Chaos onset

The Heaviside step ftinction 0(e+Ae) in Eq.(26) symbolically emphasizes the fact that the FO VP diffusion description is valid when the initial energy level lies above the chaos onset limit s =... 

Once this equation is solved for Ach, it also determines the critical energy cch and the critical value of the interaction V, fj for the chaos onset. Eq.(30) can be rephrased in terms of a condition which insures that the energy level e lies in the chaotic region ... 

In tills chapter we shall examine how such temporal and spatial stmctures arise in far-from-equilibrium chemical systems. We first examine spatially unifonn systems and develop tlie tlieoretical tools needed to analyse tlie behaviour of systems driven far from chemical equilibrium. We focus especially on tlie nature of chemical chaos, its characterization and the mechanisms for its onset. We tlien turn to spatially distributed systems and describe how regular and chaotic chemical patterns can fonn as a result of tlie interjilay between reaction and diffusion. 

We shall describe some of tire common types of chemical patterns observed in such experiments and comment on tire mechanisms for tlieir appearance. In keeping witli tire tlieme of tliis chapter we focus on states of spatio-temporal chaos or on regular chemical patterns tliat lead to such turbulent states. We shall touch only upon tire main aspects of tliis topic since tliere is a large variety of chemical patterns and many mechanisms for tlieir onset [2,3, 5,31]. 

Turbulence is generally understood to refer to a state of spatiotemporal chaos that is to say, a state in which chaos exists on all spatial and temporal scales. If the reader is unsatisfied with this description, it is perhaps because one of the many important open questions is how to rigorously define such a state. Much of our current understanding actually comes from hints obtained through the study of simpler dynamical systems, such as ordinary differential equations and discrete mappings (see chapter 4), which exhibit only temporal chaosJ The assumption has been that, at least for scenarios in which the velocity field fluctuates chaotically in time but remains relatively smooth in space, the underlying mechanisms for the onset of chaos in the simpler systems and the onset of the temporal turbulence in fluids are fundamentally the same. 

A theoretical framework for considering how the behavior of dynamical systems change as some parameter of the system is altered. Poincare first apphed the term bifurcation for the splitting of asymptotic states of a dynamical system. A bifurcation is a period-doubling, -quadrupling, etc., that precede the onset of chaos and represent the sudden appearance of a qualitatively different behavior as some parameter is varied. Bifurcations come in four basic varieties flip bifurcations, fold bifurcations, pitchfork bifurcations, and transcritical bifurcations. In principle, bifurcation theory allows one to understand qualitative changes of a system change to, or from, an equilibrium, periodic, or chaotic state. 

The title chaos is an unfortunate misnomer. As shown above, there is considerable structure displayed in the onset and interruption of chaotic behaviour. Even within the chaotic regime the value of x evolves according to completely defined rules—one value explicitly determines the next, with 100 per cent certainty. There is no randomness, no element of chance uncertainty or irregularity. If we know the rules and can measure a given starting condition exactly, even a chaotic pattern can be predicted exactly. 

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. 

There is a voluminous literature on steady-state multiplicity, oscillations (and chaos), and derivation of bifurcation points that define the conditions that lead to onset of these phenomena. For example, see Morbidelli et al. [ Reactor Steady-State Multiplicity and Stability, in Chemical Reaction and Reactor Engineering, Carberry and Varrria (eds), Marcel Dekker, 1987], Luss [ Steady State Multiplicity and Uniqueness... 

Wu YR, Chen CM, Chao CY, Ro LS, Lyu RK, Chang KH, Lee-Chen GJ (2007) Glucocerebrosidase gene mutation is a risk factor for early onset of Parkinson disease among Taiwanese. J Neurol Neurosurg Psychiatry 78 977-979... 

The plan of Chapter 5 is the following. In order to get a feeUng for the dynamics of the kicked molecule, we approximate it by a one-dimensional schematic model by restricting its motion to rotation in the x, z) plane and ignoring motion of the centre of mass. In this approximation the kicked molecule becomes the kicked rotor, probably the most widely studied model in quantum chaology. This model was introduced by Casati et al. in 1979. The classical mechanics of the kicked rotor is discussed in Section 5.1. Section 5.2 presents Chirikov s overlap criterion, which can be applied generally to estimate analytically the critical control parameter necessary for the onset of chaos. We use it here to estimate the onset of chaos in the kicked rotor model. The quantum mechanics of the kicked rotor is discussed in Section 5.3. In Section 5.4 we show that the results obtained for the quantum kicked rotor model are of immediate... 

The magnitude of the critical control parameter Kc can be estimated analytically with the help of Chirikov s criterion. This criterion is naturally derived in the context of the kicked rotor, and is introduced in the following section. The Chirikov criterion is also the basis for estimating the onset of chaos in many other chaotic atomic physics systems. Examples are presented in Chapters 6 and 7. 

Before we conclude this section we have to ask an important question Is it possible to observe experimentally the onset of global chaos predicted to occur at K = Kc On the classical level the answer to this... 

On the basis of the phase-space portraits presented in Fig. 7.5, we conclude that the ionization thresholds displayed in Fig. 7.3 do indeed correlate with the onset of chaos. The equation ionization thresholds = chaos thresholds is therefore justified. 

Fig. 20. Largest Liapunov exponent versus heat transfer coefficient for gas-phase coupling K for the NO/CO reaction modeled on a catalyst wafer. The wafer is composed of 100 cells with 10 randomly distributed active cells as shown on the grid. The numbers pointing at various regions indicate the onset of particular periodicities chaos is observed for 0.15 < i < 18. (From Ref. 232.)...

According to Chirikov [23J, the onset of chaos is associated with the overlap of neighboring nonlinear resonances. The overlap criterion, which bears the qualitative significance, uses the model of isolated resonances. Each resonance is characterized by its width, the maximum distance (in the action variable) from the elliptic fixed point The overlap means that the sum of the widths of two neighboring resonances is equal to the distance between two fixed points of these isolated resonances. We start with the pendulum Hamiltonian, which describes an isolated 1 N resonance under the periodic perturbation of frequency Q ... 

Before concluding this section to proceed to the next topic about a geometry that can bring the statistical nature into the system, we slightly touch upon the nonexponential behavior, which is a key to consider the nonstatistical behavior of chemical reaction. This is interesting from the viewpoint of the onset of statistical mechanics and also from the study of the role of chaos in mechanics. 

We can expect that the stagnant motions of which statistical properties are discussed in the previous section reflect a universal structure of the interface between torus and chaos. Here we discuss a conjecture concerning the universality in the stagnant layer based on the Nekhoroshev theorem, which proves the onset of a new time scale accompanied by the collapse of tori [9]. 

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