Invariant Algebraic Curves of Generalized Liénard Polynomial Differential Systems

Jaume Giné,Jaume Llibre

doi:10.3390/math10020209

Jaume Giné, Jaume Llibre

Open Access

https://doi.org/10.3390/math10020209

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Abstract

In this study, we focus on invariant algebraic curves of generalized Liénard polynomial differential systems x′=y, y′=−fm(x)y−gn(x), where the degrees of the polynomials f and g are m and n, respectively, and we correct some results previously stated.

Highlights

Consider F(x, y) = 0 an invariant algebraic curve of the differential system (1) where F(x, y) is a polynomial, there exists a polynomial K(x, y) such that the following is the case
Introduction and Statement of the MainResultsIn this work, we study the generalized Liénard polynomial differential systems of the following form:x = y, y = − fm(x)y − gn(x), (1)where the degrees of the polynomials f and g are given by the subscripts m and n, respectively
We study the generalized Liénard polynomial differential systems of the following form: x = y, y = − fm(x)y − gn(x), (1)

Summary

Introduction

Consider F(x, y) = 0 an invariant algebraic curve of the differential system (1) where F(x, y) is a polynomial, there exists a polynomial K(x, y) such that the following is the case. Under the assumptions of Theorem 1, the generalized Liénard polynomial differential system (1) has the following hyperelliptic invariant algebraic curves: (a) F(x, y) = −(b + ax)λ + (y − b − ax)2 = 0 for f0(x) = −3a/2 and g1(x) = a(b + ax − λ)/2 with a = 0; (b) F(x, y) = −Ax2 + (y − ax)2 for f0(x) = −2a and g1(x) = (a2 − A)x with aA = 0; (c) F(x, y) = −bc/(2a) − cx − acx2/(2b) + (b + ax − y)2 = 0 for f0(x) = −2a and g1(x) = (2ab − c)(b + ax)/(2b) with ab = 0.

Results

Conclusion