Turbulence spatio-temporal complexity

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. 

We recall from our earlier discussion of chaos in one-dimensional continuous systems (see section 4.1) that period-doubling is not the only mechanism by which chaos can be generated. Another frequently occurring route to chaos is intermittency. But while intermittency in low dimensional dynamical systems appears to be constrained to purely temporal behavior [pomeau80], CMLs exhibit a spatio-temporal intermittency in which laminar eddies are intermixed with turbulent regions in a complex pattern in space-time. 

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. 

S. Kai, Y. Adachi, S. Nasimo Stability diagram, defect turbulence, and new patterns in electroconvection in nematics, in RE. Cladis, P. Palffy-Muhoray (eds.) Spatio-Temporal Patterns in Nonequilibrium Complex Systems, SFI Studies in the Sciences of Complexity, Addison-Wesley, (1994). 

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