Propagation of travelling waves in an open cubic autocatalytic chemical system

The reaction-diffusion travelling waves that can be initiated in an open isothermal chemical system governed by cubic autocatalytic kinetics are discussed. The system is shown to be capable of sustaining up to three spatially uniform steady states, the (trivial) unreacted state, which is always stable (a node), and two nontrivial states, one of which is always unstable (a saddle point). The third state can change its stability through Hopf bifurcation (both subcritical and supercritical). This allows the possibility of two sorts of travelling wave being established; there are wave profiles which connect the unreacted state ahead to the nontrivial state at the rear, and wave profiles (pulse waves) which have the unreacted state at both the front and rear. The conditions under which a particular wave is initiated are considered by both a discussion of the (ordinary) differential equations governing the travelling waves and by numerical integrations of an initial-value problem. This treatment also reveals the possibility of a stable travelling wave propagating through the system, leaving behind a temporally unstable stationary state. Under these conditions, spatiotemporal chaotic behaviour is seen to develop after the passage of the wave.