Periodic Orbits

Abstract

After equilibrium points, the most interesting solutions of planar differential equations are periodic orbits. In fact, we have seen in the previous chapter the birth of periodic orbits when a nonhyperbolic equilibrium point undergoes a Poincaré-Andronov-Hopf bifurcation. There can also be periodic orbits far away from equilibrium points. The detection of such orbits is very difficult. In 1900, as part of problem sixteen of his famous list, Hilbert posed the following question: What is the number of (isolated) periodic orbits of a general polynomial system of differential equations on the plane? The problem remains unsolved even for the case where the components of the planar vector field are quadratic polynomials. Our inability to solve this basic problem exemplifies, in a striking way, the limited scope of our knowledge of periodic orbits. Despite the somber note, in this chapter we first present several basic theorems on the presence or absence of periodic orbits of planar systems. We then investigate the stability and local bifurcations of periodic orbits in terms of Poincaré maps. As an important application of these ideas, we establish the existence of a globally attracting periodic orbit of the oscillator of Van der Pol. We conclude the chapter with an example illustrating how a periodic orbit can bifurcate from a homoclinic loop.