Hopf bifurcation in a reaction-diffusion system governed by the Fitz- Hugh-Nagumo equations
The author considers the Hopf bifurcation in a reaction-diffusion system modeled by the Fitz-Hugh-Nagumo (FHN) equations. Especially numerical methods are proposed to tackle this bifurcation. To approximate the FHN system, first the \(x\)-variable is rescaled, so that a uniform grid is mapped into a non-uniform one with concentration at \(x=-1\). This set-up provides an efficient spatial discretization for forming the simulating system of ordinary differential equations (ODE's). Having found a Hopf bifurcation point by using a damped Newton method and then an Euler-Newton predictor-corrector procedure, the author discusses how to trace the bifurcate periodic solution branch. For the problems encountered in the paper, perturbations up to the fourth order are needed. This gives rise to complex computations such that the author applies the Mathematica package. Having traced out a branch of periodic solutions the author proceeds to study their stability. This is done by linearizing the ODE system around a given \(2\pi\)-periodic solution located on the traced branch. Then the stability is determined by the spectrum of the Floquet matrix obtained through integrating the linearized equations with the identity matrix as the initial matrix. In the last part an example is demonstrated in which a periodic solution becomes highly unstable with period tending to infinity. This example is worked out by the proposed center manifold-Fourier spectral-pseudo-arc- length continuation, which seems to be quite suitable for studying Hopf bifurcation in FHN systems.
center manifold-Fourier spectral- pseudo-arc-length continuation
scientific article; zbMATH DE number 60652
scientific article