Chaos nonlinearly coupled dynamics

Abstract. Quantum chaos at finite-temperature is studied using a simple paradigm, two-dimensional coupled nonlinear oscillator. As an approach for the treatment of the finite-temperature a real-time finite-temperature field theory, thermofield dynamics, is used. It is found that increasing the temperature leads to a smooth transition from Poissonian to Gaussian distribution in nearest neighbor level spacing distribution. 

After an overview of the main papers devoted to chaos in lasers (Section I.A) and in nonlinear optical processes (Section I.B), we present a more detailed analysis of dynamics in a process of second-harmonic generation of light (Section II) as well as in Kerr oscillators (Section III). The last case we consider particularly in the context of coupled nonlinear systems. Finally, we present a cumulant approach to the problem of quantum corrections to the classical dynamics in second-harmonic generation and Kerr processes (Section IV). 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

Another very successful apphcation of nonlinear dynamics to the heart is through mathematical modehng. An example in which a simple model based on coupled oscillators describes the dynamics of agonist induced vasomotion is in the work of de Brouwer et al. [586], where the route to chaos in the presence of verapamil, a class IV antiarrhythmic drug, is studied. 

At the microscopic level, chemical reactions are dynamical phenomena in which nonlinear vibrational motions are strongly coupled with each other. Therefore, deterministic chaos in dynamical systems plays a crucial role in understanding chemical reactions. In particular, the dynamical origin of statistical behavior and the possibility of controlling reactions require analyses of chaotic behavior in multidimensional phase space. 

Hennig, D. and Esser, B. (1992). TYansfer dynamics of a quasiparticle in a nonlinear dimer coupled to an intersite vibration Chaos on the Bloch sphere, Phys. Rev. A46, 4569-4576. 

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. 

The subject remained dormant until the late nineteenth century, when the French mathematician Jules Henri Poincare pursued a couple of lines of study that led to different branches of topology. He looked at the relationship between the algebraic properties of an object and its geometrical properties, which gave rise to geometrical topology. He also studied physical processes in which classical mechanics seemed to be unable to describe the results, giving rise to the field of nonlinear dynamics, one of the branches that contributed to the rise of chaos theory a hundred years later. 

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