scholarly journals Extreme values for Benedicks–Carleson quadratic maps

2008 ◽  
Vol 28 (4) ◽  
pp. 1117-1133 ◽  
Author(s):  
ANA CRISTINA MOREIRA FREITAS ◽  
JORGE MILHAZES FREITAS
Keyword(s):  
Dynamical Systems ◽  
Stochastic Processes ◽  
Extreme Values ◽  
Extreme Value Distribution ◽  
Random Variable ◽  
Absolutely Continuous ◽  
Quadratic Maps ◽  
Chaotic Dynamical Systems ◽  
Stationary Stochastic Processes ◽  
Absolutely Continuous Invariant

AbstractWe consider the quadratic family of maps given by fa(x)=1−ax2 with x∈[−1,1], where a is a Benedicks–Carleson parameter. For each of these chaotic dynamical systems we study the extreme value distribution of the stationary stochastic processes X0,X1,… , given by Xn=fan, for every integer n≥0, where each random variable Xn is distributed according to the unique absolutely continuous, invariant probability of fa. Using techniques developed by Benedicks and Carleson, we show that the limiting distribution of Mn=max {X0,…,Xn−1} is the same as that which would apply if the sequence X0,X1,… was independent and identically distributed. This result allows us to conclude that the asymptotic distribution of Mn is of type III (Weibull).

DERIVING CHAOTIC DYNAMICAL SYSTEMS FROM ENERGY FUNCTIONALS

2001 ◽  
Vol 01 (03) ◽  
pp. 377-388 ◽  
Author(s):  
PAUL BRACKEN ◽  
PAWEŁ GÓRA ◽  
ABRAHAM BOYARSKY
Keyword(s):  
Differential Equation ◽  
Dynamical Systems ◽  
Order Differential Equation ◽  
Lagrange Equation ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Energy Functionals ◽  
Chaotic Dynamical Systems ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

Simple one-dimensional chaotic dynamical systems are derived by optimizing energy functionals. The Euler–Lagrange equation yields a nonlinear second-order differential equation whose solution yields a 2–1 map which admits an absolutely continuous invariant measure. The solutions of the differential equation are studied.

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

Speed of convergence to an extreme value distribution for non-uniformly hyperbolic dynamical systems

2015 ◽  
Vol 15 (04) ◽  
pp. 1550028 ◽  
Author(s):  
Mark Holland ◽  
Matthew Nicol
Keyword(s):  
Dynamical Systems ◽  
Convergence Rates ◽  
Hyperbolic Systems ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Recurrence Time ◽  
Quadratic Maps ◽  
Uniformly Hyperbolic Systems ◽  
Unique Maximum

Suppose (f, 𝒳, ν) is a dynamical system and ϕ : 𝒳 → ℝ is an observation with a unique maximum at a (generic) point in 𝒳. We consider the time series of successive maxima Mn(x) := max {ϕ(x),…,ϕ ◦ fn-1(x)}. Recent works have focused on the distributional convergence of such maxima (under suitable normalization) to an extreme value distribution. In this paper, for certain dynamical systems, we establish convergence rates to the limiting distribution. In contrast to the case of i.i.d. random variables, the convergence rates depend on the rate of mixing and the recurrence time statistics. For a range of applications, including uniformly expanding maps, quadratic maps, and intermittent maps, we establish corresponding convergence rates. We also establish convergence rates for certain hyperbolic systems such as Anosov systems, and discuss convergence rates for non-uniformly hyperbolic systems, such as Hénon maps.

CLARIFYING CHAOS III: CHAOTIC AND STOCHASTIC PROCESSES, CHAOTIC RESONANCE, AND NUMBER THEORY

1999 ◽  
Vol 09 (05) ◽  
pp. 785-803 ◽  
Author(s):  
RAY BROWN ◽  
LEON O. CHUA
Keyword(s):  
Time Series ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Stochastic Processes ◽  
Prediction Models ◽  
Prime Numbers ◽  
Chaotic Time Series ◽  
Chaotic Dynamical Systems ◽  
Estimation And Prediction ◽  
The Relationship

In this tutorial we continue our program of clarifying chaos by examining the relationship between chaotic and stochastic processes. To do this, we construct chaotic analogs of stochastic processes, stochastic differential equations, and discuss estimation and prediction models. The conclusion of this section is that from the composition of simple nonlinear periodic dynamical systems arise chaotic dynamical systems, and from the time-series of chaotic solutions of finite-difference and differential equations are formed chaotic processes, the analogs of stochastic processes. Chaotic processes are formed from chaotic dynamical systems in at least two ways. One is by the superposition of a large class of chaotic time-series. The second is through the compression of the time-scale of a chaotic time-series. As stochastic processes that arise from uniform random variables are not constructable, and chaotic processes are constructable, we conclude that chaotic processes are primary and that stochastic processes are idealizations of chaotic processes. Also, we begin to explore the relationship between the prime numbers and the possible role they may play in the formation of chaos.

BRANCHING PROCESSES AND COMPUTATIONAL COLLAPSE OF DISCRETIZED UNIMODAL MAPPINGS

2002 ◽  
Vol 12 (12) ◽  
pp. 2847-2867
Author(s):  
P. DIAMOND ◽  
I. VLADIMIROV
Keyword(s):  
Dynamical Systems ◽  
Branching Process ◽  
Invariant Measures ◽  
Transition Probabilities ◽  
Branching Processes ◽  
Absolutely Continuous ◽  
Logistic Mapping ◽  
Finite Set ◽  
Absolutely Continuous Invariant ◽  
Unimodal Mappings

In computer simulations of smooth dynamical systems, the original phase space is replaced by machine arithmetic, which is a finite set. The resulting spatially discretized dynamical systems do not inherit all functional properties of the original systems, such as surjectivity and existence of absolutely continuous invariant measures. This can lead to computational collapse to fixed points or short cycles. The paper studies loss of such properties in spatial discretizations of dynamical systems induced by unimodal mappings of the unit interval. The problem reduces to studying set-valued negative semitrajectories of the discretized system. As the grid is refined, the asymptotic behavior of the cardinality structure of the semitrajectories follows probabilistic laws corresponding to a branching process. The transition probabilities of this process are explicitly calculated. These results are illustrated by the example of the discretized logistic mapping.

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.

Pseudorandom Number Generators Based on Chaotic Dynamical Systems

2001 ◽  
Vol 08 (02) ◽  
pp. 137-146 ◽  
Author(s):  
Janusz Szczepański ◽  
Zbigniew Kotulski
Keyword(s):  
Dynamical Systems ◽  
Communication Systems ◽  
Initial Conditions ◽  
Pseudorandom Number ◽  
New Approach ◽  
Chaotic Dynamical Systems ◽  
Pseudorandom Number Generators ◽  
Transmission Of Information ◽  
Extreme Sensitivity ◽  
Contemporary Technology

Pseudorandom number generators are used in many areas of contemporary technology such as modern communication systems and engineering applications. In recent years a new approach to secure transmission of information based on the application of the theory of chaotic dynamical systems has been developed. In this paper we present a method of generating pseudorandom numbers applying discrete chaotic dynamical systems. The idea of construction of chaotic pseudorandom number generators (CPRNG) intrinsically exploits the property of extreme sensitivity of trajectories to small changes of initial conditions, since the generated bits are associated with trajectories in an appropriate way. To ensure good statistical properties of the CPRBG (which determine its quality) we assume that the dynamical systems used are also ergodic or preferably mixing. Finally, since chaotic systems often appear in realistic physical situations, we suggest a physical model of CPRNG.

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