Abstract
A class of autonomous systems of three ordinary differential equations is studied, some of whose examples arise from models of interacting populations. This class of systems always has a unique equilibrium, and the Jacobian matrix at that equilibrium has one negative eigenvalue and two complex eigenvalues with positive real part. It is shown that all solutions not on the stable manifold of the equilibrium become unbounded for large times, spiraling outward from the equilibrium. The argument of the proof is based on Poincare maps and Lyapunov functions.
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