Lienard-type Equations Research Articles

On Truly Nonlinear Oscillator Equations of Ermakov-Pinney Type

In this paper we present a general class of differential equations of Ermakov-Pinney type which may serve as truly nonlinear oscillators. We show the existence of periodic solutions by exact integration after the phase plane analysis. The related quadratic Lienard type equations are examined to show for the first time that the Jacobi elliptic functions may be solution of second-order autonomous non-polynomial differential equations.

Symmetry analysis for a second-order ordinary differential equation

In this article, we apply the Lie symmetry analysis to a second-order nonlinear ordinary differential equation, which is a Lienard-type equation with quadratic friction. We find the infinitesimal generators under certain parametric conditions and apply them to construct canonical variables. Also we present some formulas for the first integral for this equation.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/85/abstr.html

Response to “Comment on ‘Classification of Lie point symmetries for quadratic Liénard type equation ẍ + f(x)ẋ2 + g(x) = 0’” [J. Math. Phys. 61, 044101 (2020)

We respond to the comment on ‘Classification of Lie point symmetries for quadratic Lienard type equation ẍ + f(x)ẋ2 + g(x) = 0’ [J. Math. Phys. 61, 044101 (2013)] by Iacono regarding linearizability and isochronicity. We assert here that the condition for linearization of the equation ẍ + f(x)ẋ2 + g(x) = 0 given by us in our paper is correct with the condition g1=ω02>0. We present the explicit form of local and nonlocal transformations that transform the quadratic Lienard equation x+F+11−xx2+x(1−x)(1+Dx)=0 into the harmonic oscillator equation for the four cases mentioned in the comment and confirm the statements given in our paper are all valid.

On discrete groups and solvable equations of nonlinear oscillations

The autonomous ordinary differential equation of Lienard is considered. This equation and its generalizations arise in the theory of nonlinear oscillations, dynamical systems, etc. Currently for equations of the Lienard type, the problem of finding exact solutions is relevant. In view of this, the problem is to find connections in Lienard type equations with investigated nonlinear equations. These problems are solved in this article for the Lienard equation with three-parametrical coefficients of polynomial type. We found a transformation that reduces the equation under consideration of the well-studied generalized Emden-Fowler equation. We indicated the method of finding exact solutions of Lienard equations with various parameters. The transformations discrete group of dihedron for the class of generalized Emden-Fauler equations induces a discrete group of transformations for the corresponding class of Lienard equations. Likewise, known solutions of some particular generalized Emden-Fauler equations induce solutions of the Lienard equations. As examples, we found exact solutions in elementary (including polynomials) and special functions of several Lienard equations with certain parameters.

Numeric-Analytic Solutions for Nonlinear Oscillators via the Modified Multi-Stage Decomposition Method

This work deals with a new modified version of the Adomian-Rach decomposition method (MDM). The MDM is based on combining a series solution and decomposition method for solving nonlinear differential equations with Adomian polynomials for nonlinearities. With application to a class of nonlinear oscillators known as the Lienard-type equations, convergence and error analysis are discussed. Several physical problems modeled by Lienard-type equations are considered to illustrate the effectiveness, performance and reliability of the method. In comparison to the 4th Runge-Kutta method (RK4), highly accurate solutions on a large domain are obtained.

Some modifications on RCAM for getting accurate closed-form approximate solutions of Duffing- and Lienard-type equations

In this work, authors propose some modifications Adomian decomposition method to get some accurate closed form approximate or exact solutions of Duffing- and Li´enard-type nonlinear ordinary differential equations.Results obtained by the revised scheme have been exploited subsequently to derive constraints among parameters to get the solutions to be bounded. The present scheme appears to be efficient and may be regarded as the confluence of apparently different methods for getting exact solutions for a variety of nonlinear ordinary differential equations appearing as mathematical models in several physical processes.

