Infinite Dynamical Systems and Time Evolution: Rigorous Results

Assuming that the unpredictability associated with many dynamical systems is an artefact of the differential treatment of their time evolution, we propose here an integral treatment as an alternative. We make the assumption that time is two-dimensional, and that the time distribution in the past of observables characterizing the dynamical system, is some characteristic "projection" of its time distribution in the future. We show here how this method can be used to predict the time evolution of several dynamically complex systems over long time intervals. The present work can be considered as the natural next step to the assumption of nonderivability for subatomic dynamical systems to explain the connection between Quantum Mechanics and General Relativity. Here we propose that matter and space–time are not only nonderivable but also show structural discontinuity. Starting with this premise we use continuity and derivability, but only as a first order approximation to reality. Extrapolation to very large or very small scales, or to predictions over long time scales for many natural systems on intermediate scales (human scales), may lead to chaotic behavior, or to nondeterministic or probabilistic theories.

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ON DYNAMICAL SYSTEMS GENERATED BY TWO ALTERNATING VECTOR FIELDS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127496001302 ◽

1996 ◽

Vol 06 (11) ◽

pp. 2015-2030 ◽

Cited By ~ 5

Author(s):

A. KLÍČ ◽

P. POKORNÝ

Keyword(s):

Dynamical Systems ◽

Fixed Points ◽

Periodic Solutions ◽

Time Evolution ◽

Autonomous System ◽

Vector Fields ◽

Numerical Study ◽

Method Of Averaging ◽

Controlling Of Chaos

Dynamical systems with time evolution determined by two alternating vector fields are investigated both analytically and numerically. When the two vector fields are related by an involutory diffeomorphism G then the fixed points of G (either isolated or non-isolated) are shown to give rise to branches of periodic solutions of the resulting non-autonomous system. The method of averaging is used for small switching periods. Detailed numerical study of both conservative (“blinking vortex”) and dissipative (“blinking nodes”, “blinking cycles” and “blinking Lorenz”) systems shows that the technique of blinking can be used to initiating and controlling of chaos.

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Quasistatic dynamical systems

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2016.9 ◽

2016 ◽

Vol 37 (8) ◽

pp. 2556-2596 ◽

Cited By ~ 4

Author(s):

NEIL DOBBS ◽

MIKKO STENLUND

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Time Evolution ◽

Chaotic Maps ◽

Martingale Problem ◽

Initial Distribution ◽

Initial State ◽

Statistical Behavior ◽

Long Time ◽

Well Posed

We introduce the notion of a quasistatic dynamical system, which generalizes that of an ordinary dynamical system. Quasistatic dynamical systems are inspired by the namesake processes in thermodynamics, which are idealized processes where the observed system transforms (infinitesimally) slowly due to external influence, tracing out a continuous path of thermodynamic equilibria over an (infinitely) long time span. Time evolution of states under a quasistatic dynamical system is entirely deterministic, but choosing the initial state randomly renders the process a stochastic one. In the prototypical setting where the time evolution is specified by strongly chaotic maps on the circle, we obtain a description of the statistical behavior as a stochastic diffusion process, under surprisingly mild conditions on the initial distribution, by solving a well-posed martingale problem. We also consider various admissible ways of centering the process, with the curious conclusion that the ‘obvious’ centering suggested by the initial distribution sometimes fails to yield the expected diffusion.

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Cellular Automata

Journal of Organizational and End User Computing ◽

10.4018/joeuc.2017010103 ◽

2017 ◽

Vol 29 (1) ◽

pp. 42-50 ◽

Cited By ~ 2

Author(s):

Rupali Bhardwaj ◽

Anil Upadhyay

Keyword(s):

Dynamical Systems ◽

Cellular Automata ◽

Empirical Study ◽

Time Evolution ◽

Discrete Dynamical Systems ◽

The State ◽

Dimensional Lattice ◽

Elementary Cellular Automata

Cellular automata (CA) are discrete dynamical systems consist of a regular finite grid of cell; each cell encapsulating an equal portion of the state, and arranged spatially in a regular fashion to form an n-dimensional lattice. A cellular automata is like computers, data represented by initial configurations which is processed by time evolution to produce output. This paper is an empirical study of elementary cellular automata which includes concepts of rule equivalence, evolution of cellular automata and classification of cellular automata. In addition, explanation of behaviour of cellular automata is revealed through example.

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Time Evolution of Local Complexity Measures and Aperiodic Perturbations of Nonlinear Dynamical Systems

NATO ASI Series - Measures of Complexity and Chaos ◽

10.1007/978-1-4757-0623-9_18 ◽

1989 ◽

pp. 155-171 ◽

Cited By ~ 2

Author(s):

Gottfried Mayer-Kress ◽

Alfred Hübler

Keyword(s):

Dynamical Systems ◽

Time Evolution ◽

Nonlinear Dynamical Systems ◽

Complexity Measures ◽

Nonlinear Dynamical ◽

Local Complexity

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Intermittent quasistatic dynamical systems: weak convergence of fluctuations

Nonautonomous Dynamical Systems ◽

10.1515/msds-2018-0002 ◽

2018 ◽

Vol 5 (1) ◽

pp. 8-34 ◽

Cited By ~ 1

Author(s):

Juho Leppänen

Keyword(s):

Dynamical Systems ◽

Weak Convergence ◽

Time Evolution ◽

Martingale Problem ◽

Statistical Properties ◽

Time Dependent ◽

Stochastic Diffusion ◽

Interval Maps ◽

Well Posed ◽

Over Time

Abstract This paper is about statistical properties of quasistatic dynamical systems. These are a class of non-stationary systems that model situations where the dynamics change very slowly over time due to external influences. We focus on the case where the time-evolution is described by intermittent interval maps (Pomeau-Manneville maps) with time-dependent parameters. In a suitable range of parameters, we obtain a description of the statistical properties as a stochastic diffusion, by solving a well-posed martingale problem. The results extend those of a related recent study due to Dobbs and Stenlund, which concerned the case of quasistatic (uniformly) expanding systems.

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