)]≠∫02π‍p(t)sin(θ+nt)dt, for allθ∈[0,2π],2[g(+∞)-g(-∞)]-2nsin(nτ)[F(+∞)-F(-∞)]≠∫02π‍p(t)sin(θ+nt)dt, for allθ∈[0,2π],2n[F(+∞)-F(-∞)]-2sin(nτ)[g(+∞)-g(-∞)]≠∫02π‍p(t)sin(θ+nt)dt, for allθ∈[0,2π].

On necessary conditions for global asymptotic stability of equilibrium for the Liénard equation

In [1], necessary and sufficient conditions for the global asymptotic stability of the trivial solution of the Lienard equation ẍ+f(x)ẋ+g(x)=0, g(0)=0, were obtained under the condition $$x\int_0^x {f(s)ds \geqslant 0.} $$ In [1], the following problem was also posed: To determine whether condition (A) is a necessary condition for the global asymptotic stability of the trivial solution of the Lienard equation. The present paper answers this question, and the answer is negative, i.e., condition (A) is not a necessary condition.

Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems

This paper is concerned with bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Lienard systems. By analyzing the multiplicities of the zeroes of the slow divergence integrals and their complete unfolding, the upper bounds of canard limit cycles bifurcating from the suitable limit periodic sets through respectively the generic Hopf breaking mechanism, the generic jump breaking mechanism and a succession of the Hopf and jump mechanisms in these polynomial Lienard systems are obtained.

Periodic solutions of Liénard equations with resonant isochronous potentials

In this paper, we study the existence and multiplicity of periodic solutions of Lienard equations $$ x''+f(x)x'+V'(x)+g(x)=p(t), $$ where $V$ is a $2\pi/n$-isochronous potential. When $F(F(x)=\int_0^xf(s)ds)$ and $g$ are bounded, we provide new sufficient conditions to ensure the existence of periodic solutions of this equations. Moreover, we prove the multiplicity of periodic solutions of the given equations under certain bounded conditions by using topological degree method. When $F, g$ satisfy a certain class of unbounded conditions, we also give sufficient conditions to ensure the existence of $2\pi$-periodic solutions of the given equations.

Application of ( G G' )-expansion method to nonlinear Lienard equation

In this paper, we obtain the exact solutions of Lienard equation using ( G'/ G )-expansion method. The solutions obtained here are expressed in hyperbolic functions. Our work is motivated by the fact that the ( G'/ G )-expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in obtained wider set of solutions as special values then some of previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.

Application of ( G G )-expansion method to nonlinear Lienard equation

Application of ( G G )-expansion method to nonlinear Lienard equation

Local bifurcations of critical periods for quartic Lié nard equations with quintic damping

In this article, we study the local bifurcation of critical periods near a nonde-generate center of the quartic Lie nard equation with quintic damping and prove that at most two local critical periods can be produced from either a weak center of finite order or the isochronous center. MSC: 34C05; 34C07.

EXTENSION OF CHICONE'S METHOD FOR PERTURBATION SYSTEMS OF THREE PARAMETERS WITH APPLICATION TO THE LIENARD SYSTEM

In this paper, Chicone's method has been extended for perturbation systems with three parameters. By applying Diliberto's theorem and variational lemma, some conditions for continuation of periodic solutions of a three-parametric autonomous perturbation system has been introduced. Also, the continuation region in the parameter space has been found. As an application of this extension, we applied it to the Lienard system.

Hilbert’s Sixteenth Problem for Polynomial Liénard Equations

This article reports on the survey talk ‘Hilbert’s Sixteenth Problem for Lienard equations,’ given by the author at the Oberwolfach Mini-Workshop ‘Algebraic and Analytic Techniques for Polynomial Vector Fields.’ It is written in a way that it is accessible to a public with heterogeneous mathematical background. The article reviews recent developments and techniques used in the study of Hilbert’s 16th problem where the main focus is put on the subclass of polynomial vector fields derived from the Lierd equations.

On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory

We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Lienard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincare problem for some families is also approached.

Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems

In [1] and [2] upperbounds have been given for the number of large amplitude limit cycles in polynomial Liénard systems of type $(m,n)$ with $m<2n+1$, $m$ and $n$ odd. In the current paper we improve the upperbounds from [1] and [2] by one unity, obtaining sharp results. We therefore introduce the 'method of cloning variables' that might be useful in other cyclicity problems.

