Spatio-temporal chaos, oscillations

Non-linear steady state bistability in Economics and Physiology Oscillations spatio-temporal oscillatory features in population Chaos and highly complex time series in Sociology and Economics Fractal growth of cities... 

Coming back to limit cycle oscillations shown by systems of ordinary differential equations, this simple mode of motion still seems to deserve some more attention, especially in relation to its role as a basic functional unit from which various dynamical complexities arise. This seems to occur in at least two ways. As mentioned above, one may start with a simple oscillator, increase [x, and obtain complicated behaviors this forms, in fact, a modern topic. However, another implication of this dynamical unit should not be left unnoticed. We should know that a limit cycle oscillator is also an important component system in various self-organization phenomena and also in other forms of spatio-temporal complexity such as turbulence. In this book, particular emphasis will be placed on this second aspect of oscillator systems. This naturally leads to the notion of the many-body theory of limit cycle oscillators we let many oscillators contact each other to form a field , and ask what modes of self-organiza-tion are possible or under what conditions spatio-temporal chaos arises, etc. A representative class of such many-oscillator systems in theory and practical application is that of the fields of diffusion-coupled oscillators (possibly with suitable modifications), so that this type of system will primarily be considered in this book. 

Reaction-diffusion systems are expected to show spatio-temporal chaos in various circumstances. A few specific cases will be discussed. They include the turbuhzation of uniform oscillations, of propagating wave fronts and of rotating spiral waves. 

Fig. 24. Turing-Hopf spatio-temporal chaos . Snapshots (a) and (c) are separated by 15 seconds corresponding to a half-period for the oscillation in the Hopf-hole . (b) Time-averaged image over one period of oscillation. The small scale mosaic corresponds to the Turing mode while the larger uniformly gray patches correspond to Hopf-holes . View size 4.2 mm x 4.2 mm. Experimental conditions as in Figure 23 but with [CH2(C00H)2] = 8.5 X 10 M.

Chaos control of spatio-temporal oscillations in resonant tunneling diodes. . 158... 

Equilibrium state —Linear steady state close to equilibrium —Steady state —> Non-linear steady state — Bifurcation phenomena —> Multi-stability —> Temporal and spatio-temporal oscillations —> More complex situations (chaos, turbulence, pattern formation, fractal growth). All these stages have been discussed in different chapters of the book. 

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