On the Asymptotic Stability of Nonlinear Time-Varying Switched Systems

This paper considers a class of nonlinear systems with switching from one nonstationary Lienard-type equation to another. For such a system, the asymptotic stability of a given equilibrium is studied using the method of Lyapunov functions and theory of differential inequalities. The switching law constraints that guarantee the desired property of the system are established. As shown below, these constraints generally depend on the rate of change of time-varying system parameters. Some illustrative examples are also given.

Integrable Nonautonomous Liénard-Type Equations

We study a family of nonautonomous generalized Lienard-type equations. We consider the equivalence problem via the generalized Sundman transformations between this family of equations and type-I Painleve–Gambier equations. As a result, we find four criteria of equivalence, which give four integrable families of Lienard-type equations. We demonstrate that these criteria can be used to construct general traveling-wave and stationary solutions of certain classes of diffusion–convection equations. We also illustrate our results with several other examples of integrable nonautonomous Lienard-type equations.

Theory of Exact Trigonometric Periodic Solutions to Quadratic Liénard Type Equations

A mathematical theory is developed through generalized Sundman transformation to show the existence of classes of quadratic Lienard type equations which admit exact and explicit general trigonometric solutions but with amplitude-dependent frequency. The application of the theory to compute also exact and explicit general periodic solutions to nonlinear differential equations like inverted Painleve-Gambier equations in terms of trigonometric or Jacobian elliptic functions is highlighted by some illustrative examples.

On the General Solution to the Bratu and Generalized Bratu Equations

This work shows that the Bratu equation belongs to a general class of Lienard-type equations for which the general solution may be exactly and explicitly computed within the framework of the generalized Sundman transformation. In this perspective the exact solution of the Bratu nonlinear two-point boundary value problem as well as of some well-known Bratu-type problems have been determined.

Analytical Solutions for Nonlinear Convection–Diffusion Equations with Nonlinear Sources

Nonlinear equations of the convection–diffusion type with nonlinear sources are used for the description of many processes and phenomena in physics, mechanics, and biology. In this work, we consider the family of nonlinear differential equations which are the traveling wave reductions of the nonlinear convection–diffusion equation with polynomial sources. The question of the construction of the general analytical solutions to these equations is studied. The steady-state and nonstationary cases with and without account of convection are considered. The approach based on the application of nonlocal transformations generalizing the Sundman transformation is applied for construction of the analytical solutions. It is demonstrated that in the steady-state case without account of convection, the general analytical solution can be found without any constraints on the equation parameters and it is expressed in terms of the Weierstrass elliptic function. In the general case, this solution has a cumbersome form; the constraints on the parameters for which it has a simple form are found, and the corresponding analytical solutions are obtained. It is shown that in the nonstationary case, both with and without account of convection, the general solution to the studied equations can be constructed under certain constraints on the parameters. The integrability criteria for the Lienard-type equations are used for this purpose. The corresponding general analytical solutions to the studied equations are explicitly constructed in terms of exponential or elliptic functions.

Lie symmetry analysis and exact solution of certain fractional ordinary differential equations

A systematic investigation of finding Lie point symmetries of certain fractional linear and nonlinear ordinary differential equations is presented. More precisely, Lie point symmetries of fractional Riccati equation, nonhomogeneous fractional linear ordinary differential equation with variable coefficients and quadratic fractional Lienard-type equation in the sense of Riemann–Liouville fractional derivative are derived. Using the obtained Lie point symmetries, we derive exact solution of the above-mentioned fractional ordinary differential equations wherever possible.

On Stability of Periodic Solutions of Lienard Type Equations

We use the Floquet theory to analyze the stability of periodic solutions of Lienard type equations under the asymptotic linear growth of restoring force in this paper. We find that the existence and the stability of periodic solutions are determined primarily by asymptotic behavior of damping term. For special type of Lienard equation, the uniqueness and stability of periodic solutions are obtained. Furthermore, the sharp rate of exponential decay of the stable periodic solutions is determined under suitable conditions imposed on restoring force.