Constructing path curve for a class of generalized phase tracks of canard system

A class of generalized phase tracks of canard system is obtained. Firstly, the solutions to the generalized Lienard system are considered. Then the possessed head canard solutions are constructed, and illustrated the examples to construct the canard. Using this same method, we may construct also more extensive canards.

Limit cycle bifurcations of a general Liénard system with polynomial restoring and damping functions

Applying a canonical system with field rotation parameters and using geometric properties of the spirals filling the interior and exterior domains of limit cycles, we solve the limit cycle problem for a general Lienard system with polynomial restoring and damping functions.

New results on the existence of periodic solutions for a higher-order Li é nard type p -Laplacian differential equation

In this paper, by using the continuation theorem of coincidence degree theory, we consider the higher-order Li enard type p-Laplacian differential equation as follows (ϕp(x(m)))(m)+f(x)x′+g(t,x)=e(t). Some new results on the existence of periodic solutions for the previous equation are obtained, which generalize and enrich some known results to some extent from the literature. Copyright © 2011 John Wiley & Sons, Ltd.

Limit Cycles of a Class of Cubic Liénard Equations

In this paper, a class of polynomial Lienard systems $$\begin{array}{ll}&\dot{x}=y-(a_{1}x+a_{2}x^{2}+a_{3}x^{3}),\\&\dot{y}=-(b_{1}x+b_{2}x^2+b_{3}x^3), \end{array}$$ is considered. Some conditions of the existence, non-existence and uniqueness of limit cycles are obtained by using Filippov transformations and Zhang’s theorem. We obtain that the above system has at most one limit cycle surrounding the origin if a1a3 < 0 or b2 = 0. And, one example is given to illustrate that the system can have three limit cycles.

The projective vector field of a kind of three-dimensional quasi-homogeneous system on [formula omitted

In this paper we study the projective vector field Q T of a three-dimensional quasi-homogeneous system with weight ( 1 , 1 , α 3 ) and degree δ = 2 , α 3 ≥ 2 . Projective vector fields Q T of this kind are classified into two types. For one type, Q T has no closed orbit and at most eight singularities, which lead to a global topology of the three-dimensional system. For the other type, Q T has at most ten singularities. In addition, we show a relationship between Q T and a Lienard system of this type. For both of them we obtain some conditions for the existence of limit cycles.

An upper estimate for the number of limit cycles of even-degree Liénard equations in the focus case

We give an upper estimate for the number of limit cycles of the Lienard equations $ \dot{x} $ = y ? F(x), $ \dot{y} $ = ?x of even degree in the case where its unique singular point (0, 0) is a focus. M. Caubergh and F. Dumortier [1] obtained an explicit linear upper estimate for the number of large-amplitude limit cycles of such equations. We estimate the number of mid-amplitude limit cycles of the Lienard equations using the growth-and-zeros theorem proved by Ilyashenko and Yakovenko in [7]. Our estimate depends on four parameters: n, C, a 1, and R. Let $ F(x) = {x^n} + \sum\limits_{i = 1}^{n - 1} {{a_i}{x^i}} $ , where n = 2l is the even degree of the monic polynomial F without a constant term, |a i | < C for all i, so C is the size of a compact subset in the space of parameters, R is the size of the neighborhood of the origin, such that there are at most l limit cycles located outside of this neighborhood, |a 1 | stands for the distance from the equation linearization to the center case in the space of parameters, and 2 ? |a 1 | stands for the distance from the equation linearization to the node case in the space of parameters.

Existence and uniqueness of periodic solutions for a kind of Liénard equation with multiple deviating arguments

In this paper, a kind of Lienard equation with multiple deviating arguments of the form $$x''(t)+f(x(t))x'(t)+\sum_{k=1}^{n}g_{k}(t,x(t-\tau_{k}(t)))=p(t)$$ is considered. By using coincidence degree theory, some criteria are obtained to guarantee the existence and uniqueness of periodic solutions of this equation. The obtained results are also valid and new for the problem discussed in the previous literature.

On the uniform boundedness of solutions of Lienard type equations with multiple deviating arguments

This paper considers a Lienard type equation with multiple variable deviating arguments. We find some sufficient conditions for the solution of this equation to be uniform bounded by utilizing Lyapunov functional approach. An example is also given to show the effectiveness of our result.