Tsallis Entropy of Product MV-Algebra Dynamical Systems

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.

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Tsallis Entropy of Fuzzy Dynamical Systems

Mathematics ◽

10.3390/math6110264 ◽

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.

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Integrable two-dimensional dynamical systems and the characteristic Lie algebras

Proceedings of the Mavlyutov Institute of Mechanics ◽

10.21662/uim2007.1.023 ◽

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.

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Potential Well in Poincaré Recurrence

Entropy ◽

10.3390/e23030379 ◽

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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A DYNAMICAL SYSTEM WITH CHAOTIC BEHAVIOR

Modern Physics Letters B ◽

10.1142/s0217984989001813 ◽

1989 ◽

Vol 03 (15) ◽

pp. 1185-1188 ◽

Cited By ~ 1

Author(s):

J. SEIMENIS

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Infinite Number ◽

Chaotic Behavior ◽

Equations Of Motion ◽

Hamiltonian Dynamical Systems

We develop a method to find solutions of the equations of motion in Hamiltonian Dynamical Systems. We apply this method to the system [Formula: see text] We study the case a → 0 and we find that in this case the system has an infinite number of period dubling bifurcations.

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The Dynamics of Musical Participation

Musicae Scientiae ◽

10.1177/1029864920988319 ◽

2021 ◽

pp. 102986492098831

Author(s):

Andrea Schiavio ◽

Pieter-Jan Maes ◽

Dylan van der Schyff

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Systems Theory ◽

Dynamical Systems Theory ◽

Coordination Dynamics ◽

Musical Experience ◽

The Core ◽

Interactive Dynamics ◽

Interdisciplinary Scholarship ◽

Core Features

In this paper we argue that our comprehension of musical participation—the complex network of interactive dynamics involved in collaborative musical experience—can benefit from an analysis inspired by the existing frameworks of dynamical systems theory and coordination dynamics. These approaches can offer novel theoretical tools to help music researchers describe a number of central aspects of joint musical experience in greater detail, such as prediction, adaptivity, social cohesion, reciprocity, and reward. While most musicians involved in collective forms of musicking already have some familiarity with these terms and their associated experiences, we currently lack an analytical vocabulary to approach them in a more targeted way. To fill this gap, we adopt insights from these frameworks to suggest that musical participation may be advantageously characterized as an open, non-equilibrium, dynamical system. In particular, we suggest that research informed by dynamical systems theory might stimulate new interdisciplinary scholarship at the crossroads of musicology, psychology, philosophy, and cognitive (neuro)science, pointing toward new understandings of the core features of musical participation.

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Induced transformations of recurrent a.m.s. dynamical systems

Stochastics and Dynamics ◽

10.1142/s0219493715500100 ◽

2015 ◽

Vol 15 (02) ◽

pp. 1550010

Author(s):

Sheng Huang ◽

Mikael Skoglund

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Random Process ◽

Ergodic Theorem ◽

Finite Measure ◽

Stationary Random Process ◽

Finite State ◽

Ratio Ergodic Theorem

This note proves that an induced transformation with respect to a finite measure set of a recurrent asymptotically mean stationary dynamical system with a sigma-finite measure is asymptotically mean stationary. Consequently, the Shannon–McMillan–Breiman theorem, as well as the Shannon–McMillan theorem, holds for all reduced processes of any finite-state recurrent asymptotically mean stationary random process. As a by-product, a ratio ergodic theorem for asymptotically mean stationary dynamical systems is presented.

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Almost all $S$-integer dynamical systems have many periodic points

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385798113378 ◽

1998 ◽

Vol 18 (2) ◽

pp. 471-486 ◽

Cited By ~ 4

Author(s):

T. B. WARD

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Cellular Automata ◽

Zeta Function ◽

Periodic Points ◽

Radius Of Convergence ◽

Dynamical Zeta Function ◽

Hyperbolic Dynamical Systems ◽

Almost All

We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case almost every zeta function is irrational.We conjecture that for almost every ergodic $S$-integer dynamical system the radius of convergence of the zeta function is exactly $\exp(-h_{\rm top})<1$ and the zeta function is irrational.In an important geometric case (the $S$-integer systems corresponding to isometric extensions of the full $p$-shift or, more generally, linear algebraic cellular automata on the full $p$-shift) we show that the conjecture holds with the possible exception of at most two primes $p$.Finally, we explicitly describe the structure of $S$-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.

