Oscillatory wave bifurcation and spatiotemporal patterns in fractional subhyperbolic reaction-diffusion systems

The stability of a two-component time-fractional reaction-diffusion system during its linear stage is analyzed, revealing the emergence of a new type of instability at specific values of the fractional derivative order. With this instability, perturbations with finite wave numbers become unstable and cause spatially inhomogeneous oscillations. A comprehensive spectral study identified such type of instability across a broad range of wave numbers and system parameters. An increase in the fractional derivative order allows larger nonhomogeneous wave numbers to be linked with these unstable modes, which can trigger nonlinear nonhomogeneous oscillations and the formation of spatial oscillatory structures. The results of the linear stability analysis are confirmed by computer simulations of the fractional sub-hyperbolic Bonhoeffer-van der Pol-type reaction-diffusion system. It has been established that oscillatory-wave instability occurs also in hyperbolic systems and covers, in this limiting case, both the maximum range of wave numbers and system parameters.