Target patterns and pacemakers in reaction-diffusion systems

Pattern formation in systems far from thermal equilibrium is a fascinating phenomenon. Reaction-diffusion systems are an important type of system where pattern formation is observed. The target pattern and the associated wave source called pacemaker are typical patterns in such systems. This thesis studies pacemakers and target patterns systematically by analytical and numerical means. The underlying dynamics of the system may be oscillatory or excitable and the pacemakers may either consist of spatial heterogeneities of the medium or be self-organized, i.e. result of intrinsic processes. The investigation of heterogeneous pacemakers in oscillatory systems in the framework of the complex Ginzburg-Landau equation focuses on two aspects. First, the conditions of the creation of pacemakers and extended target patterns versus the creation of wave sinks and localized target patterns are derived systematically. In particular, inward traveling target patterns and large heterogeneities are discussed. Then, pacemakers which emit target waves with high frequencies are considered. In this case, the waves become Eckhaus unstable, causing ring-shaped amplitude defects or other complex patterns. For even larger frequencies, the amplitude defects already take place at the boundary of the heterogeneity, giving rise to a localized desynchronization phenomenon. Moreover, wave sinks can have a significant impact on the spatio-temporal dynamics of the system by breaking the waves arriving from other wave sources. It is well known that oscillatory media close to a Hopf bifurcation are not able to give rise to stable self-organized pacemakers. Therefore, to model such pacemakers, a system close to a pitchfork-Hopf bifurcation is proposed. The normal form and amplitude equations of the pitchfork-Hopf bifurcation are derived. Such a system displays birhythmicity, i.e. bistability of limit cycles, and it is demonstrated analytically that stable self-organized pacemakers are possible. Simulations confirm the existence of stable self-organized pacemakers. In the presence of a parameter gradient, such patterns drift, as shown analytically and numerically. The interaction between pacemakers is studied numerically, giving rise either to coexisting pacemakers or to a new phenomenon called global inhibition: Established pacemakers suppress new cores or merge with them. When the frequencies of the limit cycles differ strongly, the waves may become Eckhaus unstable and the pacemaker may destabilize. Furthermore, kinetic instabilities of the pacemakers are possible, creating breathing and swinging pacemakers. Self-organized pacemakers in excitable media are usually unstable. In this thesis, a three-component activator-inhibitor system on the basis of the FitzHugh-Nagumo model is proposed that gives rise to stable self-organized pacemakers in the excitable regime. The formation of such patterns is demonstrated if several conditions are fulfilled: The system is close to relaxational oscillations, the additional component is strongly diffusive, and the additional component inhibits the inhibitor. Moreover, bistability of pulse solutions is observed in such a system. Different pulses can interact and may create pacemakers. Alternatively, other complex spatio-temporal dynamics is observed. If the diffusion of the activator vanishes, the waves emitted by the wave source are unstable and spatio-temporal chaos appears. Thus, this thesis presents new results on the dynamics of pacemakers with large frequencies and demonstrates for the first time the possibility of stable self-organized pacemakers in birhythmic and excitable systems