Investigations On Dynamics Of Spiral Waves With An Antiphased Core

This dissertation is devoted to the study of dynamics of spiral waves in discrete systems. Spiral waves are the most frequently encountered pattern formation in two-dimensional systems far away from equilibrium. Spiral waves can display a number of distinct behaviors, some of which are quite complex. The van der Pol oscillator is a paradigmatic example of self-excited oscillator. In some extended systems, how possible discontinuity in spatial variables impacts on pattern formation in these systems is an interesting research subject.In this paper, we first investigate a two-dimensional model system simulating the dynamics of trapped ions, as pattern formation in ultra cold quantum systems has recently received a great deal of attention. We find a spiral wave that is rigidly rotating, but with a peculiar core region in which adjacent ions oscillate in anti-phase. Instead of the phase singularity of a normal spiral wave, this anti-phased core region plays the role of a motor to maintain the spread of this special kind of spiral wave. The formation of this spiral wave is ascribed to the excitability previously reported by Lee and Cross. The breakup of the spiral wave is probed and, especially, an extraordinary scenario of the disappearance of the spiral wave, caused by spontaneous expansion of the anti-phase core, is unveiled.We study pattern formation in coupled van der Pol oscillators, too. Starting with the stability analysis of the synchronous state and the anti-synchronous states, we investigated a pair of coupled oscillators, one-dimensional array of oscillators, and two-dimensional array of oscillators in details. We found that, for two oscillators, only two possible states, the synchronous state and the anti-synchronous states, can be realized. However, both in one-dimensional and in two-dimensional arrays, the spatial-temporal behaviors become quite rich. In two-dimensional array, we found spiral waves with an anti-phase core in which ordinary phase singularity is lost.Finally, the whole work of this article is summarized briefly. Furthermore, the direction of future research is discussed with consideration of our previous work.