On the Topology of Liapunov Functions for Dissipative Periodic Processes

Summary

The existence and nature of nonlinear oscillations for periodically forced nonlinear differential equations has historically attracted quite a bit of attention in both the pure and the applied mathematics literature. In control theory, it encompasses the study of the steady-state response of control systems to periodic inputs, generalizing the frequency domain theory that underlies classical control and its many successes. More than fifty years ago, Levinson initiated the study of dissipative periodic processes for planar systems, an approach that has since inspired the development of a general theory of dissipative systems for both lumped and distributed nonlinear systems. In the lumped case, dissipative processes have a dissipative Poincaré map $\mathcal{P}$ and a fair amount of effort has been expended determining the fixed point properties of $\mathcal{P}$, culminating in the use of a remarkable fixed point theorem of F. Browder which showed that general dissipative periodic processes always have harmonic oscillations. An alternative approach to studying dissipative periodic processes using Liapunov theory was developed by the Russian school of nonlinear analysis, pioneered by Pliss, Krasnosel’skiĭ and others. It is fair to say that the largest technical challenge arising in this approach is the lack of a general, user-friendly description of the the level and sublevel sets of these Liapunov functions. In the equilibrium case, the recent solution of the Poincaré Conjecture in all dimensions has resulted in a simple description and useful description of these sets [3], viz. the sublevel sets are always homeomorphic to a disk $\mathbb{D}^n$. Fortunately, the techniques underlying the proofs of the Poincaré conjectures have shed enough light on related classification questions that we can now also describe the topology of the level and sublevel sets of Liapunov functions for dissipative periodic process. Among the results we prove in this paper is that these sublevel sets of a Liapunov function are always homeomorphic to solid tori, $\mathbb{D}^n \times S^1$, and diffeomorphic except perhaps when n = 3. Together with recent sufficient conditions for periodic orbits proven by Brockett and the author [4], these descriptions give streamlined proofs of the existence of harmonic oscillations, and some related results. The proof of our main theorem uses the work of Wilson on the topology of Liapunov functions for attractors, the s-cobordism theorem in dimensions greater than five, the validity of the Poincaré Conjecture in dimension three and four, and a smoothing result of Kirby and Siebenmann for five-manifolds.