Relaxation cycles in a model of two weakly coupled oscillators with sign-changing delayed feedback

We consider the nonlocal dynamics of a model describing two weakly coupled oscillators with nonlinear compactly supported delayed feedback. Such models are found in applied problems of radiophysics, optics, and neurodynamics. The key assumption is that the nonlinearity is multiplied by a sufficiently large coefficient. This assumption allows using a special asymptotic method of a large parameter. Using this method, we reduce studying the existence, asymptotic behavior, and stability of relaxation cycles of the original infinite-dimensional system to studying the dynamics of the constructed finite-dimensional map. We investigate the dynamics of this map, construct the asymptotic behavior of the relaxation cycles of the original system, and conclude that the system is multistable.