Construction of Lyapunov Certificates for Systems on Torus Using Trigonometric Polynomials

Amethod for the certification of local and almost global stability of invariant sets of ordinary differential equations on torus is proposed. The method is based on using trigonometric polynomials as candidates of local Lyapunov functions or (global) Lyapunov densities (so-called dual Lyapunov functions) and converting the required local Lyapunov or dual Lyapunov conditions into semidefinite programming. We provide several examples with three and four oscillators to showcase the adaptability of our results, which can be used to certify partial or complete phase-locking/phase-cohesion and to validate the presence of an attracting/repelling set in various scenarios.