Symbolic Analysis of Dynamical systems

1. Symbolic Analysis of Dynamical systems

2. Overview • Definition an simple properties • Symbolic Image • Periodic points • Entropy • Definition • Calculation • Example • Is this method important for us?

3. Definition • Space M • Homeomorphism f • Trajectory … x-1=f-1(x), x0=x, x1 = f(x), x2 = f2(x), …

4. f(x, y) = (1- 1.4x2+0.3y, x) Two maps

5. Types of trajectories • Fixed points • Periodic points • All other

6. Applications • Prey-predator • Pendulum • Three body’s problem • Many, many other …

7. Symbolic Image

8. Background • Measuring Errors • Computation

9. Construction • Covering C = {M(i)} • Corresponding vertex «i» • Cell’s Image f(M(i)) ∩ M(j) ≠ 0 • Graph construction

10. Construction

11. Path • Sequence …, i0, … , in … is a path if ik and ik+1 connected by an edge.

12. Correspondences • Cells – points • Trajectories – paths Be careful, not paths – trajectories • i-k-l, j-k-m – paths not corresponding to trajectories

13. Periodic points

14. What we are looking for? • Fix p • Try to find all p-periodic points

15. Main idea If we have correspondences cell – vertex and trajectory – path, then to each periodic trajectory will correspond periodic path (path i1, … , ik, where i1 =ik)

16. Algorithm • Starting covering C with diameter d0. • Construct covering’s symbolic image. • Find all his periodic points. Consider union of cells. Name it Pk • Subdivide this cells. New diameter d0/2. Go to step 2.

17. Algorithm

18. Algorithm's results • Theorem. = Per(p), where Per(p) is the set of p-periodic points of the dynamical system. • So we may found Per(p) with any given precision

19. Example

20. Applications • Unfortunately we can’t guarantee the existence of p-periodic point in cell from Pk • Ussually we apply this method to get stating approximations for more precise algorithms, for example for Newton Method

21. Conclusion • What is the main stream • Formulating problem • Translation into Symbolic Image language • Applying subdivision process

22. Entropy

23. What is the reason? • Strange trajectories • We call this effect chaos

24. Intuitive definition part I • Consider finite open covering C={M(i)} • Consider trajectory {xk = fk(x),k = 0, . . .N-1}of length N • Let the sequence ξ(x) = {ik, k = 0, . . .N-1}, where xk є M(ik)be a coding • Be careful. One trajectory more than one coding

25. Intuitive definition part II • Let K(N) be number of admissible coding • Consider usually a=2 or a=e • h = 0 – simple system • h > 0 – chaotic behavior • In case h>0, K(N) = BahN, where B is a constant

26. Why exactly this? Situation. • We know N-length part of the code of the trajectory • We want to know next p symbols of the code • How many possibilities we have?

27. Why exactly this? Answer. • In average we will have K(N+p)/K(N) admissible answers • h > 0. K(N+p)/K(N) ≈ ahp • h=0. K(N) = ANαand K(N+p)/K(N) ≈ (1+p/N) α • h>0 we can’t say anything, h=0 we may give an answer for large N

28. Strong mathematical definition • Consider finite open covering C={M(i)} • Consider M(i0) • Find M(i1) such that M(i0)∩f-1(M(i1)) ≠ 0 • Find M(i2)such that M(i0)∩f-1(M(i1))∩f-2(M(i2)) ≠ 0 • And so on…

29. Strong mathematical definition • Denote by M(i0i1..iN-1) • This sequences corresponds to real trajectories • Aggregation of sets M(i0i1..iN-1)is an open covering

30. Strong mathematical definition • Consider minimal subcovering • Let ρ(CN) be number of its elements • be entropy of covering C • called topology entropy of the map f

31. Difference • Consider real line, its covering by an intervals and identical map. • All trajectories is a fixed points

32. Difference. First definition • All sequences from two neighbor intervals is admissible coding • N(K)≥n*2N • h≥1 • But identical map is really determenic

33. Difference. Second definition • M(i0i1..iN-1) may be only intervals and intersections of two neighbors • ρ(CN) = N, we may take C as a subcovering • h=0

34. Let’s start a calculation!

35. Sequences entropy • a1, … , an– symbols • Some set of sequences P • h(P) = lim log K(N)/N– entropy

36. Subdivision • Consider covering C and its Symbolic Image G1 • Consider subcoverind D and its Symbolic Image G2 • Define cells of D as M(i,k) such that M(i,k) subdivide M(i) in C • Corresponding vertices as (i,k)

37. Map s • Define map s : G2 -> G1. s(i, k) = i • Edges are mapped to edges

38. Space of vertices PG ={ξ = {vi}: vi connected to vi+1} I.e. space of admissible paths

39. S and P • Extend a map s to P2 and P1 • Denote s(P2)=P12

40. Proposition • h(P12) ≤h(P1) • h(P12) ≤h(P2)

41. Inscribed coverings • Let C0, C1, … , Ck, … be inscribed coverings • st(zt+1) = zt, for M(zt+1) M(zt)

42. Paths

43. What’s happened?

44. Theorem • Plk Plk+1 and h(Plk)≥h(Plk+1) • Set of coded trajectories Codl = ∩k>lPlk • hl=h(Codl)=limk->+∞hlk, hl grows by l • If f is a Lipshitch’s mapping then sequence hl has a finite limit h* and h(f) ≤h*

45. Example

46. Map and subcoverings • f(x, y) = (1-1.4x2+0.3y, x)

47. Result

48. Or in graphics

49. Answer • h* = 0.46 + eps • Results of other methods h(f) = 0.4651 • Quiet good result

50. Conclusion • Method is corresponding to real measuring • Method is computer-oriented • We may solve most of its problems • It is simple in simple task and may solve difficult tasks • Quiet good results