CONTROLLING CHAOS AND THE INVERSE FROBENIUS–PERRON PROBLEM: GLOBAL STABILIZATION OF ARBITRARY INVARIANT MEASURES

2000 ◽  
Vol 10 (05) ◽  
pp. 1033-1050 ◽  
Author(s):  
ERIK M. BOLLT
Keyword(s):  
Dynamical System ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Stochastic Matrix ◽  
Open Loop ◽  
Global Stabilization ◽  
Invariant Density ◽  
Controlling Chaos

The inverse Frobenius–Perron problem (IFPP) is a global open-loop strategy to control chaos. The goal of our IFPP is to design a dynamical system in ℜn which is: (1) nearby the original dynamical system, and (2) has a desired invariant density. We reduce the question of stabilizing an arbitrary invariant measure, to the question of a hyperplane intersecting a unit hyperbox; several controllability theorems follow. We present a generalization of Baker maps with an arbitrary grammar and whose FP operator is the required stochastic matrix.

A variation on the variational principle and applications to entropy pairs

1997 ◽  
Vol 17 (1) ◽  
pp. 29-43 ◽  
Author(s):  
F. BLANCHARD ◽  
E. GLASNER ◽  
B. HOST
Keyword(s):  
Dynamical System ◽  
Variational Principle ◽  
Invariant Measure ◽  
Topological Entropy ◽  
Invariant Measures ◽  
Open Cover ◽  
Topological Dynamical System ◽  
Positive Topological Entropy ◽  
Finite Open Cover

The variational principle states that the topological entropy of a topological dynamical system is equal to the sup of the entropies of invariant measures. It is proved that for any finite open cover there is an invariant measure such that the topological entropy of this cover is less than or equal to the entropies of all finer partitions. One consequence of this result is that for any dynamical system with positive topological entropy there exists an invariant measure whose set of entropy pairs is equal to the set of topological entropy pairs.

STOCHASTIC PERTURBATIONS OF POSITION DEPENDENT RANDOM MAPS

2003 ◽  
Vol 03 (04) ◽  
pp. 545-557 ◽  
Author(s):  
WAEL BAHSOUN ◽  
PAWEŁ GÓRA ◽  
ABRAHAM BOYARSKY
Keyword(s):  
Dynamical System ◽  
Invariant Measures ◽  
Invariant Density ◽  
Absolutely Continuous ◽  
Stochastic Perturbations ◽  
Random Maps ◽  
Absolutely Continuous Invariant Measures ◽  
Absolutely Continuous Invariant

A random map is a dynamical system consisting of a collection of maps which are selected randomly by means of fixed probabilities at each iteration. In this note, we consider absolutely continuous invariant measures of random maps with position dependent probabilities and prove that they are stable under small stochastic perturbations. This result depends on a new lemma which handles arbitrarily small extra partition elements that may arise from the perturbation of the random map. For perturbations satisfying additional conditions, we give precise estimates of the error in the invariant density.

CONSTRUCTING INVARIANT MEASURES FROM DATA

1995 ◽  
Vol 05 (04) ◽  
pp. 1181-1192 ◽  
Author(s):  
GARY FROYLAND ◽  
KEVIN JUDD ◽  
ALISTAIR I. MEES ◽  
DAVID WATSON ◽  
KENJI MURAO
Keyword(s):  
Dynamical System ◽  
Experimental Data ◽  
Phase Space ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Reconstruction Technique ◽  
Continuous Approximation ◽  
Absolutely Continuous ◽  
Finite Set

We present a method of approximating an invariant measure of a dynamical system from a finite set of experimental data. Our reconstruction technique automatically provides us with a partition of phase space, and we assign each set in the partition a certain weight. By refining the partition, we may make our approximation to an invariant measure of the reconstructed system as accurate as we wish. Our method provides us with both a singular and an absolutely continuous approximation, so that the most suitable representation may be chosen for a particular problem.

scholarly journals INVARIANT MEASURES OF STOCHASTIC PERTURBATIONS OF DYNAMICAL SYSTEMS USING FOURIER APPROXIMATIONS

2011 ◽  
Vol 21 (01) ◽  
pp. 113-123 ◽  
Author(s):  
MD SHAFIQUL ISLAM ◽  
PAWEL GÓRA
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Invariant Measures ◽  
Invariant Density ◽  
Stochastic Perturbations ◽  
Fourier Approximation ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation ◽  
Frobenius Operator

We consider dynamical system τ : [0, 1] → [0, 1] and its stochastic perturbations [Formula: see text], N ≥ 1. Using Fourier approximation, we construct a finite dimensional approximation PN to a perturbed Perron–Frobenius operator. Let [Formula: see text] be an invariant density of τ and [Formula: see text] be a fixed point of PN. We show that [Formula: see text] converges in L1 to [Formula: see text].

