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), x 0 =x, x 1 = f(x), x 2 = f 2 (x), …

4 Two maps f(x, y) = (1- 1.4x y, x)

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 i k and i k+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 i 1, …, i k, where i 1 = i k )

16 Algorithm 1. Starting covering C with diameter d Construct covering’s symbolic image. 3. Find all his periodic points. Consider union of cells. Name it Pk 4. Subdivide this cells. New diameter d 0 /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 P k 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 {x k = f k (x),k = 0,...N-1} of length N Let the sequence ξ(x) = {i k, k = 0,...N-1}, where x k є M(i k ) 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) = Ba hN, 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) ≈ a hp h=0. K(N) = AN α and 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(i 0 ) Find M(i 1 ) such that M(i 0 ) ∩ f -1 (M(i 1 )) ≠ 0 Find M(i 2 ) such that M(i 0 ) ∩ f -1 (M(i 1 )) ∩ f -2 (M(i 2 )) ≠ 0 And so on…

29 Strong mathematical definition Denote by M(i 0 i 1..i N-1 ) This sequences corresponds to real trajectories Aggregation of sets M(i 0 i 1..i N-1 ) is an open covering

30 Strong mathematical definition Consider minimal subcovering Let ρ(C N ) 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*2 N h≥1 But identical map is really determenic

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

34 Let’s start a calculation!

35 Sequences entropy a 1, …, a n – symbols Some set of sequences P h(P) = lim log K(N)/N – entropy

36 Subdivision Consider covering C and its Symbolic Image G 1 Consider subcoverind D and its Symbolic Image G 2 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 : G 2 -> G 1. s(i, k) = i Edges are mapped to edges

38 Space of vertices P G ={ξ = {v i }: v i connected to v i+1 } I.e. space of admissible paths

39 S and P Extend a map s to P 2 and P 1 Denote s(P 2 )=P 1 2

40 Proposition h(P 1 2 ) ≤h(P 1 ) h(P 1 2 ) ≤h(P 2 )

41 Inscribed coverings Let C 0, C 1, …, C k, … be inscribed coverings s t (z t+1 ) = z t, for M(z t+1 ) M(z t )

42 Paths

43 What’s happened?

44 Theorem P l k P l k+1 and h(P l k )≥h(P l k+1 ) Set of coded trajectories Cod l = ∩ k>l P l k h l =h(Cod l )=lim k->+∞ h l k, h l grows by l If f is a Lipshitch’s mapping then sequence h l has a finite limit h* and h(f) ≤h*

45 Example

46 Map and subcoverings f(x, y) = (1-1.4x y, x)

47 Result

48 Or in graphics

49 Answer h* = eps Results of other methods h(f) = 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

51 Thank you for your attention

52 Applause

53 It is a question time