Phase equation with nonlinear excitation for nonlocally coupled oscillators

Abstract

Some reaction-diffusion systems feature nonlocal interaction and, near the point of Hopf bifurcation, can be represented as a system of nonlocally coupled oscillators. Phase of oscillations satisfies an evolution pde which takes different forms depending on the values of parameters. In the simplest case the equation is effectively a diffusion equation which is excitation-free. However, more complex forms are possible such as the Nikolaevskii equation and the KuramotoSivashinsky equation incorporating linear excitation. We analyse a situation when the phase equation is based on nonlinear excitation. We derive conditions on the values of the parameters leading to the situation and show that the values satisfying the conditions exist.

Finite-Difference Approach for a 6th-Order Nonlinear Phase Equation with Self-Excitation

“…The exact physical meaning of the phase u is the departure from the uniformly increasing phase of the oscillation c 2 t (we use the notations from [4]); thus the oscillators behave as a sin ϕ with ϕ = c 2 t + u , where a is the amplitude of the oscillations. In [2,4] c 2 is constant in space and time.…”

“…As pointed out in [2,4] the phase equations in question may be relevant to certain real-life systems such as cellular slime molds and the Belousov-Zhabotinsky reaction dispersed in water-in-oil aerosol OT microemulsion. A real physical system of any kind is never uniform, in particular, the parameter c 2 may vary (naturally fluctuate) in space.…”

“…A real physical system of any kind is never uniform, in particular, the parameter c 2 may vary (naturally fluctuate) in space. In the ideal case of a constant rate c 2 the phase satisfies the equation (35) from [4],…”

“…This is done intentionally by an appropriate selection of the original parameters. The analysis which takes into account the magnitude of u shows (we refer to [4] for details) that the magnitude of the small parameter ε 2 must be ε 2 = ε 20/3 1 since all the terms in (5) need to be of the same order of magnitude. Finally, neglecting ε 2 c (1) 2 inside the square brackets in comparison to c (0) 2 and returning to the unscaled operators ∇ = ε 1 ∇ 1 and ∂ t = ε 2 1 ∂ t 1 we arrive at the forced equation…”

“…However, as is well-established [1,2], under certain conditions the combined effect of the diffusion and self-excitation (caused by reactions) may produce more complicated dynamics, where this difference never disappears. Strunin derived [3,4] a form of such an equation incorporating nonlinear self-excitation, ∂ t u = −g∇ 2 u(∇u) 2 + b(∇u) 4 + c∇ 6 u ,…”