The Study On Coupled Spiral Pattern Dynamics

Spatiotemporal pattern, which appears in systems far away from thermodynamic equilibrium conditions, is ubiquitous in nature. Pattern dynamics is a subject which discusses the origin and rule of these patterns, and is one of the important branches in nonlinear science. Since the time when famous British scientist Turing brought forward the Turing pattern theory in the early1950s, the rule of the pattern dynamics in two dimensional systems is gradually clear, and studying the rules and characteristics of pattern dynamics in coupled systems, which is much closer to our real world, becomes very important. In this thesis, we numerically and theoretically study the dynamic behaviors of coupled spiral wave patterns in both oscillatory and excitable reaction-diffusion systems. The first chapter serves as a brief introduction on some related concepts: history of pattern and pattern dynamics, research background, and current situation of the pattern dynamics in coupled systems. Here, we introduce the different kinds of spiral wave patterns and the features of the spiral wave patterns in different reaction-diffusion systems.The third chapter studies the interaction of two spiral waves with independent frequencies in a bilayer oscillatory medium. In most of the chapter, we consider symmetric coupling. If the frequencies of the two spirals are different, the faster spiral is almost unaffected by the slower one, and depending on e, the slower one can show a variety of behaviors, for example, in order of increasing e, phase drifting, amplitude modulation, amplitude domination, and phase synchronization. This high-frequency dominance and the asymmetric driving-response effect under the condition of a symmetric coupling are generic and independent of the rotating direction of the spirals. If the frequencies of the spirals are identical, then depending on the relative rotation direction of the two spirals and their initial separation distance, the two spirals may show complete synchronization, parallel drift, or circular drift. We also make some comparisons with coupled spirals in monolayer media, previous works on coupled spirals in bilayer systems, and coupled phase oscillators.In the fourth chapter, we study the interaction of two identical excitable spiral waves in a bilayer system. We find that if the coupling strength is sufficiently large, the two spiral waves can be completely synchronized. Prior to the complete synchronization, we find a new type of weak synchronization between the two coupled spirals. For example, the two spirals have the same geometric shape, however, the amplitude of the spiral wave in the driven system is much lower than that of the spiral in the driving system. We call this general behavior projective synchronization of two spiral waves; it is similar to the extensively studied projective synchronization of two coupled nonlinear oscillators. By studying the pulse collision in one-dimensional systems, the underlying mechanism is uncovered.The fifth chapter studies the interaction of spiral waves in a coupled twolayer oscillatory system. We find that prior to the complete synchronization of the two spirals, under the condition of strongly asymmetric coupling, a novel type of spiral wave, the amplitude spiral wave, exists in the driven system. Different from the usual phase spirals which rotate around the central tip points, the amplitude spiral wave shows the spiral structure in the amplitude without a singularity point (tip), however, the amplitude of the usual phase spirals vanishes at the tip point and is uniform far away from tips.The sixth chapter summarizes the whole thesis.