Analysis of Lienard II-type oscillator equation by symmetry-transformation methods

Özlem Orhan,Teoman Özer

doi:10.1186/s13662-016-0966-4

Özlem Orhan, Teoman Özer

Open Access

https://doi.org/10.1186/s13662-016-0966-4

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Abstract

In this study, we consider a Lienard II-type harmonic nonlinear oscillator equation as a nonlinear dynamical system. Firstly, we examine the first integrals in the form $A(t,x)\dot{x}+B(t,x)$ , the corresponding exact solutions and the integrating factors. In addition, we analyze other types of the first integrals via the λ-symmetry approach. We show that the equation can be linearized by means of a nonlocal transformation, the so-called Sundman transformation. Furthermore, using the modified Prelle-Singer approach, we point out that explicit time-independent first integrals can be identified for the Lienard II-type harmonic nonlinear oscillator equation.

Highlights

Mathematical modeling of many problems in physics and engineering sciences involve nonlinear ordinary differential equations
4.2 The exact solution of the equation using the λ-symmetries based on a linearization method we examine another method to investigate symmetries of the nonlinear equations
We introduce a first integral and an exact solution of the nonlinear oscillator harmonic equation by using λ-symmetry ( . ), which is found by linearization method

Summary

Introduction

Mathematical modeling of many problems in physics and engineering sciences involve nonlinear ordinary differential equations. A detailed review for the available generalizations and recent contributions can be found in the references [ , ] Another method to solve nonlinear differential equations is to obtain λ-symmetries of the equations. The Prelle-Singer method guarantees that if a first-order ordinary differential equation has a first integral in terms of elementary functions, this first integral can be found. This method has been generalized to incorporate the integrals with nonelementary functions This theory is generalized to obtain general solutions for second- and higher-order ordinary differential equations without any integration [ ]. In Section , we apply the modified Prelle-Singer method to the Lienard II-type harmonic nonlinear oscillator equation to obtain Lie symmetries, the first integrals, λsymmetries, the integrating factors and the Lagrangian-Hamiltonian functions.

The λ-symmetries and the integrating factors

The λ-symmetry and the nonlocal transformation pair

The extended Prelle-Singer method and λ-symmetry relation