Abstract:
The recently discovered chimera state involves the coexistence of synchronized and desynchronized states for a group of identical oscillators. This fascinating chimera state has until now been found only in non-local or globally coupled oscillator systems. In this work, we for the first time show numerical evidence of the existence of spiral wave chimeras in reaction-diffusion systems where each element is locally coupled by diffusion. This spiral wave chimera rotates inwardly, i.e., coherent waves propagate toward the phase randomized core. A continuous transition from spiral waves with smooth core to spiral wave chimeras is found as we change the local dynamics of the system. Our findings on the spiral wave chimera in locally coupled oscillator systems largely improve our understanding of the chimera state and suggest that spiral chimera states may be found in natural systems which can be modeled by a set of oscillators indirectly coupled by a diffusive environment.

Abstract:
Spiral waves are ubiquitous in two-dimensional systems of chemical or biological oscillators coupled locally by diffusion. At the center of such spirals is a phase singularity, a topological defect where the oscillator amplitude drops to zero. But if the coupling is nonlocal, a new kind of spiral can occur, with a circular core consisting of desynchronized oscillators running at full amplitude. Here we provide the first analytical description of such a spiral wave chimera, and use perturbation theory to calculate its rotation speed and the size of its incoherent core.

Abstract:
Nonlocally coupled oscillator systems can exhibit an exotic spatiotemporal structure called chimera, where the system splits into two groups of oscillators with sharp boundaries, one of which is phase-locked and the other is phase-randomized. Two examples of the chimera states are known: the first one appears in a ring of phase oscillators, and the second one is associated with the two-dimensional rotating spiral waves. In this article, we report yet another example of the chimera state that is associated with the so-called Ising walls in one-dimensional spatially extended systems, which is exhibited by a nonlocally coupled complex Ginzburg-Landau equation with external forcing.

Abstract:
We report some qualitatively new features of emergence, competition and dynamical stabilization of dissipative rotating spiral waves (RSWs) in the cellular-automaton model of laser-like excitable media proposed in arXiv:cond-mat/0410460v2 ; arXiv:cond-mat/0602345 . Part of the observed features are caused by unusual mechanism of excitation vorticity when the RSW's core get into the surface layer of an active medium. Instead of the well known scenario of RSW collapse, which takes place after collision of RSW's core with absorbing boundary, we observed complicated transformations of the core leading to regeneration (nonlinear "reflection" from the boundary) of the RSW or even to birth of several new RSWs in the surface layer. Computer experiments on bottlenecked evolution of such the RSW-ensembles (vortex matter) are reported and a possible explanation of real experiments on spin-lattice relaxation in dilute paramagnets is proposed on the basis of an analysis of the RSWs dynamics. Chimera states in RSW-ensembles are revealed and compared with analogous states in ensembles of nonlocally coupled oscillators. Generally, our computer experiments have shown that vortex matter states in laser-like excitable media have some important features of aggregate states of the usual matter.

Abstract:
Two-dimensional lattices of N synchronized oscillators with reactive coupling are considered as high-precision frequency sources in the case where a spiral pattern is formed. The improvement of the frequency precision is shown to be independent of N for large N, unlike the case of purely dissipative coupling where the improvement is proportional to N, but instead depends on just those oscillators in the core of the spiral that acts as the source region of the waves. Our conclusions are based on numerical simulations of up to N=29929 oscillators, and analytic results for a continuum approximation to the lattice in an infinite system. We derive an expression for the dependence of the frequency precision on the reactive component of the coupling constant, depending on a single parameter given by fitting the frequency of the spiral waves to the numerical simulations.

Abstract:
Spiral waves arise in many biological, chemical, and physiological systems. The kinematical model can be used to describe the motion of the spiral arms approximated as curves in the plane. For this model, there appeared some results in the literature. However, these results all are based upon some simplification on the model or prior phenomenological assumptions on the solutions. In this paper, we use really full kinematic model to classify a generic kind of steadily rotating spiral waves, i.e., with positive (or negative) curvature. In fact, using our results (Theorem 8), we can answer the following questions: Is there any steadily rotating spiral wave for a given weakly excitable medium? If yes, what kind of information we can know about these spiral waves? e.g., the tip's curvature, the tip's tangential velocity, and the rotating frequency. Comparing our results with previous ones in the literature, there are some differences between them. There are only solutions with monotonous curvatures via simplified model but full model admits solutions with any given oscillating number of the curvatures.

Abstract:
An oscillatory system can have clockwise and anticlockwise senses of rotation. We propose a general rule how to obtain counter-rotating oscillators from the definition of a dynamical system and then investigate synchronization. A type of mixed synchronization emerges in counter-rotating oscillators under diffusive scalar coupling when complete synchronization and antisynchronization coexist in different state variables. Stability conditions of mixed synchronization are obtained analytically in Rossler oscillator and Lorenz system. Experimental evidences of mixed synchronization are given for limit cycle as well as chaotic oscillators in electronic circuits.

Abstract:
We investigate the dynamical one-armed spiral instability in differentially rotating stars with both eigenmode analysis and hydrodynamic simulations in Newtonian gravity. We find that the one-armed spiral mode is generated around the corotation radius of the star, and the distribution of angular momentum shifts inwards the corotation radius during the growth of one-armed spiral mode. We also find by investigating the distribution of the canonical angular momentum density that the low T/|W| dynamical instability for both m=1 and m=2 mode, where T is the rotational kinetic energy and W is the gravitational potential energy, is generated around the corotation point. Finally, we discuss the feature of gravitational waves generated from these modes.

Abstract:
In this paper, we report mixed synchronization between two counter rotating chaotic oscillators. We describe a procedure how to obtain a counter rotating oscillator for generalized oscillators. We elaborate the method with numerical examples of the Sprott system, Pikovsky-Rabinovich (PR) circuit model. Noise-induced mixed synchronization is also reported in PR circuit model. The physical realization of mixed synchronization in an electronic circuit of two counters-rotating Sprott systems also shown.

Abstract:
We investigate the dynamical instability of the one-armed spiral m=1 mode in differentially rotating stars by means of 3+1 hydrodynamical simulations in Newtonian gravitation. We find that both a soft equation of state and a high degree of differential rotation in the equilibrium star are necessary to excite a dynamical m=1 mode as the dominant instability at small values of the ratio of rotational kinetic to potential energy, T/|W|. We find that this spiral mode propagates outward from its point of origin near the maximum density at the center to the surface over several central orbital periods. An unstable m=1 mode triggers a secondary m=2 bar mode of smaller amplitude, and the bar mode can excite gravitational waves. As the spiral mode propagates to the surface it weakens, simultaneously damping the emitted gravitational wave signal. This behavior is in contrast to waves triggered by a dynamical m=2 bar instability, which persist for many rotation periods and decay only after a radiation-reaction damping timescale.