Dynamics Of Spiral Waves Driven By Complex External Force

Spiral waves are typical examples of spatiotemporal construction or evolution in nonlinear science and it is a form of patterns far from thermodynamic equilibrium. The form exists extensively in excitable media, oscillation media and bi-stable system and they have been observed in many actual systems including physics, mechanics, medicine, biology, astronomy and chemistry etc. There is a significant theoretical significance and practical application value. The dynamical behaviors of spiral waves driven by a external force are a significant direction in the study of spiral waves. This paper will study the dynamical behaviors of excitable media driven by a external spiral force and the main work is as follows:Firstly, we study the dynamical behaviors in the driven system subjected directly to a complex external force (this work is writen in the second chapter ). The increasing of the driving intensity will lead to the dynamical behaviors in the driven system change along the order of"unexcited– spatiotemporal turbulence–integrated excitable wave". We have also observed that the form of the integrated excitable wave depends on the excitable parameters of the driving and driven systems, in the range of the driving intensity supporting integrated driving wave. For different excitable parameters, there are several spiral forms including multi-arm spiral waves, single-arm spiral of excitable state being wider than the driving wave, and single-arm spiral of excitable state with the equal width with the driving wave. The generating mechanism and stabilization criterion of multi-arm spiral waves are analysed making use of the natures of the excitable media and the contribution of the direct driven item. The synchronous behavior of driving wave and driven wave are also discussed in this paper and we defined a physical quantity describing their similarity. It is very insteresting that, in the range of driving intensity supporting integral driving wave, the higher similarity appear under a smaller driving intensity. This result is reverse with the intuition, and is also indentified through the studies of the tip dynamics on the driven spiral waves for different driving intensities and excitable parameters. Certainly, we give logical interpretations about the above results.Secondly, we research the dynamical behaviors in the driven system when a complex external force is introduced by the formation of difference (this work is writen in the second and third sections of the second chapter ).The model is also seen as a bilayer of unidirectional coupled term. The upper limit of the driving intensity corresponding to unexcited behavors in driven system is sharply higher than the one of the directly driving scheme. Another important character is that the form of a integral wave in driven system is only single-arm spiral waves , and the width of its excitable state is almostly equal to the one of driving wave. The absence of multi-arm spiral waves can be interpreted through the structure, properties of spiral waves and the synchronization effect of the driven term with .the formation of difference. The quantitative description of the similarity of the driven and driving waves showed that, for a larger driving intensity, the driven wave is closer to the driving wave. This is qualitatively consistent to the dynamical behaviors of a bi-directional-coupled two-layer system. The effect of the external complex forces on the tip dynamics behaviors of spiral wave, under different driving intensities and excitable parameters of the driven and driving systems, is also studied as well. Finally, we make a linear stability analysis of the driven system under the two driving schemes, and give the corresponding characteristic equations and eigenvalues. From this perspective, Some numerical results is attemptly discussed(this work is writen in the forth section of the third chapter).