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 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.

A New Approach to the Control Design of Fuzzy Dynamical Systems

2011 ◽  
Vol 133 (6) ◽  
Author(s):  
Ye-Hwa Chen
Keyword(s):  
Dynamical Systems ◽  
Optimization Problem ◽  
Closed Form Solution ◽  
Control Design ◽  
Form Solution ◽  
Bottom Line ◽  
New Approach ◽  
Fuzzy Dynamical System ◽  
The Cost ◽  
Fuzzy Dynamical Systems

A new approach to the control design for fuzzy dynamical systems is proposed. For a fuzzy dynamical system, the uncertainty lies within a fuzzy set. The desirable system performance is twofold: one deterministic and one fuzzy. While the deterministic performance assures the bottom line, the fuzzy performance enhances the cost consideration. Under this setting, a class of robust controls is proposed. The control is deterministic and is not if-then rules-based. An optimal design problem associated with the control is then formulated as a constrained optimization problem. We show that the problem can be solved and the solution exists and is unique. The closed-form solution and cost are explicitly shown. The resulting control is able to guarantee the prescribed deterministic performance and minimize the average fuzzy performance.

scholarly journals Tsallis Entropy of Product MV-Algebra Dynamical Systems

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 589 ◽  
Author(s):  
Dagmar Markechová ◽  
Beloslav Riečan
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Mathematical Modelling ◽  
Tsallis Entropy ◽  
Entropy Measure ◽  
Mv Algebra

This paper is concerned with the mathematical modelling of Tsallis entropy in product MV-algebra dynamical systems. We define the Tsallis entropy of order α, where α∈(0, 1)∪(1, ∞), of a partition in a product MV-algebra and its conditional version and we examine their properties. Among other, it is shown that the Tsallis entropy of order α, where α > 1, has the property of sub-additivity. This property allows us to define, for α > 1, the Tsallis entropy of a product MV-algebra dynamical system. It is proven that the proposed entropy measure is invariant under isomorphism of product MV-algebra dynamical systems.

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 Entropy of Fuzzy Partitions and Entropy of Fuzzy Dynamical Systems

Entropy ◽  
2016 ◽  
Vol 18 (1) ◽  
pp. 19 ◽  
Author(s):  
Dagmar Markechová ◽  
Beloslav Riečan
Keyword(s):  
Dynamical Systems ◽  
Fuzzy Partitions ◽  
Fuzzy Dynamical Systems

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 $G$.

Thermodynamic Modeling of Large-Scale Interconnected Systems

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Thermodynamic Modeling ◽  
Energy Flow ◽  
Large Scale ◽  
Type Inequality ◽  
Dynamical System Theory ◽  
Flow Models ◽  
Energy Equipartition ◽  
Definition Of

This chapter describes the thermodynamic modeling of large-scale interconnected dynamical systems. Using compartmental dynamical system theory, it develops energy flow models possessing energy conservation and energy equipartition principles for large-scale dynamical systems. It then gives a deterministic definition of entropy for a large-scale dynamical system that is consistent with the classical definition of entropy and shows that it satisfies a Clausius-type inequality leading to the law of nonconservation of entropy. It also introduces the notion of ectropy as a measure of the tendency of a dynamical system to do useful work and grow more organized. It demonstrates how conservation of energy in an isolated thermodynamic large-scale system leads to nonconservation of ectropy and entropy. Finally, the chapter uses the system ectropy as a Lyapunov function candidate to show that the large-scale thermodynamic energy flow model has convergent trajectories to Lyapunov stable equilibria determined by the system initial subsystem energies.

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.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

scholarly journals Potential Well in Poincaré Recurrence

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 379
Author(s):  
Miguel Abadi ◽  
Vitor Amorim ◽  
Sandro Gallo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Recurrence Time ◽  
Potential Well ◽  
Exponential Approximation ◽  
Mixing Processes ◽  
System Perspective ◽  
Return Times ◽  
Error Terms

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For ϕ-mixing and ψ-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar.

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