Deterministic chaotic systems

Recent studies have indicated that fluidized beds may be deterministic chaotic systems (Daw etal.,1990 Daw and Harlow, 1991 Schouten and van den Bleek, 1991 van den Bleek and Schouten, 1993). Such systems are characterized by a limited ability to predict their evolution with time. If fluidized beds are deterministic chaotic systems, the scaling laws should reflect the restricted predictability associated with such systems. 

For motion in a diss tive deterministic chaotic system, deterministic chaotic diffiision along a coordinate x is developed according to the following condition (5P) ... 

For the past three decades deterministic classical systems with chaotic dynamics have been the subject of extensive study (Chirikov, 1979)-(Sagdeev et. al., 1988). Dynamical chaos is a phenomenon peculiar to the deterministic systems, i.e. the systems whose motion in some state space is completely determined by a given interaction and the initial conditions. Under certain initial conditions the behaviour of these systems is unpredictable. 

Marek, M., and Schreiber, I., Chaotic Behaviour of Deterministic Dissipative Systems . 2nd edn., Cambrige University Press, Cambridge (1995). 

This approach needs modification as soon as multiple attracting periodic trajectories exist for a particular set of operating parameters. A conceptually different modification will be necessary to account for attractors which are not simply periodic. Quasi-periodic solutions, characterized by multiple frequencies, are the first type one should expect these are by no means exotic but occur generally in several periodically forced systems. Deterministic chaotic situations, arising from the system nonlinearities (and not the stochastic responses due to random noise) need not be discarded as intractable (Wolf et al., 1986 Shaw, 1981). 

In many cases ordinary differential equations (ODEs) provide adequate models of chemical reactors. When partial differential equations become necessary, their discretization will again lead to large systems of ODEs. Numerical methods for the location, continuation and stability analysis of periodic and quasi-periodic trajectories of systems of coupled nonlinear ODEs (both autonomous and nonautonomous) are extensively used in this work. We are not concerned with the numerical description of deterministic chaotic trajectories where they occur, we have merely inferred them from bifurcation sequences known to lead to deterministic chaos. Extensive literature, as well as a wide choice of algorithms, is available for the numerical analysis of periodic trajectories (Keller, 1976,1977 Curry, 1979 Doedel, 1981 Seydel, 1981 Schwartz, 1983 Kubicek and Hlavacek, 1983 Aluko and Chang, 1984). 

M. Marek and I. Schreiber, Chaotic Behavior in Deterministic Dissipative Systems... 

In this chapter we have reviewed the development of unimolecular reaction rate theory for systems that exhibit deterministic chaos. Our attention is focused on a number of classical statistical theories developed in our group. These theories, applicable to two- or three-dimensional systems, have predicted reaction rate constants that are in good agreement with experimental data. We have also introduced some quantum and semiclassical approaches to unimolecular reaction rate theory and presented some interesting results on the quantum-classical difference in energy transport in classically chaotic systems. There exist numerous other studies that are not considered in this chapter but are of general interest to unimolecular reaction rate theory. 

In the absence of deterministic chaos in the time evolution of the wave functions of bounded systems, the focus of quantum chaos research shifted towards the identification of the fingerprints of classical chaos in the properties of -0- The usual procedure is to start with a classically chaotic system, quantize it canonically, and then try to identify those characteristics of V in the semiclassical limit (ft -) 0) that give away the chaoticity of the underlying classically chaotic system. 

Chaotic time series have been obtained from a wide variety of experimental systems. Figures 16 and 17 show examples of the irregular time series found in various cases. It is somewhat problematic to assign the term chaotic to these reported time series because it was rarely investigated whether the time series were deterministically chaotic in the strict sense of the word. Therefore, when we use the expression chaotic, we are well aware that there is, in many cases, no proof for chaos in the oscillation patterns. The few reports wherein a thorough analysis has been performed (64,104) do though show the existence of deterministically chaotic oscillations. It would be beyond the scope of this review to describe the methods... 

Chaotic behavior and synchronization in heterogeneous catalysis are closely related. Partial synchronization can lead to a complex time series, generated by superposition of several periodic oscillators, and can in some cases result in deterministically chaotic behavior. In addition to the fact that macroscopically observable oscillations exist (which demonstrates that synchronization occurs in these systems), a number of experiments show the influence of a synchronizing force on all the hierarchical levels mentioned earlier. Sheintuch (294) analyzed on a general level the problem of communication between two cells. He concluded that if the gas-phase concentration is the autocatalytic variable, then synchronization is attained in all cases. However, if the gas-phase concentration were the nonautocatalytic variable, then this would lead to symmetry breaking and the formation of spatial structures. When surface variables are the model variables, the existence of synchrony is dependent upon the size scale. Only two-variable models were analyzed, and no such strict analysis has been provided for models with two or more surface concentrations, mass balances, or heat balances. There are, however, several studies that focused on a certain system and a certain synchronization mechanism. 

It is true that the hyperbolic system is an ideal dynamical system to understand from where randomness comes into the completely deterministic law and why the loss of memory is inevitable in the chaotic system, but generic physical and chemical systems do not belong strictly to such ideal systems. They are not uniformly hyperbolic, meaning that invariant structures are heterogeneously distributed in phase space, and there may not exist a lower bound of instability. It is believed that dynamical systems of such classes are certainly to be explored for our understanding of dynamical aspects of all relevant physical and chemical phenomena. 

Can chaotic systems be differentiated from random fluctuations Yes, even though the dynamics are complex and resemble a stochastic system, they can be differentiated from a truly stochastic system. Figure 11.15 compares the plots of N = 10,001 and a selection of points chosen randomly from 13,000 to 0. Note that after approximately 10 time intervals, the dynamics of both are quite wild and it would be difficult to distinguish one from another as far as one is deterministic and the other chaotic. However, there is a simple way to differentiate these two alternatives the phase-space plot. 

Plots of the population size vs. the population size at a specific time interval reveals the structure of a chaotic system. In A, derived from the deterministic yet stochastic-looking dynamics, a pattern readily forms that is characteristic of the underlying equation. In B, a shotgun blast or random pattern is revealed. 

M.Marek and LSchreiber, Chaotic Behavior of Deterministic Dissipative Systems, Cambridge University Press, 1995. 

That is, the system is completely deterministic there is no randomness at all. However, the evolution of the values of the variables is exquisitely sensitive to their exact initial values. That is, very slightly different initial values of the variables will soon evolve in very different ways. Because these initial conditions can only be specified with finite accuracy, the exact long-term behavior of the system is unpredictable, although its statistical properties can often be determined. Thus, chaotic systems are deterministic, yet unpredictable in the long run. 

Chaotic system - A complex system whose behavior is governed by deterministic laws but whose evolution can vary drastically when small changes are made in the initial conditions. Charge - See Electric charge. 

Ruelle D. (1989). Chaotic evolution and strange attractors the statistical analysis of time series for deterministic nonlinear systems. Cambridge University Press, Cambridge, U.K. 

Chaotic systems do not give random outputs, but they have a random appearance. The output is predictable (deterministic) if the underlying relationship is known. 

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