scholarly journals Graph Labelings and Continuous Factors of Dynamical Systems

10.37236/2213 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Stephen M. Shea
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Topological Entropy ◽  
Directed Graphs ◽  
Sufficient Conditions ◽  
Perfect Graph ◽  
Shift Space ◽  
Finite Set ◽  
Vertex Set ◽  
Definition Of

A labeling of a graph is a function from the vertex set of the graph to some finite set.  Certain dynamical systems (such as topological Markov shifts) can be defined by directed graphs.  In these instances, a labeling of the graph defines a continuous, shift-commuting factor of the dynamical system.  We find sufficient conditions on the labeling to imply classification results for the factor dynamical system.  We define the topological entropy of a (directed or undirected) graph and its labelings in a way that is analogous to the definition of topological entropy for a shift space in symbolic dynamics.  We show, for example, if $G$ is a perfect graph, all proper $\chi(G)$-colorings of $G$ have the same entropy, where $\chi(G)$ is the chromatic number of $G$.

Li-Yorke chaos in models with backward dynamics

2016 ◽  
Vol 20 (5) ◽  
Author(s):  
David R. Stockman
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Overlapping Generations ◽  
Sufficient Conditions ◽  
Economic Models ◽  
Limited Commitment ◽  
Inverse Limits ◽  
Overlapping Generations Model ◽  
Definition Of ◽  
Cash In Advance

AbstractSome economic models like the cash-in-advance model of money, the overlapping generations model and a model of credit with limited commitment may have the property that the dynamical system characterizing equilibria in the model are multi-valued going forward in time, but single-valued going backward in time. Such models or dynamical systems are said to have backward dynamics. In such instances, what does it mean for a dynamical system (set-valued) to be chaotic? Furthermore, under what conditions are such dynamical systems chaotic? In this paper, I provide a definition of chaos that is in the spirit of Li and Yorke for a dynamical system with backward dynamics. I utilize the theory of inverse limits to provide sufficient conditions for such a dynamical system to be Li-Yorke chaotic.

scholarly journals What is the dynamical hypothesis?

1998 ◽  
Vol 21 (5) ◽  
pp. 633-634 ◽  
Author(s):  
Nick Chater ◽  
Ulrike Hahn
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Turing Machines ◽  
Definition Of ◽  
Dynamical Hypothesis

Van Gelder's specification of the dynamical hypothesis does not improve on previous notions. All three key attributes of dynamical systems apply to Turing machines and are hence too general. However, when a more restricted definition of a dynamical system is adopted, it becomes clear that the dynamical hypothesis is too underspecified to constitute an interesting cognitive claim.

scholarly journals Surjunctivity and topological rigidity of algebraic dynamical systems

2017 ◽  
Vol 39 (3) ◽  
pp. 604-619 ◽  
Author(s):  
SIDDHARTHA BHATTACHARYA ◽  
TULLIO CECCHERINI-SILBERSTEIN ◽  
MICHEL COORNAERT
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Sufficient Conditions ◽  
Countable Group ◽  
Continuous Group ◽  
Continuous Map ◽  
Algebraic Dynamical Systems ◽  
Metrizable Group ◽  
Group Automorphisms

Let$X$be a compact metrizable group and let$\unicode[STIX]{x1D6E4}$be a countable group acting on$X$by continuous group automorphisms. We give sufficient conditions under which the dynamical system$(X,\unicode[STIX]{x1D6E4})$is surjunctive, i.e. every injective continuous map$\unicode[STIX]{x1D70F}:X\rightarrow X$commuting with the action of$\unicode[STIX]{x1D6E4}$is surjective.

Thermodynamic Modeling for Discrete-Time Large-Scale Dynamical Systems

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Thermodynamic Modeling ◽  
Discrete Time ◽  
Large Scale ◽  
Type Inequality ◽  
Second Law Of Thermodynamics ◽  
Second Law ◽  
Thermodynamic Models ◽  
Definition Of

This chapter describes the thermodynamic modeling of discrete-time large-scale dynamical systems. In particular, it develops nonlinear discrete-time compartmental models that are consistent with thermodynamic principles. Since thermodynamic models are concerned with energy flow among subsystems, the chapter constructs a nonlinear compartmental dynamical system model characterized by conservation of energy and the first law of thermodynamics. It then provides a deterministic definition of entropy for a large-scale dynamical system that is consistent with the classical thermodynamic definition of entropy and shows that it satisfies a Clausius-type inequality leading to the law of entropy nonconservation. The chapter also considers nonconservation of entropy and the second law of thermodynamics, nonconservation of ectropy, semistability of discrete-time thermodynamic models, entropy increase and the second law of thermodynamics, and thermodynamic models with linear energy exchange.

scholarly journals Tsallis Entropy of Fuzzy Dynamical Systems

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 264
Author(s):  
Dagmar Markechová
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Conditional Entropy ◽  
Tsallis Entropy ◽  
Positive Real ◽  
Fuzzy Dynamical System ◽  
Fuzzy Partitions ◽  
Definition Of ◽  
Mathematical Behavior ◽  
Fuzzy Dynamical Systems

