Construction of Lyapunov density for nonautonomous dynamical systems on hypertorus

We present a semidefinite programming framework for constructing time-varying Lyapunov densities for nonautonomous dynamical systems on a hypertorus. The formulation leverages Gram matrix representations of hybrid (real-trigonometric) polynomials. In addition, we introduce a novel block decomposition of these Gram representations to confine the blow-up of the resulting density to a prescribed set. The results are then applied to establish the almost global synchronization of a time-varying Kuramoto model and the robust almost-global stability of a parameter-varying nonautonomous system. These examples demonstrate the applicability of the proposed method and validate the theoretical results. All computational results are obtained using an open-source MATLAB implementation, as referenced in the text, thereby facilitating reproducibility of the reported examples.