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On the Analysis of Time-Periodic Nonlinear Hamiltonian Dynamical Systems

Volume 3A: 15th Biennial Conference on Mechanical Vibration and Noise — Vibration of Nonlinear, Random, and Time-Varying Systems ◽

10.1115/detc1995-0277 ◽

1995 ◽

Author(s):

Eric A. Butcher ◽

S. C. Sinha

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Normal Form ◽

Linear Part ◽

Normal Form Theory ◽

Mathieu’S Equation ◽

Form Theory ◽

Hamiltonian Dynamical Systems ◽

Time Invariant ◽

Time Periodic

Abstract In this paper, some analysis techniques for general time-periodic nonlinear Hamiltonian dynamical systems have been presented. Unlike the traditional perturbation or averaging methods, these techniques are applicable to systems whose Hamiltonians contain ‘strong’ parametric excitation terms. First, the well-known Liapunov-Floquet (L-F) transformation is utilized to convert the time-periodic dynamical system to a form in which the linear pan is time invariant. At this stage two viable alternatives are suggested. In the first approach, the resulting dynamical system is transformed to a Hamiltonian normal form through an application of permutation matrices. It is demonstrated that this approach is simple and straightforward as opposed to the traditional methods where a complicated set of algebraic manipulations are required. Since these operations yield Hamiltonians whose quadratic parts are integrable and time-invariant, further analysis can be carried out by the application of action-angle coordinate transformation and Hamiltonian perturbation theory. In the second approach, the resulting quasilinear time-periodic system (with a time-invariant linear part) is directly analyzed via time-dependent normal form theory. In many instances, the system can be analyzed via time-independent normal form theory or by the method of averaging. Examples of a nonlinear Mathieu’s equation and coupled nonlinear Mathieu’s equations are included and some preliminary results are presented.

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Theory of hybrid dynamical systems and its applications to biological and medical systems

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2010.0237 ◽

2010 ◽

Vol 368 (1930) ◽

pp. 4893-4914 ◽

Cited By ~ 65

Author(s):

Kazuyuki Aihara ◽

Hideyuki Suzuki

Keyword(s):

Nonlinear Dynamics ◽

Dynamical Systems ◽

Mathematical Modelling ◽

Systems Theory ◽

Control Systems ◽

Hybrid Systems ◽

Hybrid Dynamical Systems ◽

Medical Systems ◽

Theme Issue ◽

And Control

In this introductory article, we survey the contents of this Theme Issue. This Theme Issue deals with a fertile region of hybrid dynamical systems that are characterized by the coexistence of continuous and discrete dynamics. It is now well known that there exist many hybrid dynamical systems with discontinuities such as impact, switching, friction and sliding. The first aim of this Issue is to discuss recent developments in understanding nonlinear dynamics of hybrid dynamical systems in the two main theoretical fields of dynamical systems theory and control systems theory. A combined study of the hybrid systems dynamics in the two theoretical fields might contribute to a more comprehensive understanding of hybrid dynamical systems. In addition, mathematical modelling by hybrid dynamical systems is particularly important for understanding the nonlinear dynamics of biological and medical systems as they have many discontinuities such as threshold-triggered firing in neurons, on–off switching of gene expression by a transcription factor, division in cells and certain types of chronotherapy for prostate cancer. Hence, the second aim is to discuss recent applications of hybrid dynamical systems in biology and medicine. Thus, this Issue is not only general to serve as a survey of recent progress in hybrid systems theory but also specific to introduce interesting and stimulating applications of hybrid systems in biology and medicine. As the introduction to the topics in this Theme Issue, we provide a brief history of nonlinear dynamics and mathematical modelling, different mathematical models of hybrid dynamical systems, the relationship between dynamical systems theory and control systems theory, examples of complex behaviour in a simple neuron model and its variants, applications of hybrid dynamical systems in biology and medicine as a road map of articles in this Theme Issue and future directions of hybrid systems modelling.

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Categories as Mathematical Models

10.1093/oso/9780198748991.003.0016 ◽

2018 ◽

Cited By ~ 1

Author(s):

David I. Spivak

Keyword(s):

Dynamical Systems ◽

Mathematical Modelling ◽

Mathematical Models ◽

Category Theory ◽

Vector Spaces ◽

Mathematical Objects ◽

Working Definition ◽

Modelling Framework ◽

Categorical Models ◽

Definition Of

Category theory is presented as a mathematical modelling framework that highlights the relationships between objects, rather than the objects in themselves. A working definition of model is given, and several examples of mathematical objects, such as vector spaces, groups, and dynamical systems, are considered as categorical models.

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