1 / 53

Symbolic Analysis of Dynamical systems

Symbolic Analysis of Dynamical systems. Overview. Definition an simple properties Symbolic Image Periodic points Entropy Definition Calculation Example Is this method important for us ?. Definition. Space M Homeomorphism f

juliahoward
Download Presentation

Symbolic Analysis of Dynamical systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related