Chaos with coupled oscillators

The above discussion focused on the suppression of chaos by periodic oscillations. As shown by the bifurcation diagram in fig. 6.3, other dynamic transitions may be brought about by the coupling of two populations endowed with distinct d5mamic properties the global behaviour... 

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. 

As a consequence of the collective motion of the neutral system across the homogeneous magnetic field, a motional Stark term with a constant electric field arises. This Stark term inherently couples the center of mass and internal degrees of freedom and hence any change of the internal dynamics leaves its fingerprints on the dynamics of the center of mass. In particular the transition from regularity to chaos in the classical dynamics of the internal motion is accompanied in the center of mass motion by a transition from bounded oscillations to an unbounded diffusional motion. Since these observations are based on classical dynamics, it is a priori not clear whether the observed classical diffusion will survive quantization. From both the theoretical as well as experimental point of view a challenging question is therefore whether quantum interference effects will lead to a suppression of the diffusional motion, i.e. to quantum localization, or not. 

In the absence of coupling each autonomous oscillator, e R", follows its own local (predator-prey) dynamics x = F(xi, Xi) which we assume to be either a limit cycle or phase coherent chaos. The oscillators are coupled by local dispersal with strength e over a predefined set Ni of next neighbours and using the diagonal coupling matrix C = diag(ci, C2. .., c ). 

The nonlinearity of chemical processes received considerable attention in the chemical engineering literature. A large number of articles deal with stand-alone chemical reactors, as for example continuously stirred tank reactor (CSTR), tubular reactor with axial dispersion, and packed-bed reactor. The steady state and dynamic behaviour of these systems includes state multiplicity, isolated solutions, instability, sustained oscillations, and exotic phenomena as strange attractors and chaos. In all cases, the main source of nonlinearity is the positive feedback due to the recycle of heat, coupled with the dependence of the reaction rate versus temperature. 

An example of such a situation was considered at the end of the preceding chapter the system with two oscillatory isozymes (fig. 3.23) contains two instability mechanisms coupled in parallel. Compared with the model based on a single product-activated enzyme, new behavioural modes may be observed, such as birhythmicity, hard excitation and multiple oscillatory domains as a function of a control parameter. The modes of dynamic behaviour in that model remain, however, limited, because it contains only two variables. For complex oscillations such as bursting or chaos to occur, it is necessary that the system contain at least three variables. 

In line with this explanation, complex periodic oscillations, birhyth-micity and chaos disappear in the model when the concentration of the substrate is held constant in the course of time. The system then admits a unique oscillatory mechanism based on the coupling between selfamplification in cAMP synthesis and its sole limitation by receptor desensitization. 

Rul kov, N.F., A.R. Volkovskii, A. Rodriguez-Lozano, E. Del Rio M.G. Velarde. 1992. Mutual synchronization of chaotic self-oscillators with dissipative coupling. Int. J. Bif. Chaos 2 669-76. 

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.)...

The Ml manifestation of chaotic behavior requires that at least Mee equations of motion are coupled. This is not a strong requirement for a mechanical system. Each degree of freedom gives rise to two (Hamilton) equations of motion (or one, but second-order, Newton equation). So two coupled (anharmonic) oscillators can already exhibit chaotic behavior. Solving the trajectory for an atom colliding with a diatom requires six equations. If there are only two variables, one can get oscillatory solutions but not chaos. 

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