Blow Up of a Cycle in Lotka-Volterra Type Equations with Competition-Cooperation Terms and Quasi-Linear Systems

We consider one scenario of transformation of a cycle of a differential equation, which we call the blow up. In this scenario, a cycle born via the Hopf bifurcation grows to infinity as a parameter ranges over a finite segment. From another perspective, in the product of the phase space and the parameter axis there is a branch of cycles connecting the equilibrium and infinity. We first discuss the existence of such a branch for planar differential equations where the proof can be based on the Poincare theorem. A Lotka-Volterra type system with a competition-cooperation term is considered as an example. Then we discuss the existence of a branch of cycles stretching from zero to infinity for a class of higher order quasilinear equations: this theorem continues the results of Refs. 1,2. The results for planar systems are presented in the next section. Section 3 contains the main result for higher order equations. We briefly sketch some points of the proofs.