Contraction Criteria for Existence, Stability, and Robustness of a Limit Cycle

Stable periodic behavior is the desired behavior of many robotic systems, such as walking robots, swimming robots, and other biologically inspired systems. Designing and verifying controllers for stable periodic behaviour is significantly more challenging than for equilibria. We study pointwise contraction conditions for an autonomous system which imply the existence and global stability of a limit cycle in a compact manifold. Unlike methods based on Lyapunov theory, our condition does not require knowledge of the location of the limit cycle in state-space. This allows straightforward extension to robustness via parameter-dependent certificates, and compositions of systems in feedback and serial interconnections. This new criterion also has interesting connections to contraction theory for trajectories, orbital stability analysis via transverse linearization, and topological facts about regions of attraction of limit cycles.