A characterization of the generalized Liénard polynomial differential systems having invariant algebraic curves. (May 2022)

• Record Type: Journal Article
• Authors: Giné, Jaume; Llibre, Jaume
• Is Part Of: Chaos, solitons and fractals. Volume 158(2022)
• Publication Date: 2022-05
• Subjects: Liénard polynomial differential systems -- Invariant algebraic curve -- First integrals; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2022.112075
• Languages: English
• ISSNs: 0960-0779

Abstract

Abstract: The generalized Liénard polynomial differential systems are the differential systems of the form x ′ = y, y ′ = − f ( x ) y − g ( x ), where f and g are polynomials. We characterize all the generalized Liénard polynomial differential systems having an invariant algebraic curve. We show that the first four higher coefficients of the polynomial in the variable y, defining the invariant algebraic curve, determine completely the generalized Liénard polynomial differential system. This fact does not hold for arbitrary polynomial differential systems. 2010 mathematics subject classification: Primary 34A05. Secondary 34C05, 37C10. Highlights: The characterization of the generalized Li\'enard polynomial systems having an invariant algebraic curves are given. Such systems are determined for low degrees of invariant curves.