Relaxation modes of a system of diffusion coupled oscillators with delay

• Published in: Communications in nonlinear science & numerical simulation 2021-02, Vol.93, p.105488, Article 105488
• Main Author: Kashchenko, A.A.
• Format: Article
• Language: English
• Subjects: Asymptotic methods; Delay differential equations; Differential equations; Dimensional stability; Initial conditions; Multistability; Nonlinear equations; Nonlinear systems; Oscillators; Parameters; Periodic solution; Relaxation mode
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2020.105488

•Relaxation cycles of the nonlinear system of DDE with a large parameter are found.•Nonlocal dynamics of DDE is determined by the dynamics of the finite-dimensional map.•Multistability in the considered model is shown. In this paper we study nonlocal dynamics of a singular perturbed system of two nonlinear delay differential equations simulating two coupled oscillators. Using a special asymptotic method of a large parameter, we reduce studying the existence and stability of relaxation cycles of the original infinite-dimensional system with a large parameter to studying the dynamics of the constructed finite-dimensional map without large or small parameters. We investigate the dynamics of this map, and by its robust stable cycles we obtain initial conditions for exponentially orbitally stable relaxation periodic solutions of the original system. It is shown that small coupling leads to the loss of stability by a homogeneous relaxation cycle and to the emergence of stable inhomogeneous relaxation cycles.