Relaxation and Spontaneous Ordering in Systems with Competition

Spontaneous order happens in non-equilibrium systems composed of interacting elements. This phenomenon manifests in both the formation of space-time patterns in reaction-diffusion systems and collective rhythmic behaviors in coupled oscillators. In this thesis, we present the results of two studies:

1. The response of a multi-species predator-prey system to perturbation.
2. The features of a rich attractor space in a system of repulsively coupled Kuramoto oscillators.

In the first part, we address this question: how does a complex coarsening system with non-trivial in-domain dynamics respond to perturbations? We choose a cyclic predator-prey model with six species each attacking three others. As a result of this interaction network, two competing domains form, while inside each domain three species play a rock-paper-scissors game which results in the formation of spirals inside the domains. We perturb the system by changing the interaction scheme which leads to a change of alliances and therefore a different spatial pattern. As expected, perturbing a complex space-time pattern results in a complex response.

In the second part, we explore the attractor space of a system of repulsively coupled oscillators with non-homogeneous natural frequencies on a hexagonal lattice. Due to the negative coupling and the particular choice of geometry, some of the links between oscillators become frustrated. Coupled oscillators with frustration show similar features as frustrated magnetic systems. We use the parameters of the system like the coupling constant and the width of the frequency distribution to understand the system's attractor space. Further, we study the effects of external noise on the system. We also identify the breaking of time-translation invariance in the absence of external noise, in our system.