Dynamics of active systems with nonlinear excitation of the phase

Dynamics of active systems with nonlinear excitation of the phase

Many physical and chemical systems exhibit self-oscillatory dynamics, for example systems involving the Belousov-Zhabotinsky reaction and systems used for material synthesis by solid-phase combustion, known as self-propagating high-temperature synthesis. Phase of oscillators crucially depend on diffusion (or thermal conductivity), which is reflected in the partial differential equation governing the phase of oscillations. At first sight, the role of diffusion is to equalise the phase in space. However, more complex situations are possible; for example the phase equation may involve self-excitation such as anti-diffusion in the (Kuramoto-Sivashinsky equation). In this research we investigate a version of the phase equation based on a nonlinear self-excitation. Previously it was shown that nonlinear self-excitation can arise in chemical systems with non-local interaction.

In the present research, we analyse this kind of system 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. They are also used to verify the numerical code. In the numerical experiments, we use a range of sizes of spatial domain.

Lastly, different forms of the nonlinearly excited phase equation are investigated based on different types of dynamical balance.