scholarly journals Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 80
Author(s):  
Sergey Kryzhevich ◽  
Viktor Avrutin ◽  
Nikita Begun ◽  
Dmitrii Rachinskii ◽  
Khosro Tajbakhsh
Keyword(s):  
Risk Management ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Interval Exchange ◽  
Management Model ◽  
Closing Lemma ◽  
Symbolic Model ◽  
Metric Properties ◽  
Ergodic Measures ◽  
Maximal Invariant

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and demonstrated that an ITM, endowed with such a measure, is metrically conjugated to an interval exchange map (IEM). This allowed us to extend some properties of IEMs (e.g., an estimate of the number of ergodic measures and the minimality of the symbolic model) to ITMs. Further, we proved a version of the closing lemma and studied how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.

Model Based Control of Laminar Wake Using Fluidic Actuation

2010 ◽  
Vol 5 (4) ◽  
Author(s):  
Imran Akhtar ◽  
Ali H. Nayfeh
Keyword(s):  
Dynamical System ◽  
Stokes Equations ◽  
Control Function ◽  
Practical Importance ◽  
Nonlinear Damping ◽  
Open Loop ◽  
Pole Placement ◽  
Cylinder Surface ◽  
Control Input ◽  
Laminar Wake

Control of fluid-structure interaction is of practical importance from the perspective of wake modification and reduction of vortex-induced vibrations (VIVs). The aim of this study is to design a control to suppress vortex shedding. We perform a two-dimensional simulation of the flow past a circular cylinder using a parallel Computational Fluid Dynamics (CFD) solver. We record the velocity and pressure fields over a shedding cycle and compute the proper orthogonal decomposition (POD) modes of the divergence-free velocity and pressure, respectively. The Navier–Stokes equations are projected onto these POD modes to reduce the dynamical system to a set of ordinary-differential equations (ODEs). This dynamical system exhibits a limit cycle with negative linear damping and positive nonlinear damping. The reduced-order model is then modified by placing a pair of suction actuators and applying a control strategy using a control function method. We use the pressure POD mode distribution on the cylinder surface to optimally locate the actuators. We design a controller based on the linearized system and make it positively damped using pole-placement technique. The control-input settles to a constant value, suggesting constant suction through the actuators. We validate the results using CFD simulations in an open-loop setting and observe suppression of the hydrodynamic forces acting on the cylinder.

On Controlling Chaos in an Inflation–Unemployment Dynamical System

1999 ◽  
Vol 10 (9) ◽  
pp. 1567-1570 ◽  
Author(s):  
E. Ahmed ◽  
H ◽  
A. El-misiery ◽  
H.N. Agiza
Keyword(s):  
Dynamical System ◽  
Controlling Chaos

CHAOTIC BEHAVIOR OF HIGHER DIMENSIONAL TRANSFORMATIONS DEFINED ON COUNTABLE PARTITIONS

1993 ◽  
Vol 03 (04) ◽  
pp. 1045-1049
Author(s):  
A. BOYARSKY ◽  
Y. S. LOU
Keyword(s):  
Invariant Measure ◽  
Additional Condition ◽  
Invariant Measures ◽  
Chaotic Behavior ◽  
Finite Collection ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Higher Dimensional ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

Jablonski maps are higher dimensional maps defined on rectangular partitions with each component a function of only one variable. It is well known that expanding Jablonski maps have absolutely continuous invariant measures. In this note we consider Jablonski maps defined on countable partitions. Such maps occur, for example, in multivariable number theoretic problems. The main result establishes the existence of an absolutely continuous invariant measure for Jablonski maps on a countable partition with the additional condition that the images of all the partition elements form a finite collection. An example is given.

scholarly journals On Relatively Invariant Measures

1960 ◽  
Vol 12 ◽  
pp. 367-373
Author(s):  
Mark Mahowald
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Locally Compact Groups ◽  
Major Portion ◽  
Compact Groups ◽  
Locally Compact ◽  
Multiplier Function ◽  
Baire Measure

In this note we will discuss the question of the measurability of the multiplier function of a relatively invariant measure on a group. That is, for a group G, σ-ring S, and a measure μ defined on the sets of S, we assume: E in S, x in G implies xE is in S and μ(XE) = σ(x)μ(E) and study the measurability of the function σ(x).The problem was discussed by Halmos (1, p. 265), on locally compact groups and there the situation proved to be as nice as it could be, that is, if the measure is a non-trivial, relatively invariant Baire measure then the multiplier function is continuous. We prove two theorems for groups in which no topology is assumed. In the first theorem we assume a shearing condition and answer the question completely. The second theorem places a condition on the measure and weakens the shearing assumption. Its proof is complicated and occupies the major portion of this paper.

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