Vladimir Belykh

In this paper, we constructed a new model of the neuron membrane potential in the form of a piecewise-smooth (PWS) slow-fast system. Its fast part is a two-dimensional PWS subsystem specially “sewn” from the normal forms of a stable focus and a saddle-node bifurcation, and the slow one-dimensional equation serves to periodically force the fast one. Our approach to constructing PWS systems in this way allows them to qualitatively reproduce the phase space structure of their smooth counterpart and also provide explicit solutions. By obtaining an exact Poincaré map as a composition of solutions and examining it, we proved the complete bifurcation structure of the fast subsystem. By selecting three bifurcation routes and applying a time-periodic change of the subsystem parameter along them using the third slow equation, we obtained three different types of bursting patterns. Finally, we showed that, when coupled into a network, our PWS models form a synthetic central pattern generator that manifests a traveling bursting wave.

Synchronization phenomena in networks of globally coupled non-identical oscillators have been one of the key problems in nonlinear dynamics over the years. The main model used within this framework is the Kuramoto model. This model shows three main types of behavior: global synchronization, cluster synchronization including chimera states and totally incoherent behavior. We present new sufficient conditions for phase synchronization and conditions for an asynchronous mode in the finite-size Kuramoto model. In order to find these conditions for constant and time varying frequency mismatch, we propose a simple method of comparison which allows one to obtain an explicit estimate of the phase synchronization range. Theoretical results are supported by numerical simulations.

We consider a system of two coupled Van der Pol-Duffing oscillators with Huygens coupling as an appropriate model of two mechanical oscillators connected to a movable platform via a spring. We examine the complicated dynamics of the system and study its multistable behavior. In particular, we reveal the co-existence of several chaotic regimes and study the structure of the associated riddled basins.