On the general traveling wave solutions of some nonlinear diffusion equations

We consider a family of nonlinear diffusion equations with nonlinear sources. We assume that all nonlinearities are polynomials with respect to a dependent variable. The traveling wave reduction of this family of equations is an equation of the Lienard-type. Applying recently obtained criteria for integrability of Lienard-type equations we find some new integrable families of traveling wave reductions of nonlinear diffusion equations as well as their general analytical solutions.

On analysis of nonlinear dynamical systems via methods connected with $$\lambda $$ λ -symmetry

This study focuses on the analysis of the first integrals, the integrating factors and the solutions for some classes of nonlinear dynamical systems represented by a mass–spring–damper model. The study consists of the applications of local–nonlocal transformations and the extended Prelle–Singer approach by considering the relation with Lie symmetry groups and \(\lambda \)-symmetry. In addition, the mathematical relations between these methods are presented, and time-dependent and time-independent first integrals and the corresponding exact solutions of nonlinear dynamical systems such as general Morse oscillator equation, the path equation, the harmonic oscillator equation and the displaced harmonic oscillator equation as special cases of Lienard-type equations are investigated. Furthermore, the general forms of Lienard-type equations including general nonlinear damping and nonlinear spring functions are studied.

Lie point symmetries classification of the mixed Liénard-type equation

In this paper we develop a systematic and self-consistent procedure based on a set of compatibility conditions for identifying all maximal (eight parameter) and non-maximal (one and two parameter) symmetry groups associated with the mixed quadratic-linear Lienard-type equation, $$\ddot{x} + f(x){\dot{x}}^{2} + g(x)\dot{x}+h(x)= 0$$ , where $$f(x),\,g(x)$$ and h(x) are arbitrary functions of x. With the help of this procedure we show that a symmetry function b(t) is zero for non-maximal cases, whereas it is not so for the maximal case. On the basis of this result the symmetry analysis gets divided into two cases, (i) the maximal symmetry group $$(b\ne 0)$$ and (ii) non-maximal symmetry groups $$(b=0)$$ . We then identify the most general form of the mixed quadratic linear Lienard-type equation in each of these cases. In the case of eight-parameter symmetry group, the identified general equation becomes linearizable. In the case of non-maximal symmetry groups the identified equations are all integrable. The integrability of all the equations is proved either by providing the general solution or by constructing time-independent Hamiltonians.

Stability tests for second order linear and nonlinear delayed models

For the nonlinear second order Lienard-type equations with time-varying delays $$\ddot{x}(t)+\sum_{k=1}^m f_k(t,x(t),\dot{x}(g_k(t)))+\sum_{k=1}^ls_k(t,x(h_k(t)))=0,$$ global asymptotic stability conditions are obtained. The results are based on the new sufficient stability conditions for relevant linear equations and are applied to derive explicit stability conditions for the nonlinear Kaldor–Kalecki business cycle model. We also explore multistability of the sunflower non-autonomous equation and its modifications.

Periodic solutions for Liénard type equation with time-variable coefficient

By means of topological theory, a class of Lienard type equations with a deviating argument of the form $(\varphi_{p}(x'(t)))'+f(x(t-\tau))x'(t-\tau)+\beta(t)g(x(t-\tau))=e(t)$ is studied. It is notable that the coefficient $\beta(t)$ in front of $g(x(t-\tau))$ is allowed to change sign in this paper. Moreover, a numerical simulation is carried out to verify the validity of the obtained results. In addition, the generalized form of the above equation with time-varying delays is also discussed briefly.

On the Stability to a Functional Lienard Type Equation with Variable Delay by Fixed Points Theory

On the Stability to a Functional Lienard Type Equation with Variable Delay by Fixed Points Theory

On the Stability to a Functional Lienard Type Equation with Variable Delay by Fixed Points Theory

This paper is devoted to the mathematical analysis of stability of a scalar Li ´ enard type equation with variable delay. We use the fixed point technique under an exponentially weighted me tric to prove the stability of the zero solution. By this work , we extend and improve some related results in the literature