Diffusion-induced chaos

An intriguing possibility is the existence of spatially irregular solutions.39,40 This type of diffusion induced chaos certainly warrants further study. 

Kuramoto, Y. (1978) Diffusion-induced chaos in reaction systems. Prog. Theor. Phys. Suppl. 64, 346... 

Fig. 8. Diffusion-induced chaos obtained when integrating the reaction-diffusion system (3) with Dirichlet boundary conditions (4) and the slow manifold (7) 6 = 10 ). The model parameters are e = 0.01, D = 0.05, a = A, uo = 2.5, u = -3.1. (a) Spatio-temporal variation of the variable u x t) coded as in Figure 6 (b) phase portrait (c) Poincar map (d) ID map.

Z. Neufeld, P.H. Haynes, and T. Tel. Chaotic mixing induced transitions in reaction-diffusion systems. Chaos, 12 426-438, 2002b. 

Lengyel, I. I.R. Epstein. 1991. Diffusion-induced instability in chemically reacting systems Steady-state multiplicity, oscillation, and chaos. Chaos 1 69 76. 

Figure 12.20 Bifurcation diagram for two physically coupled Degn-Harrison oscillators. (Reprinted with permission from Lengyel, I. Epstein, I. R. 1991. Diffusion-Induced Instability in Chemically Reacting Systems Steady-State Multiplicity, Oscillation, and Chaos, Chaos /, 69-76. 1991 American Institute of Physics.)...

People often speak of chemical turbulence whereby either of two distinct chaotic phenomena may be meant. One is the spatially uniform but temporally chaotic dynamics exhibited by the concentrations of chemical species, while the other involves spatial chaos too. For chemical turbulence in the latter sense, our attention is usually focused upon systems in which the local dynamics itself is non-chaotic, while such non-chaotic elements are coupled through diffussion to produce spatio-temporal chaos. In fact, if the local elements were already chaotic, the fields composed of them would trivially exhibit spatio-temporal chaos. Hence non-trivial chemical turbulence involving spatio-temporal chaos may be called diffusion-induced chemical turbulence. 

We shall now analyze the effect of fluctuations in the regime of diffusion-induced spatio-temporal chaos. The simplest model exhibiting this behavior is the Brusselator introduced already in Section 3, Equations (16), in the region of parameter values for which inequality (20) is violated. One can readily verify that this is indeed the case if >2 = 0 and a = 2. In the sequel... 

In this paper, we have given a brief summary of our recent work on heat conduction in one dimensional systems. We have shown that strong chaos is sufficient but not strictly necessary for the validity of the Fourier heat law. Indeed linear mixing can be sufficient to induce a diffusive process which ensures normal heat conductivity. 

These simple and mostly intuitive arguments apply mainly to chemical and biological reaction engineering systems. For other systems such as fluid flow systems, the sources and causes of bifurcation and chaos can be quite different. It is well established that the transition from laminar flow to turbulent flow is a transition from nonchaotic to chaotic behavior. The synergetic interaction between hydrodynamically induced bifurcation and chaos and that resulting from chemical and biological reaction/diffusion is not well studied and calls for an extensive multidisciplinary research effort. 

---