Scroll Waves And Spiral Waves Under The Control Of Electric Fields

Wave patterns are ubiquitous in various physical, chemical and biological systems which are far away from the equilibrium. Recently, the discovery that the spiral waves and spiral wave turbulence play an essential role in cardiac arrhythmia and fibrillation makes the research about controlling such wave patterns popular. There are several instability mechanisms for two-dimensional spiral waves, such as Eckhaus instability induced by long-wave perturbation of traveling waves and meandering instability induced by Doppler effect. Three-dimensional spiral waves, also known as scroll waves, are extensions of two-dimensional spiral waves in the third dimension.Connecting phase singularities of two-dimensional spiral waves in different layers forms the filament line of three-dimensional scroll waves. Any mechanism that can induce instability of two-dimensional spiral waves can also induce instability of three-dimensional scroll waves. However, in the parameter regions where two-dimensional spiral waves are stable and rigidly rotating, scroll waves may develop into turbulence due to negative tension of the filament. Such negative tension-induced instability of scroll waves has been observed in the Belousov-Zhabotinsky (BZ)chemical reaction systems.In this thesis, we propose a method to restabilize scroll wave turbulence caused by negative tension in three-dimensional chemical excitable media using a circularly polarized external field. The stabilization mechanism is analyzed in terms of phase-locking caused by the external field, which makes the effective filament tension positive. The phase-locked scroll waves that have positive tension and higher frequency defy the turbulence and finally restore order. A linear theory based on Response function theory is presented for the change of filament tension caused by a generic rotating external field and its predictions closely agree with numerical simulations. It is believed that ventricular fibrillation is related to the negative tension-induced turbulence of scroll waves in three-dimensional excitable media, so it is of great significance to find ways of eliminating such spatiotemporal chaos.Next, we study spiral wave’s directional drift under different external electric fields. The drifting of spiral waves are key and basic problems in pattern dynamics theoretically, numerically, and experimentally. Spiral waves are shown to undergo a directional drift in the presence of alternating current and polarized electric fields when the frequencies of the electric fields are twice the spiral frequency, yet an analytical and quantitative explanation for this mechanism is still lacking. Here, using the response function theory, we propose a quantitative description for spiral wave drift induced by electric fields. The theory provides explicit equations for spiral wave drift speed and drift direction. Numerical simulations are performed demonstrating quantitative agreement with analytical results in both weakly and highly excitable media.Since alternating current and polarized electric fields have already been realized in BZ reaction systems, the results we obtained in this thesis are possible to be confirmed by experiments.