Dynamics of Active Systems with Nonlinear Excitation of the Phase

In this book, we investigate a version of the phase equation based on a nonlinear self-excitation. We also analyze systems that exhibit self-oscillatory dynamics in order to determine the validity range of the nonlinearly excited phase equation in the parametric space. Specifically, we numerically evaluate the values of the parameters that guarantee the assumptions of slow variations of the phase in space and time and, simultaneously, the key role of the nonlinear self-excitation. We also numerically solve the phase equation with nonlinear self-excitation in two spatial dimensions by finite-difference discretization in space and subsequent numerical integration of a system of ordinary differential equation in time. Irregular dynamics intermitting with periods of slow evolution are revealed and discussed. As a separate task, we derive a forced variant of the phase equation and present selected exact solutions - stationary and oscillatory. Lastly, different forms of the nonlinearly excited phase equation are investigated based on different types of dynamical balance.