Lienard Equation Research Articles

On the Homotopy-First Integral Method for Non-conservative Oscillators

This paper presents a ready-to-use formula for determining the number and approximate location of periodic orbits in second-order Lienard systems. As a result of the exact closed-form derived in [16], in which an ordinary differential equation (ODE) must be solved to determine the existence and location of periodic orbits for general non-conservative oscillators, a homotopy functional is defined for Lienard-type systems. This provides a closed-form and ready-to-use polynomial formula with roots as an approximation of the periodic orbit's amplitude. In addition, some examples are analyzed, along with conclusions and future plans.

S-asymptotically ω-Periodic Solutions of generalized Liénard

The S-asymptotically ω-periodic functions are a continuous and bounded functions from the real axis to a Banach space that converges to a periodic functions as tends to infinity. Starting from the zero solution, we prove in this work the existence and uniqueness of S-asymptotically ω-periodic solution of generalized Liénard's differential Equation. We stady after that the regular dependence of this solution with a certain parameter in Banach space, present in our equation, and with the forcing term hwo possesses a similar nature as the later. For this, our approach will be to use a perturbation method around an equilibrium. More precisely when the forcing is a Sasymptotically ω-periodic function we study the differentiable dependence of the S-asymptotically ω-periodic solution of Liénard equation. In this study, we changed our initial objective, which was to search for na S asymptotically ω-periodic solution for our Lienard equation, a problem relating to dynamical systems, towards an approach based on functional analysis. Concretely, we adopted a strategy consisting of using the implicit function theorem on a specific operator that we defined in our workspaces. This approach allowed us to achieve the objective stated in our main theorem. To realize our aim, we use the Nemytskii operators (also called superposition operators) and state some properties on these operators. We have also extended the well-established result on the almost periodic function of the derivative of an almost periodic function to the context of S-asymptotically ω-periodic cases. Finally, and to close our work, we give a corollary which presents a particular case of our main theorem.

An improvement of Massera's theorem for the Lienard equation

An improvement of Massera's theorem for the Lienard equation

Geometry of non-standard Hamiltonian structures of Liénard equations and contact structure

We start with a self-contained brief review of the construction of non-standard Lagrangian and Hamiltonian structures for the Liénard equations satisfying Chiellini condition, we apply it to a special polynomial class of Liénard equation and explore the integrability structure. Then after a brief exposition of the contact geometry and its connection with the non-standard Hamiltonian structures, we describe the time evolution of the contact Hamiltonian and Liénard equation. We present the formulation of the Liénard equation in terms of General Equation for the Non-Equilibrium Reversible–Irreversible Coupling (GENERIC) method and also study the gradient-type flow. Finally, we describe the generalized Liénard equation by using the conformal Hamiltonian mechanics and illustrate our construction using spatio-temporal and autocatalysis systems.

The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron

This paper proposes an approximation algorithm based on the Legendre and Chebyshev artificial neural network to explore the approximate solution of fractional Lienard and Duffing equations with a Caputo fractional derivative. These equations show the oscillating circuit and generalize the spring–mass device equation. The proposed approach transforms the given nonlinear fractional differential equation (FDE) into an unconstrained minimization problem. The simulated annealing (SA) algorithm minimizes the mean square error. The proposed techniques examine various non-integer order problems to verify the theoretical results. The numerical results show that the proposed approach yields better results than existing methods.

Some Qualitative Properties of Solutions For Multi-Delay Nonautonomous Stochastic Lienard Equation

Some Qualitative Properties of Solutions For Multi-Delay Nonautonomous Stochastic Lienard Equation

The Extended Non-Elementary Amplitude Functions as Solutions to the Damped Pendulum Equation, the Van der Pol Equation, the Damped Duffing Equation, the Lienard Equation and the Lorenz Equations

The Extended Non-Elementary Amplitude Functions as Solutions to the Damped Pendulum Equation, the Van der Pol Equation, the Damped Duffing Equation, the Lienard Equation and the Lorenz Equations

On a Hypothetical Model with Second Kind Chebyshev’s Polynomial–Correction: Type of Limit Cycles, Simulations, and Possible Applications

In this article, we explore a new extended Lienard-type planar system with “corrections” of the second kind Chebyshev’s polynomial Un. The number and type of limit cycles are also studied. The discussion on the y(t)—component of the solution of the Lienard system is connected to searching for the solution of the synthesis of filters and electrical circuits. Numerical experiments, depicting our outcomes using CAS MATHEMATICA, are presented.

