Limit cycle behavior in three or higher dimensional nonlinear systems: the Lotka-Volterra example

Abstract

The existence of limit cycle behavior in three or higher dimensional nonlinear systems is studied. The procedure by which the existence of limit cycles is established consists of two steps: 1) the boundedness of the system states is established; and 2) all equilibrium points of the system are destabilized. This procedure is applied to a three dimensional Lotka-Volterra system to determine sets of parameters for which the system exhibits limit cycle behavior. The procedure by which the existence of limit cycle behavior in nonlinear systems is established, is applicable to planar or higher dimensional systems conveniently. However, establishing the boundedness of the system states (step 1) may be difficult for some systems. Moreover, this procedure does not exclude the possibility of quasi-periodic or chaotic behaviors.