Planar polynomial vector fields having first integrals and algebraic limit cycles

Abstract

With the help of Abel differential equations we obtain a new class of Darboux integrable planar polynomial differential systems, which have degenerate infinity. Moreover such integrable systems may have algebraic limit cycles. Also we present the explicit expressions of these algebraic limit cycles for quintic systems.

The limit cycles of a class of quintic polynomial vector fields

“…The main interest for studying the limit cycles of the planar polynomial differential systems is due to the 16-th Hilbert problem, see for instance [13] and [15]. Many recent papers are also dedicated to the study of the limit cycles, see for instance the papers [2,3,6,12,16] which are more related with our present work.…”

The hyperelliptic limit cycles of the Liénard systems

“…If a limit cycle is contained in a hyperelliptic invariant curve, then it is called a hyperelliptic limit cycle, see for instance [1,5,11,22]. In 1964 Wilson [26] constructed the following Liénard system…”

“…In [17,18] Llibre et al had studied the upper bound of algebraic limit cycles under some special conditions. Cao and Jiang [1] investigated the relation between the existence of first integrals and algebraic limit cycles. In [23] Llibre and Zhao provided polynomial differential systems with algebraic limit cycles.…”

A Class of Non-Linear Oscillators with Non-Autonomous First Integrals and Algebraic Limit Cycles

“…In this case, this set may contain ovals, which can be algebraic limit cycles. For more details about first integrals, invariant algebraic curves and algebraic limit cycles, see [4,6,8,10,21,23,32] and Chapter 8 of [11] and the references therein.…”