Evolution of State Densities in Dynamical Systems: Entropy Analysis

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

2. EVOLUTION OF THE STATE DENSITIES AND THE ENTROPIES OF DYNAMICAL SYSTEMS Ferit Acar SAVACI Izmir Institute of Technology Dept. of Electrical Electronics Engineering Urla 35430, Izmir acarsavaci@iyte.edu.tr Serkan GÜNEL Dokuz Eylül University Dept. of Electrical Electronics Engineering Buca, 35160, Izmir serkan.gunel@eee.deu.edu.tr

3. 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

4. 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.

5. Frobenius—Perron Operators • Definition

6. 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

7.  Infinitesimal Operator of Frobenius—Perron Operator AFP : D(X)D(X) D(X): Space of state probability densities FPK equation in noiseless case

8. Stationary Solutions of FPK Eq. Reduced Fokker—Planck—Kolmogorov Equ.

9. X S(n-2) S x1 xn-1 xn S x0 fn D(X) P f1 fn-1 P Pn-2 f0 Frobenius—Perron Operator

10. S differentiable & invertible Calcutating FPO

11. Logistic Map • α=4

12. i=1,...,n d Observation vector : Estimating Densities from Observed Data • Parzen’s Estimator }  = 1

13. Logistic Map — Parzen’s Estimation

14. Logistic Map a =4

15. Chua’s Circuit E -E

16. Chua’s Circuit — Dynamics

17. Chua’s Circuit — The state densities p(x) Limit Cycles a x Double Scroll Period-2 Cycles Details Scrolls

18. Q : Energy transfered to the systemT : Temprature (Average Kinetic Energy) The 2nd Law of Thermodynamics & Information Entropy = 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

19. Entropy

20. Estimated Entropy – Logistic Map

21. Estimated Entropy — Chua’s Circuit

22. Estimated Entropy — Chua’s Circuit II

23. x(t)p(x) 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 

24. Entropy of Control Systems II • Mutual Information • Theorem

25. Uncertain v.s. Certain Controller • Theorem • Theorem

26. Principle of Maximum Entropy • Theorem

27. Optimal Control with Uncertain Controller II Select p(u) to maximize subject to

28. Optimal Control with Uncertain Controller III

29. Optimal Control with Uncertain Controller IV

30. Optimal Control with Uncertain Controller V • Theorem

31. 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.

32. 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.