| Abstract: |
| It is important to study spatially periodic traveling wave solutions for many partial differential equations in order to understand the mechanism of pattern formation in the higher-dimensional problems. In this study, we introduce a reaction-diffusion system for excitable media to mimic the cardiac cell activities. We investigate numerically the existence and stability of periodic traveling wave solutions in a two-dimensional parameter plane. Our results show two types of stability change in the periodic traveling waves: Eckhaus type and Hopf type. There are two families of periodic traveling waves: fast and slow. The fast family is stable in the case of standard FitzHugh-Nagumo system. However, we observe that the fast family becomes unstable in our model. As a result, it bifurcates to an ``oscillating PTW". We explain this phenomenon by calculating the essential spectra of the periodic traveling wave solutions numerically. In two-dimensions, we show spiral wave breakup in a one-parameter family of solutions as a consequence of the stability of periodic traveling wave solutions. |
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