scholarly journals Weak convergence to extremal processes and record events for non-uniformly hyperbolic dynamical systems

2017 ◽  
Vol 39 (4) ◽  
pp. 980-1001
Author(s):  
MARK HOLLAND ◽  
MIKE TODD
Keyword(s):  
Dynamical System ◽  
Time Series ◽  
Dynamical Systems ◽  
Weak Convergence ◽  
Record Values ◽  
Process Approach ◽  
Record Times ◽  
Skorokhod Space ◽  
Extremal Process ◽  
Hyperbolic Dynamical Systems

For a measure-preserving dynamical system $({\mathcal{X}},f,\unicode[STIX]{x1D707})$, we consider the time series of maxima $M_{n}=\max \{X_{1},\ldots ,X_{n}\}$ associated to the process $X_{n}=\unicode[STIX]{x1D719}(f^{n-1}(x))$ generated by the dynamical system for some observable $\unicode[STIX]{x1D719}:{\mathcal{X}}\rightarrow \mathbb{R}$. Using a point-process approach we establish weak convergence of the process $Y_{n}(t)=a_{n}(M_{[nt]}-b_{n})$ to an extremal process $Y(t)$ for suitable scaling constants $a_{n},b_{n}\in \mathbb{R}$. Convergence here takes place in the Skorokhod space $\mathbb{D}(0,\infty )$ with the $J_{1}$ topology. We also establish distributional results for the record times and record values of the corresponding maxima process.

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.

DYNAMICAL SYSTEMS AND TESSELATIONS: DETECTING DETERMINISM IN DATA

1991 ◽  
Vol 01 (04) ◽  
pp. 777-794 ◽  
Author(s):  
ALISTAIR I. MEES
Keyword(s):  
Dynamical System ◽  
Time Series ◽  
Dynamical Systems ◽  
State Space ◽  
Scientific Theory ◽  
Invariant Manifolds ◽  
Geometric Information ◽  
Topological Information ◽  
Low Dimensional ◽  
Dynamical Information

Data measurements from a dynamical system may be used to build triangulations and tesselations which can — at least when the system has relatively low-dimensional attractors or invariant manifolds — give topological, geometric and dynamical information about the system. The data may consist of a time series, possibly reconstructed by embedding, or of several such series; transients can be put to good use. The topological information which can be found includes dimension and genus of a manifold containing the state space. Geometric information includes information about folds, branches and other chaos generators. Dynamical information is obtained by using the tesselation to construct a map with stated smoothness properties and having the same dynamics as the data; the resulting dynamical model may be tested in the way that any scientific theory may be tested, by making falsifiable predictions.

scholarly journals On Max-Semistable Laws and Extremes for Dynamical Systems

Entropy ◽  
2021 ◽  
Vol 23 (9) ◽  
pp. 1192
Author(s):  
Mark P. Holland ◽  
Alef E. Sterk
Keyword(s):  
Dynamical System ◽  
Time Series ◽  
Dynamical Systems ◽  
Regular Variation ◽  
Extreme Value Distribution ◽  
Regularity Conditions ◽  
Linear Scaling ◽  
Asymptotic Statistics ◽  
Generalised Extreme Value Distribution ◽  
Semistable Laws

Suppose (f,X,μ) is a measure preserving dynamical system and ϕ:X→R a measurable observable. Let Xi=ϕ∘fi−1 denote the time series of observations on the system, and consider the maxima process Mn:=max{X1,…,Xn}. Under linear scaling of Mn, its asymptotic statistics are usually captured by a three-parameter generalised extreme value distribution. This assumes certain regularity conditions on the measure density and the observable. We explore an alternative parametric distribution that can be used to model the extreme behaviour when the observables (or measure density) lack certain regular variation assumptions. The relevant distribution we study arises naturally as the limit for max-semistable processes. For piecewise uniformly expanding dynamical systems, we show that a max-semistable limit holds for the (linear) scaled maxima process.

scholarly journals Unstable entropy and unstable pressure for random partially hyperbolic dynamical systems

2020 ◽  
pp. 2150021
Author(s):  
Xinsheng Wang ◽  
Weisheng Wu ◽  
Yujun Zhu
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Variational Principle ◽  
Topological Entropy ◽  
Equilibrium States ◽  
Metric Entropy ◽  
Unstable Foliation ◽  
Hyperbolic Dynamical Systems ◽  
Partially Hyperbolic

Let [Formula: see text] be a [Formula: see text] random partially hyperbolic dynamical system. For the unstable foliation, the corresponding unstable metric entropy, unstable topological entropy and unstable pressure via the dynamics of [Formula: see text] on the unstable foliation are introduced and investigated. A version of Shannon–McMillan–Breiman Theorem for unstable metric entropy is given, and a variational principle for unstable pressure (and hence for unstable entropy) is obtained. Moreover, as an application of the variational principle, equilibrium states for the unstable pressure including Gibbs [Formula: see text]-states are investigated.

Dynamic Ridge Polynomial Higher Order Neural Network

2010 ◽  
pp. 255-268 ◽  
Author(s):  
Rozaida Ghazali ◽  
Abir Hussain ◽  
Nazri Mohd Nawi
Keyword(s):  
Neural Network ◽  
Dynamical System ◽  
Neural Networks ◽  
Time Series ◽  
Dynamical Systems ◽  
Dynamic Structure ◽  
Higher Order ◽  
System Control ◽  
Internal States ◽  
The Brain

This chapter proposes a novel Dynamic Ridge Polynomial Higher Order Neural Network (DRPHONN). The architecture of the new DRPHONN incorporates recurrent links into the structure of the ordinary Ridge Polynomial Higher Order Neural Network (RPHONN) (Shin & Ghosh, 1995). RPHONN is a type of feedforward Higher Order Neural Network (HONN) (Giles & Maxwell, 1987) which implements a static mapping of the input vectors. In order to model dynamical functions of the brain, it is essential to utilize a system that is capable of storing internal states and can implement complex dynamic system. Neural networks with recurrent connections are dynamical systems with temporal state representations. The dynamic structure approach has been successfully used for solving varieties of problems, such as time series forecasting (Zhang & Chan, 2000; Steil, 2006), approximating a dynamical system (Kimura & Nakano, 2000), forecasting a stream flow (Chang et al, 2004), and system control (Reyes et al, 2000). Motivated by the ability of recurrent dynamic systems in real world applications, the proposed DRPHONN architecture is presented in this chapter.

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.

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