Uniform-ultimate boundedness of solutions to vector lienard equation with delay

Uniform-ultimate boundedness of solutions to vector lienard equation with delay

Sparse Identification of Variable Star Dynamics

Variable stars play a crucial role as standard candles and provide valuable insights into stellar physics. They can be modeled either through fully fledged hydrodynamical simulations or analytically as systems of coupled differential equations describing the evolution of relevant physical quantities. Typically, such equations are arrived at by simplified physical assumptions concerning the conservation laws governing stellar interiors. Here we apply a data-driven technique—sparse identification of nonlinear dynamics (SINDy)—to automatically learn governing equations from observed light curves. We apply SINDy to 3100 light curves of three different variable types from the Catalina Sky Survey. The success rate depends systematically on variable type, with possible implications for variable star classification; however, it does not obviously depend on amplitude or period. Successful models can be reduced to the generalized Lienard equation . Members of the Lienard class of ordinary differential equations, such as the well-studied van der Pol oscillator, already saw some application to variable star modeling. For a, b = 0 the equation can be solved exactly, and it admits both periodic and nonperiodic solutions. We find a condition on the coefficients of the general equation for the presence of a limit cycle, which is also observed numerically in several instances.

Exact and Sinusoidal Periodic Solutions of Lienard Equation Without Restoring Force

In this paper we study a Lienard equation without restoring force. Although this equation does not satisfy the classical existence theorems, we show, for the first time, that such an equation can exhibit harmonic periodic solutions. As such the usual existence theorems are not entirely adequate and satisfactory to predict the existence of periodic solutions.

Oscillation analysis of conformable fractional generalized Lienard equations

In this study, we investigate the oscillatory properties of solutions of a class of conformable fractional generalized Lienard equations. By using generalized Riccati technique, we present some new oscillation results for the equation. Illustrative examples are also given.

On Approximate Solutions of Nonlinear Boundary-Value Problems by the Newton–Kantorovich Method

We establish necessary and sufficient conditions for the solvability of the nonlinear boundary-value problem in the critical case and develop a scheme for the construction of solutions of this problem. By using the Newton–Kantorovich method, we propose a new iterative scheme for the determination of solutions to the weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case. As examples of application of the constructed iterative scheme, we obtain approximations to the solutions of a periodic boundary-value problem for the Duffing and Lienard equations. To check the accuracy of the established approximations to the solutions of the periodic boundary-value problem for the Duffing and Lienard equations, we use discrepancies in the original equations.

Double canard cycles in singularly perturbed planar systems

We consider double canard cycles including two canards in singularly perturbed planar systems with two canard points. Previous work studied the complex oscillations including relaxation oscillations and canard cycles in singularly perturbed planar systems with one-parameter layer equations, which have precisely one canard point, two jump points or one canard point and one jump point. Based on the normal form theory, blow-up technique and Melnikov theory, we investigate double canard cycles induced by two Hopf breaking mechanisms at two non-degenerate canard points. Finally, we apply the obtained results to a class of cubic Lienard equations with quadratic damping.

Stability analysis for the Lienard equation with discontinuous coefficients

A nonlinear mechanical system, whose dynamics is described by a vector ordinary differential equation of the Lienard type, is considered. It is assumed that the coefficients of the equation can switch from one set of constant values to another, and the total number of these sets is, in general, infinite. Thus, piecewise constant functions with infinite number of break points on the entire time axis, are used to set the coefficients of the equation. A method for constructing a discontinuous Lyapunov function is proposed, which is applied to obtain sufficient conditions of the asymptotic stability of the zero equilibrium position of the equation studied. The results found are generalized to the case of a nonstationary Lienard equation with discontinuous coefficients of a more general form. As an auxiliary result of the work, some methods for analyzing the question of sign-definiteness and approaches to obtaining estimates for algebraic expressions, that represent the sum of power-type terms with non-stationary coefficients, are developed. The key feature of the study is the absence of assumptions about the boundedness of these non-stationary coefficients or their separateness from zero. Some examples are given to illustrate the established results.

