computed by employing both zero-ux and periodic boundary conditions in the time in- terval 200≤t≤210. The corresponding phase plots and the power spectra are presented in gures 3.10(c)-(d) and 3.10(e)-(f) respectively. One can clearly see a quasi-periodic nature of the solutions emerging out of the gures. The power-spectra analysis in g- ures 3.10(e)-(f) clearly indicates the existence of more than one dominant modes. The above set of observations is a clear demonstration of the fact that by invoking periodic boundary conditions, one can make the spiral waves break [30] and merge, and repeat the phenomenon at certain intervals of time.
Chapter 4
DYNAMICS OF SPIRAL WAVES UNDER THE EFFECT OF
OBSTACLES
4.1 Introduction
Having successfully accomplished the validation of the code resulting from the recon- structed HOC scheme for the Barkley model (3.1)-(3.2) in the previous chapter, we now endeavour to examine the dynamics of the spiral wave in and without the presence of obstacles through the same code. All numerical simulations in the present chapter are performed on a square of dimension[0,128]×[0,128]on a grid of size257×257. Moreover for all simulations, we have employed the zero-ux boundary conditions and the initial conditions described by equations (3.17)-(3.18).
Winfree [115] observed that the rotating spiral waves, which are encountered in the famous chemical reaction known as B-Z reaction [11,118], do not necessarily rotate peri- odically. His rigorous inspection led to the nding that these rotating spiral waves present in the B-Z reaction could also perform non-periodic rotation. For such non-periodic ro- tation, he coined a term called "meandering" of waves; and the movement of such spiral waves are characterized by tracing the position of the spiral tip. As such the spiral tip can execute either periodic (rigid) or non-periodic (meandering) rotation. Moreover, spi-
Some part of this chapter is published in [49].
Some part of this chapter is published in [27].
ral waves are also believed to be a major form of reentry underlying common cardiac arrhythmias [39,93] which may lead to brillation, where monomorphic tachycardia may correspond to a stationary anchored spiral wave, and polymorphic tachycardia to a me- andering spiral wave. This exemplies the signicance of the role played by the motion of spiral wave tip in the spiral wave dynamics. There are many numerical techniques to determine the spiral tip location [40, 65, 101], amongst which, we exploit the method of the intersection of the isocontours of the excitable and recovery variables.
In recent years, studies on the spiral wave dynamics have been of considerable interest among the scientists from diverse elds such as chemical, mathematical, physical and biological sciences [5, 9, 17, 28, 30, 40, 45, 105, 119]; and one such study is the transition of the spiral wave from rigid to the meandering rotation, which is characterized by the motion of the spiral tip. Such transition of the spiral wave in the heart plays a crucial role as it may correspond to the transition from a stable to polymorphic electric arrhythmias and then to brillation which some time may result in cardiac death. Barkley [9], studied the spiral wave dynamics by changing the reaction parameters; and explored that the transition of a spiral wave from simple to compound rotation occurs via a supercritical Hopf bifurcation. Further, in [5], Amdjadi proposed a numerical method to investigate the change in dynamics of the spiral wave; and this method is based on approximating the Euclidean norm of the state variables. It is surprising to note that in spite of the crucial role played by the spiral tip paths in the dynamics, almost all studies utilize only the state variables in order to examine the stability of the system. The current chapter envisages to use the spiral tip as a major tool to study the dynamics of the spiral waves through a comprehensive spectral density analysis. Besides, we depict some interesting patterns formed by the evolution of wave tips, viz., outward and inward petals. In the process, we have also compared these petals with the ones obtained from the explicit scheme. Besides, we provide a brief detail of the numerical method to evaluate the position of a spiral wave tip at a particular time. We further perform an analysis of the stability of the computed rotating spiral wave solutions through a numerical method explained in [5].
Chapter 4. DYNAMICS OF SPIRAL WAVES UNDER THE EFFECT OF OBSTACLES 39
In addition to the above, there has also been a surge of studies of late amongst math- ematicians and scientists alike on the interaction between rotating excitation waves and obstacle in excitable media [22,60,78,79,83,99], leading to some very interesting observa- tions. For example, in [99], Shajahan et al. studied the interaction of an obstacle to the spiral turbulence, which eventually leads to a simple rotating spiral wave; Panlov and Keener [83], also discussed the possibility of spiral wave initiation due to the interaction of successive wavefront with an obstacle. Further, a meandering spiral wave is seen to perform a periodic rotation after being pinned to an obstacle which is very similar to the brillation like activity changes to tachycardia regime [60]. Moreover, interaction of me- andering spiral waves with obstacles may result in the spiral wave break up [79]. Because of the above reasons, we also study the eect of an obstacle to the periodic rotating spiral waves and demonstrate how it aects the dynamics of the spiral wave.