Dynamical systems randomness

Intennittency, in tire context of chaotic dynamical systems, is characterized by long periods of nearly periodic or Taminar motion interspersed by chaotic bursts of random duration [28]. Witliin tliis broad phenomenological... 

Non-Homogeneous CA a characteristic feature of all CA rules defined so far has been that of homogeneity - each cell of the system evolves according to the same rule 0. Hartman and Vichniac [hartSfi] were the first to systematically study a class of inhomogeneous CA (INCA), in which the state-transition rules are allowed to vary from cell to cell. The simplest such example is one where there are only two different 0 s, which are randomly distributed throughout the lattice. Kauffman has studied the other extreme in which the lattice is randomly populated with all 2 possible boolean functions of k inputs. The results of such studies, as well as the relationship with the dynamics of random, mappings, are covered in detail in chapter 8.3. 

It would appear that the tradeoffs between these two requirements are optimized at the phase transition. Langton also cites a very similar relationship found by Crutchfield [crutch90] between a measure of machine complexity and the (per-symbol) entropy for the logistic map. The fact that the complexity/entropy relationship is so similar between two different classes of dynamical systems in turn suggests that what we are observing may be of fundamental importance complexity generically increases with randomness up until a phase transition is reached, beyond which further increases in randomness decrease complexity. We will have many occasions to return to this basic idea. 

We all have an intuitive feel for complexity. An oil painting by Picasso is obviously more complex than the random finger-paint doodles of a three-year-old. The works of Shakespeare are more complex than the rambling prose banged out on a typewriter by the proverbial band of monkeys. Our intuition tells us that complexity is usually greatest in systems whose components are arranged in some intricate difficult-to-understand pattern or, in the case of a dynamical system, when the outcome of some process is difficult to predict from its initial state. 

DDLab is an interactive graphics program for studying many different kinds of discrete dynamical systems. Arbitrary architectures can be defined, ranging from Id, 2d or 3d CA to random Boolean networks. 

Another approach used to automate the randomization process is by embedding pregenerated randomization lists in the data collection and management system. The main disadvantage of this approach is the security of the randomization lists. This can be remedied by having the system dynamically generate randomization numbers. 

L. Arnold, Random Dynamical Systems, Springer, Berlin, 1998. 

MSN. 104. 1. Prigogine and M. Courbage, Intrinsic randomness and intrinsic irreversibility in classical dynamical systems, Proc. Natl. Acad. Sci. USA 80, 2412-2416 (1983). 

It is most remarkable that the entropy production in a nonequilibrium steady state is directly related to the time asymmetry in the dynamical randomness of nonequilibrium fluctuations. The entropy production turns out to be the difference in the amounts of temporal disorder between the backward and forward paths or histories. In nonequilibrium steady states, the temporal disorder of the time reversals is larger than the temporal disorder h of the paths themselves. This is expressed by the principle of temporal ordering, according to which the typical paths are more ordered than their corresponding time reversals in nonequilibrium steady states. This principle is proved with nonequilibrium statistical mechanics and is a corollary of the second law of thermodynamics. Temporal ordering is possible out of equilibrium because of the increase of spatial disorder. There is thus no contradiction with Boltzmann s interpretation of the second law. Contrary to Boltzmann s interpretation, which deals with disorder in space at a fixed time, the principle of temporal ordering is concerned by order or disorder along the time axis, in the sequence of pictures of the nonequilibrium process filmed as a movie. The emphasis of the dynamical aspects is a recent trend that finds its roots in Shannon s information theory and modem dynamical systems theory. This can explain why we had to wait the last decade before these dynamical aspects of the second law were discovered. 

Within non-equilibrium thermodynamics, the driving force for relaxation is provided by deviations in the local chemical potential from it s equilibrium value. The rate at which such deviations relax is determined by the dominant kinetics in the physical system of interest. In addition, the thermal noise in the system randomly generates fluctuations. We thus describe the dynamics of a step edge by the equation. 

The relationships, rather similar in sense, for smooth dynamic systems were introduced in ref. 34 (p. 220 etc.) for studying the random perturbations via a method of action functionals. Close concepts can also be found in ref. 39. 

A.D. Ventzel and M.I. Freidlin, Fluctuations in Dynamic Systems Caused by Small Random Perturbations, Nauka, Moscow, 1979 (in Russian). 

We observe that T(x)-1 = T(—x). The dynamical system satisfying (a)-(c) is also called a measure preserving flow. We can now introduce random homogeneous fields, starting from the random variable / ... 

So, apart from the regular behavior, which is either steady-state, periodic, or quasi-periodic behavior (trajectory on a torus, Figure 3.2), some dynamic systems exhibit chaotic behavior, i.e., trajectories follow complicated aperiodic patterns that resemble randomness. Necessary but not sufficient conditions in order for chaotic behavior to take place in a system described by differential equations are that it must have dimension at least 3, and it must contain nonlinear terms. However, a system of three nonlinear differential equations need not exhibit chaotic behavior. This kind of behavior may not take place at all, and when it does, it usually occurs only for a specific range of the system s control parameters 9. 

Let us now consider how we might go about simulating the stochastic time evolution of a dynamic system. If we are given that the system is in the state n (t) at time t, then essentially all we need in order to move the system forward in time are the answers to two questions when will the next random event occur, and what kind of event will it be Because of the randomness of the events, we may expect that these two questions will be answered in only some probabilistic sense. 

The Art of Modeling Dynamic Systems Forecasting for Chaos, Randomness, Determinism... 

Morrison, Foster. The Art of Modeling Dynamic Systems Forecasting for Chaos, Randomness, Determinism. John Wiley Sons, Inc., New York. 1991. 

J. Moser, Stable and Random Motions in Dynamical Systems, Annals of Mathematical Studies, Vol. 77, Princeton University Press, Princeton, 1973. 

In the rest of this chapter, we will discuss briefly the theoretical ideas and the models employed for the study of failure of disordered solids, and other dynamical systems. In particular, we give a very brief summary of the percolation theory and the models (both lattice and continuum). The various lattice statistical exponents and the (fractal) dimensions are introduced here. We then give brief introduction to the concept of stress concentration around a sharp edge of a void or impurity cluster in a stressed solid. The concept is then extended to derive the extreme statistics of failure of randomly disordered solids. Here, we also discuss the competition between the percolation and the extreme statistics in determining the breakdown statistics of disordered solids. Finally, we discuss the self-organised criticality and some models showing such critical behaviour. 

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. 

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