In the Presence of Purely Imaginary Eigenvalues

In this chapter, we investigate the stability and bifurcations of a nonhyperbolic equilibrium point of a planar differential equation in the case where the linearized vector field has purely imaginary eigenvalues. Using polar coordinates, we capture the dynamics of such a system in the neighborhood of the equilibrium point in terms of the dynamics of an appropriate nonautonomous scalar differentia] equation with periodic coefficients. For the analysis of this scalar equation, we appeal to results in Chapters 4 and 5. When the vector field is subjected to small perturbations, the original equilibrium point persists, and there can be no new equilibria in the neighborhood. However, if the eigenvalues of the linearized system move away from the imaginary axis, one expects the equilibrium point to change its stability type. Thus change is typically marked by the appearance of a small periodic orbit encircling the equilibrium point. We present a proof of this celebrated result—the Poincaré-Andronov-Hopf Theorem—and a discussion of the stability of the periodic orbit. We conclude with an exposition of computational procedures for determining bifurcation diagrams of periodic orbits bifurcating from an equilibrium point.