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EVOLUTION OF THE STATE DENSITIES AND THE ENTROPIES OF DYNAMICAL SYSTEMS

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### EVOLUTION OF THE STATE DENSITIES AND THE ENTROPIES OF DYNAMICAL SYSTEMS

This research has been supported in part by European Commission FP6 IYTE-Wireless Project (Contract No: 017442)

Ferit Acar SAVACI

Izmir Institute of Technology

Dept. of Electrical Electronics Engineering

Urla 35430, Izmir

Serkan GÜNEL

Dokuz Eylül University

Dept. of Electrical Electronics Engineering

Buca, 35160, Izmir

Contents

- Deterministic and indeterministic systems under influence of uncertainty...
- Evolution of state probability densities
- Transformations on probability densities Markov Operators & Frobenius—Perron Operators
- Estimating state probability densities using kernel density estimators
- Parzen’s density estimator
- Density estimates for Logistic Map and Chua’s Circuit
- The 2nd Law of Thermodynamics and Entropy
- Estimating Entropy of the system using kernel density estimations
- Entropy Estimates for Logistic Map and Chua’s Circuit
- Entropy in terms of Frobenius—Perron Operators
- Entropy and Control
- Maximum Entropy Principle
- Effects of external disturbance and observation on the system entropy
- Controller as a entropy changing device
- Equivalence of Maximum Entropy minimization to Optimal Control

Motivation

- Thermal noise effects all dynamical systems,
- Exciting the systems by noise can alter the dynamics radically causing interesting behavior such as stochastic resonances,
- Problems in chaos control with bifurcation parameter perturbations,
- Possibility of designing noise immune control systems
- Densities arise whenever there is uncertainty in system parameters, initial conditions etc. even if the systems under study are deterministic.

Frobenius—Perron Operators

- Definition

Evolution of The State Densities of The Stochastic Dynamical Systems

- i’s are 1D Wiener Processes

Fokker—Planck—KolmogorovEqu.

- p0(x) : Initial probability density of the states

Infinitesimal Operator of Frobenius—Perron Operator

AFP : D(X)D(X)

D(X): Space of state probability densities

FPK equation in noiseless case

Stationary Solutions of FPK Eq.

Reduced Fokker—Planck—Kolmogorov Equ.

Logistic Map

- α=4

Q : Energy transfered to the systemT : Temprature (Average Kinetic Energy)

The 2nd Law of Thermodynamics & InformationEntropy = Disorder of the system = Information gained by observing the system

Classius

Shannon

Boltzman

n: number of events pi: probability of event “i”

Thermodynamics

Information Theory

e(t)p(e)

x(t)p(x)

y(t)p(y)

Entropy in Control Systems I- External Effects

Change in entropy :

If State transition transformation is measure preserving, then

- Observer Entropy

Entropy of Control Systems II

- Mutual Information
- Theorem

Principle of Maximum Entropy

- Theorem

Summary I

- The state densities of nonlinear dynamical systems can be estimated using kernel density estimators using the observed data which can be used to determine the evolution of the entropy.
- Important observation : Topologically more complex the dynamics results in higher stationary entropy
- The evolution of uncertainty is a trackable problem in terms of Fokker—Planck—Kolmogorov formalism.
- The dynamics in the state space are converted to an infinite dimensional system given by a linear parabolic partial diff. equation (The FPK Equation),
- The solution of the FPK can be reduced to finding solution of a set of nonlinear algebraic equations by means of weighted residual schemes,
- The worst case entropy can be used as a performance criteria to be minimized(maximized) in order to force the system to a topologically simpler dynamics.

Summary II

- The (possibly stochastic) controller performance is determined by the information gather by the controller about the actual system state.
- A controller that reduces the entropy of a dynamical system must increase its entropy at least by the reduction to be achieved.

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