Lienard Equation Research Articles

Homoclinic and Heteroclinic Orbits for a Liénard System

A 2D dynamical system exhibiting a double-zero bifurcation with symmetry of order two is considered. This bifurcation involves the presence in the parameter space of a curve corresponding either to double homoclinic or to heteroclinic bifurcations. In this paper we derive second order approximations for the homoclinic orbits and for the curve of homoclinic bifurcation values considering the system truncated up to five order terms and parameter-dependent coefficients. These approximations were obtained using the regular perturbation method. These formulae are applied to a Lienard system, which develops a double-zero bifurcation with symmetry of order two for some parameters values. Second order approximations for the heteroclinic orbits of this system are also given. The analytical results are very accurate and they are in good accordance with the numerical ones.

Proof of Artés–Llibre–Valls's conjectures for the Higgins–Selkov and the Selkov systems

The aim of this paper is to prove Artés–Llibre–Valls's conjectures on the uniqueness of limit cycles for the Higgins–Selkov system and the Selkov system. In order to apply the limit cycle theory for Liénard systems, we change the Higgins–Selkov and the Selkov systems into Liénard systems first. Then, we present two theorems on the nonexistence of limit cycles of Liénard systems. At last, the conjectures can be proven by these theorems and some techniques applied for Liénard systems.

Positive periodic solution to indefinite singular Liénard equation

In this paper, we investigate the existence of a positive periodic solution for the following Lienard equation with a indefinite singularity $$\begin{aligned} x''+f(x)x'+\frac{b(t)}{x}=p(t), \end{aligned}$$ where $$b\in C({\mathbb {R}},{\mathbb {R}})$$ is a T-periodic sign-changing function. The novelty of the present article is that for the first time we show that a indefinite singularity enables the achievement of a new existence criterion of positive periodic solutions through a application of a topological degree theorem by Mawhin. Recent results in the literature are generalized and significantly improved, and we give the existence interval of a positive periodic solution of this equation. At last, an example is given to show applications of the theorems.

Quartic Liénard Equations with Linear Damping

In this paper we prove that the quartic Lienard equation with linear damping $$\{\dot{x}=y,\dot{y}=-(a_0+x)y-(b_0+b_1x+b_2x^2+b_3x^3+x^4)\}$$ can have at most two limit cycles, for the parameters kept in a small neighborhood of the origin $$(a_0,b_0,b_1,b_2,b_3)=(0,0,0,0,0)$$ . Near the origin in the parameter space, the Lienard equation is of singular type and we use singular perturbation theory and the family blow up. To study the limit cycles globally in the phase space we need a suitable Poincare–Lyapunov compactification.

Attractive singularity problems for superlinear Liénard equation

In this paper, we consider the following quasilinear Lienard equation with a singularity $$\begin{aligned} (\phi _p(x'(t)))'+f(x(t))x'(t)+g(t,x(t))=e(t), \end{aligned}$$where g has a attractive singularity at the origin and satisfies superlinear condition at $$x=+\infty $$. By using Manasevich–Mawhin continuous theorem, we prove that this equation has at least one positive T-periodic solution. We solve a difficulty to estimate it a priori bounds of a periodic solution for quasilinear Lienard equation in the case that superlinear condition. At last, example and numerical solution (phase portrait and time series portrait of the positive periodic solution of example) are given to show applications of the theorem.

New $\phi^{6}$ ϕ 6 -model expansion method and its applications to the resonant nonlinear Schrödinger equation with parabolic law nonlinearity

With the aid of symbolic computation, the new $\phi^{6}$ -model expansion method is applied, in this article, for the first time to the resonant nonlinear Schrodinger equation with parabolic law nonlinearity to find families of Jacobi elliptic function solutions. In particular, when the modulus of the Jacobi elliptic functions tends to one or to zero, we can get the hyperbolic and trigonometric function solutions, respectively. This new method presents a wider applicability for handling the nonlinear partial differential equations. Comparison of our new results with the well-known results are given. At the end of this paper, we use the solutions of the Lienard equation to find more different solutions for the proposed resonant nonlinear Schrodinger equation mentioned above.