This article deals with the mathematical modeling of Tsallis entropy in fuzzy dynamical systems. At first, the concepts of Tsallis entropy and Tsallis conditional entropy of order where is a positive real number not equal to 1, of fuzzy partitions are introduced and their mathematical behavior is described. As an important result, we showed that the Tsallis entropy of fuzzy partitions of order satisfies the property of sub-additivity. This property permits the definition of the Tsallis entropy of order of a fuzzy dynamical system. It was shown that Tsallis entropy is an invariant under isomorphisms of fuzzy dynamical systems; thus, we acquired a tool for distinguishing some non-isomorphic fuzzy dynamical systems. Finally, we formulated a version of the Kolmogorov–Sinai theorem on generators for the case of the Tsallis entropy of a fuzzy dynamical system. The obtained results extend the results provided by Markechová and Riečan in Entropy, 2016, 18, 157, which are particularized to the case of logical entropy.

scholarly journals Connecting ergodicity and dimension in dynamical systems

1990 ◽  
Vol 10 (3) ◽  
pp. 451-462 ◽  
Author(s):  
C. D. Cutler
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Probability Measures ◽  
Pointwise Dimension ◽  
Continuous Maps ◽  
Borel Probability ◽  
The Relationship

AbstractIn this paper we make precise the relationship between local or pointwise dimension and the dimension structure of Borel probability measures on metric spaces. Sufficient conditions for exact-dimensionality of the stationary ergodic distributions associated with a dynamical system are obtained. A counterexample is provided to show that ergodicity alone is not sufficient to guarantee exactdimensionality even in the case of continuous maps or flows.

Remarks on Topological Entropy of Nonautonomous Dynamical Systems

2015 ◽  
Vol 25 (12) ◽  
pp. 1550158 ◽  
Author(s):  
Zhiming Li
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Topological Entropy ◽  
Nonautonomous Dynamical Systems ◽  
Nonautonomous Dynamical System

In this paper, we give several classical definitions of topological entropy (on a noncompact and noninvariant subset) for nonautonomous dynamical system. Furthermore, their relationships are established.

Algorithmic complexity of points in dynamical systems

1993 ◽  
Vol 13 (4) ◽  
pp. 807-830 ◽  
Author(s):  
Homer S. White
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Topological Entropy ◽  
Invariant Probability Measure ◽  
Algorithmic Complexity ◽  
Maximal Entropy ◽  
Property A ◽  
Open Set ◽  
Unique Measure ◽  
Set Of Points

AbstractThis work is based on the author's dissertation. We examine the algorithmic complexity (in the sense of Kolmogorov and Chaitin) of the orbits of points in dynamical systems. Extending a theorem of A. A. Brudno, we note that for any ergodic invariant probability measure on a compact dynamical system, almost every trajectory has a limiting complexity equal to the entropy of the system. We use these results to show that for minimal dynamical systems, and for systems with the tracking property (a weaker version of specification), the set of points whose trajectories have upper complexity equal to the topological entropy is residual. We give an example of a topologically transitive system with positive entropy for which an uncountable open set of points has upper complexity equal to zero. We use techniques from universal data compression to prove a recurrence theorem: if a compact dynamical system has a unique measure of maximal entropy, then any point whose lower complexity is equal to the topological entropy is generic for that unique measure. Finally, we discuss algorithmic versions of the theorem of Kamae on preservation of the class of normal sequences under selection by sequences of zero Kamae-entropy.

Asymptotic Independence and Uniform Distribution of Quantization Errors for Spatially Discretized Dynamical Systems

1998 ◽  
Vol 08 (07) ◽  
pp. 1479-1490 ◽  
Author(s):  
P. Diamond ◽  
I. Vladimirov
Keyword(s):  
Dynamical System ◽  
Computer Simulation ◽  
Dynamical Systems ◽  
Computer Arithmetic ◽  
Original System ◽  
Unit Interval ◽  
Asymptotic Independence ◽  
System Study ◽  
Polynomial Mappings ◽  
Finite Set

Computer simulation of dynamical systems involves a state space which is the finite set of computer arithmetic. Restricting state values to this grid produces roundoff effects which can be studied by replacing the original system with a spatially discretized dynamical system. Study of the deviation of the discretized trajectories from those of the original system reduces to that of appropriately defined quantization errors. As the grid is refined, the asymptotic behavior of these quantization errors follows probabilistic laws. These results are applied to discretized polynomial mappings of the unit interval.

scholarly journals New Results on Controllability for a Class of Fractional Integrodifferential Dynamical Systems with Delay in Banach Spaces

2021 ◽  
Vol 5 (3) ◽  
pp. 89
Author(s):  
Daliang Zhao
Keyword(s):  
Dynamical Systems ◽  
Banach Spaces ◽  
Dynamic Systems ◽  
Operator Theory ◽  
Lipschitz Continuity ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Complete Space ◽  
Definition Of

The present work addresses some new controllability results for a class of fractional integrodifferential dynamical systems with a delay in Banach spaces. Under the new definition of controllability , first introduced by us, we obtain some sufficient conditions of controllability for the considered dynamic systems. To conquer the difficulties arising from time delay, we also introduce a suitable delay item in a special complete space. In this work, a nonlinear item is not assumed to have Lipschitz continuity or other growth hypotheses compared with most existing articles. Our main tools are resolvent operator theory and fixed point theory. At last, an example is presented to explain our abstract conclusions.

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