The Existence and Stability of Limit Cycle of Some Special Types of 2D Autonomous Dynamic Systems

Abstract

This article is focusing on the problem about limit cycles of 2D autonomous dynamics, which is a part of the famous Hilbert’s 16th problem. There are mainly three results on this paper. The first two results discuss the cases with sum of squares of the variables. Through polar substitution and the mathematics analysis on the explicit limit cycle obtained by Cramer’s Rule, the first result reveals the uniqueness and stability of the limit cycle of a specific form for 2D autonomous dynamic system. For the second part of the results, by changing the coefficients of the system of the first result, it is found that the stability of limit cycle remains unchanged on the phase portrait. The third part gives a special polynomial with terms of general odd degrees for Liénard equation, which can be rewritten as a case of 2D autonomous dynamic system. Without explicit expression of the limit cycle, the Poincaré–Bendixson method and the Liénard theorem help to give the conclusion on both uniqueness and stability of this system.