Periodic Solutions of a System of Delay Differential Equations for a Small Delay

Abstract

We prove the existence of an asymptotically stable periodic solution of a system of delay differential equations with a small time delay t > 0. To achieve this, we transform the system of equations into a system of perturbed ordinary differential equations and then use perturbation results to show the existence of an asymptotically stable periodic solution. This approach is contingent on the fact that the system of equations with t = 0 has a stable limit cycle. We also provide a comparative study of the solutions of the original system and the perturbed system. This comparison lays the ground for proving the existence of periodic solutions of the original system by Schauder's fixed point theorem.

Stability and Persistence of Synchronization in a System with a Diffusive-Time-Lag Coupling

“…S ynchronization phenomena of oscillators and diffusively coupled oscillators has been a subject of great interest by physicist, engineers, and mathematicians, see for instance (Afraimovich, et.al., 1986;Chow and Liu, 1997;Fujisaka and Yamada, 1983;Hale, 1997;Wasike, 2002;Wasike, 2003;Wasike and Rotich (2007).…”

“…The two questions have been done for diffusively coupled systems without a time lag in the coupling. Normal hyperbolicity and the generalized Lyapunov exponents have been used to establish conditions for the stability and persistence of synchronization manifold for lattice dissipative systems each with a compact global attractor, see for example (Chow and Liu, 1997;Wasike, 2002;Wasike, 2003;Josica, K, 2000;Wasike and Rotich, 2007).…”