On the Study Of Asymptotically Almost Periodic Solutions of a Class of Impulsive Population Models

Based on the Mawhin continuous theorem, the existence of strictly positive asymptotically almost periodic solutions of a class of impulsive population models is studied. The conclusion generalizes the conclusion of the existing literatures. Since the Mawhin continuous theorem is only used to prove the existence of periodic solutions or almost periodic solutions of equations (for example:impulsive differential equation, functional differential equation, integral equation, Lienard equation, P-Laplacian equation), the main result is innovative.

Qualitative Analysis of Crossing Limit Cycles in a Class of Discontinuous Liénard Systems with Symmetry

In this paper, we investigate some qualitative properties of crossing limit cycles for a discontinuous symmetric Lienard system with two zones separated by a straight line. In each zone, it is a smooth Lienard system. Firstly, by Poincare mapping method and geometrical analysis, we provide two criteria concerning the existence, uniqueness and stability of a crossing limit cycle. Secondly, we consider the position problem of the unique crossing limit cycle. Several lemmas are given to obtain an explicit upper bound of amplitude of the limit cycle. Finally, an application to van der Pol model with discontinuous vector field is given, and Matlab simulations are presented to illustrate the obtained theoretical results.

On the independent perturbation parameters and the number of limit cycles of a type of Liénard system

In this paper, we study a type of polynomial Liénard system of degree m(m≥2) with polynomial perturbations of degree n. We prove that the first order Melnikov function of such system has at most n+1−[n+1m+1] independent perturbation parameters which can be used to simplify this kind of systems. As an application, we study a type of Lienard systems for m=4,n=19,28 and obtain the new lower bounds of the maximal number of limit cycles.

Autonomous Noether Boundary-Value Problems not Solved with Respect to the Derivative

In monographs of N. V. Azbelev, A. M. Samoilenko, and A. A. Boichuk, constructive methods of study of Noether boundary-value problems have been developed. These methods continue the investigation of periodic problems stated by H. Poincare, A. M. Lyapunov, N. M. Krylov, N. N. Bogolyubov, I. G. Malkin, and O. Veivoda by the methods of small parameter. We propose an improved scheme of study of autonomous Noether boundary-value problems for nonlinear systems in critical cases. In the case of multiple roots of the equation for generating constants, we obtain sufficient conditions of existence of solutions to an autonomous boundary-value problem not solved with respect to the derivative. The effectiveness of the scheme proposed is illustrated by an example of the periodic problem for the Lienard equation.

Existence and uniqueness of limit cycles in planar system of Liénard type

This paper is a study of one of the most beautiful phenomena in dynamical systems limit cycle. In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory having the property that at least one other trajectory spirals into it as time approaches either positive or negative infinity. In this paper, existence and uniqueness of limit cycles for a generalized Lienard system will be studied. Moreover, some examples will be presented to illustrate our results.

The $$\left( \frac{\boldsymbol{G}^{\prime }}{\boldsymbol{G}},\frac{\boldsymbol{1}}{\boldsymbol{G}}\right)$$ G ′ G , 1 G -expansion method and its applications for constructing many new exact solutions of the higher-order nonlinear Schrödinger equation and the quantum Zakharov–Kuznetsov equation

In this article, we apply the two variable $$\left( \frac{G^{\prime }}{G}, \frac{1}{G}\right)$$ -expansion method with the aid of symbolic computation to construct many new exact solutions for two higher-order nonlinear partial differential equatuions (PDEs) namely, the higher-order nonlinear Schro dinger (NLS) equation with derivative non-kerr nonlinear terms describing pulse of the propagation beyond ultrashort range in optical communication systems and the higher-order nonlinear quantum Zakharov–Kuznetsov (QZK) equation which arises in quantum magneto plasma . Also, based on Lienard equation, we find many other diffrent new soliton solutions of the above NLS equation. Soliton solutions, periodic solutions, rational functions solutions and Jacobi elliptic functions solutions are obtained. Comparing our new solutions obtained in this article with the well-known solutions are given.

