Limit cycles in the plane. An equivalence class of homogeneous systems

Abstract

A set theoretic presentation of the necessary and sufficient conditions for the occurrence of limit cycle behaviour in open homogeneous systems is given for the specific case of two time-dependent components. The results are obtained by using several theorems due to Zubov in the framework of Lyapunov's stability theory. The starting point in the analysis is the definition of a closed invariant set which depicts the hypothesized or experimentally observed limit cycle. Classes of kinetics which evolve to the defined limit cycle may then be generated. When kinetics of a particular polynomic form are presumed, necessary conditions on the rate constants and external constraints are generated. Time reversal generates an equivalence class of kinetics which has the previously defined limit cycle set as an asymptotic stability domain boundary. This concept may be of use in the consideration of the turning on and off of biological clocks. Introductory results regarding the admissible kinetics giving rise to elliptical limit cycles are presented, and the use of the theory in treating experimentally observed limit cycles is discussed.