Mixing in Oscillatory Media

Chemical reactions in the atmosphere form a large system of interacting chemical species described by nonlinear kinetics. It has been shown that certain components of this system can also exhibit oscillatory dynamics in some range of the parameters (see e.g. Poppe and Lustfeld (1996)). When the typical period of these oscillations 

Oscillations can also arise from the nonlinear interactions present in population dynamics (e.g. predator-prey systems). Mixing in this context is relevant for oceanic plankton populations. Phytoplankton-zooplankton (PZ) and other more complicated plankton population models often exhibit oscillatory solutions (see e.g. Edwards and Yool (2000)). Huisman and Weissing (1999) have shown that oscillations and chaotic fluctuations generated by the plankton population dynamics can provide a mechanism for the coexistence of the huge number of plankton species competing for only a few key resources (the plankton paradox ). In this chapter we review theoretical, numerical and experimental work on unsteady (mainly oscillatory) systems in the presence of mixing and stirring. 

---

Thus in the oscillator death state (Mirollo and Strogatz, 1990), the stirring completely inhibits the chemical oscillations, transforming the unstable steady state of the local dynamics into a stable attractor of the reaction-advection-diffusion system. This arises from the competition between the non-uniform frequencies that lead to the dispersion of the phase of the oscillations, while mixing tends to homogenize the system and bring it back to the unstable steady state. Therefore this behavior is only possible in oscillatory media with spatially non-uniform frequencies. The phase diagram of the non-uniform oscillatory system in the plane of the two main control parameters (v, 5) is shown in Fig. 8.4. 

---