Anisotropic Geodesic Fluid in Non-Comoving Spherical Coordinates

We start with a recently introduced spherically symmetric geodesic fluid model (arXiv: 1601.07030) whose energy-momentum tensor in the comoving frame is dust-like with nontrivial energy flux. In the non-comoving energy frame (vanishing energy flux) the same EMT contains besides dust only radial pressure. We present Einstein's equations together with the matter equations in static spherically symmetric coordinates. These equations are self-contained (four equations for four unknowns). We solve them analytically except for a resulting nonlinear ordinary differential equation (ODE) for the gravitational potential. This ODE can be rewritten as a Lienard differential equation which, however, may be transformed into a rational Abel differential equation of the first kind. Finally we list some open mathematical problems and outline possible physical applications (galactic halos, dark energy stars) and related open problems.

Fink type conjecture on affine-periodic solutions and Levinson's conjecture to Newtonian systems

This paper concerns the existence of affine-periodic solutions for differential systems (including functional differential equations) and Newtonian systems with friction. This is a kind of pattern solutions in time-space, which may be periodic, anti-periodic, subharmonic or quasi periodic corresponding to rotation motions. Fink type conjecture is verified and Lyapunov's methods are given. These results are applied to study gradient systems and Newtonian (including Rayleigh or Lienard) systems. Levinson's conjecture to Newtonian systems is proved.

Study of the dynamics of a Lienard system

Study of the dynamics of a Lienard system

Limit cycles in Liénard systems with saturation

Abstract In this paper we present an extension of existing results on limit cycles for Lienard systems and formulate sufficient conditions for existence and uniqueness of limit cycles for Lienard systems with non-differentiable vector fields. As an application we consider the example of a linear systems with the saturation nonlinearity.

Simplified Liénard Equation by Homotopy Analysis Method

The homotopy analysis method (HAM) has been shown to approximate solutions to nonlinear problems of various types. The method is useful because of the flexibility in defining the auxiliary parameter $$\hbar $$ and that no assumption concerning relative size of parameters is required. We use the HAM to approximate frequency and solution of a conservative nonlinear oscillator which has many applications including lasers, epidemics, and microparasites. The oscillator is modeled by the simplified Lienard Equation $$\ddot{x}+\epsilon {\dot{x}}x+x=0$$ . We compare our results for the weakly-nonlinear case to those gained from a more traditional perturbation technique, the Poincare–Lindstedt method. We then show results from the HAM in the strongly-nonlinear regime and discuss some benefits and limitations of the method including conditions for convergence.

Studies of the Petrov Module for a Family of Generalized Liénard Integrable Systems

In this article we use the Lambert function in order to study a family of integrable generalized Lienard equations Xf which display a center. We first prove a conjugation lemma inside a continuum of nested periodic orbits. Then we deduce an explicit operator of Gelfand–Leray associated with the Hamiltonian of equation Xf. Afterwards, we provide a generating family for the associated Petrov module. Finally, by using the Lambert function, we study the monotonicity of the Abelian integral of this generating family’s elements.

The domain of existence of a limit cycle of Liénard system

In this paper, the Lienard system of differential equations is considered. The domain of location of the limit cycle of this system is estimated. The hypothesis under which the Lienard system has the unique stable limit cycle are also stated.

Estimate for the amplitude of the limit cycle of the Liénard equation

We consider the nonlinear Lienard equation \(\ddot x\left( t \right) + f\left( x \right)\dot x\left( t \right) + g\left( x \right) = 0\). Lienard obtained sufficient conditions on the functions f(x) and g(x) under which this equation has a unique stable limit cycle. Under additional conditions, we prove a theorem that permits one to estimate the amplitude (the maximum value of x) of this limit cycle from above. The theorem is used to estimate the amplitude of the limit cycle of the van der Pol equation \(\ddot x\left( t \right) + \mu \left[ {{x^2}\left( t \right) - 1} \right]\dot x\left( t \right) + x\left( t \right) = 0\).