Traveling waves in an ensemble of excitable oscillators: The interplay of memristive coupling and noise.

Using methods of numerical simulation, we demonstrate the constructive role of memristive coupling in the context of the traveling wave formation and robustness in an ensemble of excitable oscillators described by the FitzHugh-Nagumo neuron model. First, the revealed aspects of the memristive coupling action are shown in an example of the deterministic model where the memristive properties of the coupling elements provide for achieving traveling waves at lower coupling strength as compared to non-adaptive diffusive coupling. In the presence of noise, the positive role of memristive coupling is manifested as significant, increasing a noise intensity critical value corresponding to the noise-induced destruction of traveling waves as compared to classical diffusive interaction.

Feigenbaum scenario without parameters.

Typically, the period-doubling bifurcations exhibited by nonlinear dissipative systems are observed when varying systems' parameters. In contrast, the period-doubling bifurcations considered in the current research are induced by changing the initial conditions, whereas parameter values are fixed. Thus, the studied bifurcations can be classified as the period-doubling bifurcations without parameters.

Subcritical Andronov-Hopf scenario for systems with a line of equilibria.

Using numerical simulation methods and analytical approaches, we demonstrate hard self-oscillation excitation in systems with infinitely many equilibrium points forming a line of equilibria in the phase space. The studied bifurcation phenomena are equivalent to the excitation scenario via the subcritical Andronov-Hopf bifurcation observed in classical self-oscillators with isolated equilibrium points. The hysteresis and bistability accompanying the discussed processes are shown and explained.

Andronov-Hopf bifurcation with and without parameter in a cubic memristor oscillator with a line of equilibria.

The model of a memristor-based oscillator with cubic nonlinearity is studied. The considered system has infinitely many equilibrium points, which build a line of equilibria in the phase space. Numerical modeling of the dynamics is combined with the bifurcational analysis.