Work

First and Second Integrals of Hopf–Langford-Type Systems

Year: 2024

Type: article

Abstract: In the present work, we consider a seven-parameter, three-dimensional, autonomous, cubic polynomial system of nonlinear ordinary differential equations, generalizing the recently studied quadratic polynomial Hopf–Langford type systems. First, by introducing polar coordinates in its phase space, we show that the regarded system can be reduced to a two-dimensional Liénard system, which corresponds to a second-order Liénard equation. Then, we present (in explicit form) polynomial first and second integrals of Liénard systems of the considered type identifying those values of their parameters for which these integrals exist. It is also proved that a generic Liénard equation is factorizable if and only if the corresponding Liénard system admits a second integral of a special form. It is established that each Liénard system corresponding to a Hopf–Langford system of the considered type admits such a second integral and hence the respective Liénard equation is factorizable.

Source: Axioms

DOI: 10.3390/axioms14010008

Authors: Vassil Mitev Vassilev, Svetoslav Ganchev Nikolov

Institutions: Bulgarian Academy of Sciences, Institute of Mechanics, Todor Kableshkov University of Transport

Language: English

Topic: Advanced Differential Equations and Dynamical Systems

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $2,728

Funder: Ministry of Education and Science