Anomalous dispersion in the Belousov–Zhabotinsky reaction: Experiments and modeling

Nonlinear Instability of Periodic Traveling Waves

“…Our main result is a step in this direction with Theorem 7 which concludes that if an initial perturbation activates an appropriate subset of σ (L) ∩ {z ∈ C | Re z > 0}, then that initial perturbation is unstable. This result applies to many reaction-diffusion type systems with periodic equilibrium solutions, including but not limited to scalar reaction diffusion, FitzHugh-Nagumo [14], the Klausmeier model for vegetation stripe formulation [18,17], and the Belousov-Zhabotinskii reaction [4]. This methodology is robust enough that in Theorem 10 we show how it may be extended to dissipative systems of conservation laws such as Kuramoto-Sivashinsky [3,10] and the St. Venant equation [2,15].…”

“…Recall that our main Theorem 7 was proven in the context of reaction-diffusion type systems of the form (1.1): specifically for systems with no derivatives in the nonlinearity. Some examples of such systems would be scalar reaction diffusion, FitzHugh-Nagumo [14], the Klausmeier model for vegetation stripe formulation [18,17], and the Belousov-Zhabotinskii reaction [4]. However, our general methodology is sufficiently robust enough that it can apply more widely to dissipative systems of conservation laws.…”

Instability of periodic traveling wave solutions in a modified FitzHugh–Nagumo model for excitable media

“…In 1973, Kopell and Howard were first studied periodic traveling wave solutions (PTWs) by using a coupled reaction-diffusion equations [7] for oscillatory systems. The PTWs were also observed in biological [8,9], physical [10][11][12], chemical [13][14][15] and ecological systems [16][17][18] etc. The determination of the PTWs instability for excitable media is important for researchers because of the complex spatiotemporal patterns exhibited by the instability.…”

Periodic Pattern Formation Analysis Numerically in a Chemical Reaction-Diffusion System

“…In paper [1], authors first studied the PTW solutions of a coupled oscillatory R-D system. The PTW solutions have also been noticed in ecological [2,3,4], biological [5,6], physical [7,8,9]and chemical [10,11,12] systems. Method of continuation is a powerful method to study the PTW solutions of a R-D system [13].…”