The dynamic behavior of spirals and the spatiotemporal organization in nonlocally coupled oscillatory media are explored in terms of the coupled complex Ginzburg-Landau equations. Chirality transition and tip splitting of spiral wave are found. Both the dispersion relation analysis and numerical simulations reveal that the spiral chirality transition corresponds to a change of the sign of the wave number. Moreover, the underlying mechanism of spiral tip splitting is found to originate from the competition between the local and the nonlocal couplings. The tip splitting behavior and the near-core breakup of spiral wave are found to agree with experimental observations in Belousov-Zhabotinsky reactions.
Adaptive coupling schemes among interacting elements are ubiquitous in real systems ranging from physics, chemistry to neuroscience, which have attracted much attention during recent years. Here, we extend the Kuramoto model by considering a particular adaptive scheme in a system of globally coupled oscillators. The homogeneous coupling is correlated with the global coherence of the population that is weighted by the generic nonlinear feedback function of the amplitude of the order parameter. The studied model is analytically tractable that generalizes the theory of Kuramoto for synchronization transition. We develop a mean-field theory by establishing the self-consistent equation describing the stationary dynamics in the thermodynamic limit. Importantly, the Landau damping effect, which turns out to be far more generic, is revealed in the framework of the linear stability analysis of the resonant pole theory. Furthermore, the relaxation rate of the order parameter in the subcritical region is obtained from a universal formula. Our study can deepen the understanding of synchronization transitions and other related collective dynamics in networked oscillators with adaptive interaction schemes.
A new type of bi-stable spiral waves called "stepped spiral waves", is investigated in this study in an oscillatory medium exhibiting period-doubling bifurcations. Prior to the period-doubling bifurcation of this system, the stepped spiral waves are produced by an unwanted phase trajectory event; the loss of symmetry takes the form of synchronization defect lines, where the trajectory in the local oscillation phase space changes into two different ways. The formation principle of this type of bi-stable spiral wave and the internal structure and geometry of these synchronization defects are studied, and several potential categories of stepped spiral waves are discussed.
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