Factorization method for some inhomogeneous Lienard equations

The factorization of inhomogeneous Li\'enard equations is performed showing that through the factorization conditions involved in the method one can obtain forcing terms for which closed-form solutions exist. Because of the reduction of order feature of factorization, the solutions are simultaneously solutions of first-order differential equations with polynomial nonlinearities. Several illustrative examples of such solutions are presented, generically having rational parts and consequently singularities.

Damping parameters indentification of Lienard polynomial system

In many applications of physics, biology, and other sciences, an approach based on the concept of model equations is used as an approximate model of complex nonlinear processes. The basis of this concept is the provision that a small number of characteristic types movements of simple mathematical models inherent in systems give the key to understanding and exploring a huge number of different phenomena. In particular, it is well known that the complex oscillatory motion can be modeled by a system consisting of one or more coupled nonlinear oscillators that governs by differential equation of a second-order. A Lienard system, namely $ \ddot x(t)+f(x(t))\dot x(t)+g(x(t)) = 0$, is a generalization of the such models. Here $f(x(t))$ and $g(x(t))$ are functions that represent various nonlinear phenomena. The typical sources of nonlinearities in Lienard systems are as follows: large displacements of the structure provoking geometric nonlinearities, a nonlinear material behavior, complex law of damping dissipation, etc. In fact, parameter identification is the base of several engineering tasks: identification can be used for the following: (i) to gain knowledge about the process behavior, (ii) to validate theoretical models, (iii) to tune controller parameters, (iv) to design adaptive control algorithms, (v) to process supervision and fault detection, (vi) to on-line optimization. Hence, in order to represent these nonlinearities, identifying the parameters characterizing their behaviors is essential. The problem of constructing globally convergent identificator for polynomial representation of damping force in general Lienar oscillator is addressed. The method of invariant relations is used for identification scheme design. This aproach is based on dynamical extension of original system and construct of appropriate invariant relations, from which the unknowns parameters can be expressed as a functions of the known quantities on the trajectories of extended system. The final synthesis is carried out from the condition of obtaining asymptotic estimates of unknown parameters. It is shown that an asymptotic estimate of the unknown states can be obtained by rendering attractive an appropriately selected invariant manifold in the extended state space.

Invariant Algebraic Curves and Hyperelliptic Limit Cycles of Liénard Systems

In this paper we deal with the Lienard system $$\dot{x}=y, \dot{y}=-f_m(x)y-g_n(x),$$ where $$f_m(x)$$ and $$g_n(x)$$ are real polynomials of degree m and n, respectively. We call this system the Lienard system of type (m, n). For this system, we proved that if $$m+1\le n\le [\frac{4m+2}{3}]$$ , then the maximum number of hyperelliptic limit cycles is $$n-m-1$$ , and this bound is sharp. This result indicates that the Lienard system of type $$(m,m+1)$$ has no hyperelliptic limit cycles. Secondly, we present examples of irreducible algebraic curves of arbitrary high degree for Lienard systems of type $$(m,2m+1)$$ . Moreover, these systems have a rational first integral. Finally, we proved that the Lienard system of type (2, 5) has at most one hyperelliptic limit cycle, and this bound is sharp.

On the existence of a periodic solution of the Lienard system

The Li\'{e}nard system $\frac{dx}{dt}=y,\ \frac{dy}{dt}=-f(x)y-g(x)$ is considered. Under some assumptions on functions $f(x)$ and $g(x)$, we prove the existence of a periodic solution of this system.

Multiple results to $$\phi $$-Laplacian singular Liénard equation and applications

In this paper, we investigate the existence of a positive periodic solution for the following $$\phi $$ -Laplacian generalized Lienard equation with a singularity: $$\begin{aligned} (\phi (u'))'+f(t,u)u'+ \frac{b(t)}{u^\rho }=h(t)u^m, \end{aligned}$$ where $$\rho $$ is a positive constant and m is a constant. Our proof is based on the Manasevich–Mawhin continuation theorem, and the results are applicable to weak and strong singularities of attractive type (or repulsive type, attractive-repulsive type). Besides, these results in the literature are generalized and significantly improved, and we give the existence interval of a positive periodic solution of this equation. As applications, we study the existence of positive periodic solutions for a Rayleigh–Plesset equation. Finally, three examples are given to show applications of these theorems.