Oscillatory Waves in an Inhomogeneous Excitable Medium

An excitable medium is a nonlinear dissipative dynamical system able to sustain undamped wave propagation. Examples of excitable media include cardiac tissue and neural media. This thesis investigates a class of novel dynamical phenomena in excitable media, known collectively as oscillatory waves. The examples of oscillatory waves considered here are Tango waves, breathers and pulse generators. Tango waves, also known as oscillatory fronts, are characterized by a back-and-forth motion of a wavefront. Breathers, also known as oscillatory pulses, are characterized by an expanding and contracting motion of a pulse-shaped pattern. Pulse generators are breathers that periodically emit traveling pulses. These oscillatory waves are caused by the presence of either a timeindependent forcing or spatial variation in a model parameter. The main point of this thesis is that oscillatory fronts or pulses can be understood as emerging from the Hopf bifurcation of a stationary front or pulse, respectively. The secondary point of this thesis is to investigate secondary bifurcations connected to the Hopf bifurcation. Possible applications of oscillatory waves will also be discussed. Analytical and numerical methods are used to compute the equilibrium solutions and the spectrum of their linearizations for piecewise linear and cubic Fitzhugh-Nagumo models of excitable media. Combining this with numerical simulations allows us link oscillatory waves with a Hopf bifurcation. A center manifold reduction allows us to also connect the oscillatory waves to a generalized Hopf bifurcation.