Swapnil Tripathi

Swapnil Tripathi, Mahmut Kudeyt, Alkım Gökçen et al.

A class of Kuramoto models with a general coupling function that can be expressed in terms of a finite number of harmonics, each comprising sinusoidal terms, is studied. We propose a novel approach for certifying local phase synchronization in this class for all initial conditions lying on an arc. The trace parametrization property and Gram matrix representation of a trigonometric polynomial are utilized along with Putinar's Positivstellensatz to obtain semidefinite programming certificates for the stability of the phase-difference system, which in turn implies synchronization of the original system. The results can be extended to any system of coupled oscillators where the forward-invariance on arcs can be established.

Alkım Gökçen, Savaş Şahin, Mahmut Kudeyt et al.

This paper proposes a novel approach for the synchronization problem of nonlinear dynamical systems, integrating dual Lyapunov stability analysis with polynomial optimization. A comprehensive review of the relevant scientific literature on synchronization methods is conducted, with a particular focus on classical Lyapunov-based methods for chaotic systems. In this study, the Rössler system is synchronized by employing dual Lyapunov-based closed-loop synchronization method. This method uses semidefinite programming and sum-of-squares polynomials to compute a nonlinear state feedback function which synchronize a chaotic system to a selected reference model. It is aimed that chaotic behavior is destroyed and, instead, a limit cycle becomes attracting. Simulation works are performed for randomly selected 100 different initial conditions to show that synchronization process is successfully performed. Furthermore, bifurcation diagrams and phase portraits are evaluated to analyze the system dynamics. The paper discusses results and how new constraints should be employed and adapted to more complex systems.