LIMIT THEOREMS FOR SAMPLED DYNAMICAL SYSTEMS

2003 ◽  
Vol 03 (04) ◽  
pp. 477-497 ◽  
Author(s):  
NADINE GUILLOTIN-PLANTARD ◽  
DOMINIQUE SCHNEIDER
Keyword(s):  
Dynamical System ◽  
Random Walk ◽  
Dynamical Systems ◽  
Limit Theorems ◽  
Probability Space ◽  
Convergence In Distribution

Let [Formula: see text] be a dynamical system where [Formula: see text] is a probability space and T an invertible transformation preserving the measure μ. Let (Sk)k≥0 be a transient ℤ-random walk. Let f ∈ L2(μ) and H ∈ ]0,1[, we study the convergence in distribution of the sequence [Formula: see text] We also study the case when the random walk (Sk)k≥0 is replaced by an increasing deterministic subsequence of integers.

scholarly journals Limit theorems for a random walk with memory perturbed by a dynamical system

2020 ◽  
Vol 61 (10) ◽  
pp. 103303
Author(s):  
Cristian F. Coletti ◽  
Lucas R. de Lima ◽  
Renato J. Gava ◽  
Denis A. Luiz
Keyword(s):  
Dynamical System ◽  
Random Walk ◽  
Limit Theorems ◽  
With Memory

Limit theorems for stochastically perturbed dynamical systems

1995 ◽  
Vol 32 (2) ◽  
pp. 459-469 ◽  
Author(s):  
Krzysztof Łoskot ◽  
Ryszard Rudnicki
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Time ◽  
Classical Limit ◽  
Limit Theorems ◽  
Polish Space ◽  
Recurrence Formula ◽  
Stationary Measure ◽  
Random Elements ◽  
Stochastically Perturbed

We consider a discrete-time stochastically perturbed dynamical system on the Polish space given by the recurrence formula Xn = S(Xn–1, Yn), where Yn are i.i.d. random elements. We prove the existence of unique stationary measure and versions of classical limit theorems for the process (Xn).

Limit theorems for stochastically perturbed dynamical systems

1995 ◽  
Vol 32 (02) ◽  
pp. 459-469 ◽  
Author(s):  
Krzysztof Łoskot ◽  
Ryszard Rudnicki
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Time ◽  
Classical Limit ◽  
Limit Theorems ◽  
Polish Space ◽  
Recurrence Formula ◽  
Stationary Measure ◽  
Random Elements ◽  
Stochastically Perturbed

We consider a discrete-time stochastically perturbed dynamical system on the Polish space given by the recurrence formulaXn=S(Xn–1,Yn),whereYnare i.i.d. random elements. We prove the existence of unique stationary measure and versions of classical limit theorems for the process (Xn).

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.

A DYNAMICAL SYSTEM WITH CHAOTIC BEHAVIOR

1989 ◽  
Vol 03 (15) ◽  
pp. 1185-1188 ◽  
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.

scholarly journals The Dynamics of Musical Participation

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.

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

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.

On the age distribution of a Markov chain

1978 ◽  
Vol 15 (1) ◽  
pp. 65-77 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Limit Theorems ◽  
Age Distribution ◽  
Weak Limit ◽  
Absorbing State ◽  
Absorbing Markov Chain ◽  
Continuous Random Walk ◽  
A Chain ◽  
Watson Process

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain, obtained by restarting the original chain at a fixed state after each absorption. The limiting age, A(j), is the weak limit of the time given Xn = j (n → ∞).A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems for A (J) (J → ∞) are given for these examples.

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

1998 ◽  
Vol 18 (2) ◽  
pp. 471-486 ◽  
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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