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

Two maps

f(x, y) = (1- 1.4x2+0.3y, 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 ik and ik+1 connected by an edge.

i j

k

l

m

n

12

Correspondences Cells – points Trajectories – paths

Be careful, not paths – trajectories

i-k-l, j-k-m – paths not corresponding to trajectories

i

i j

k

lm

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

1. Starting covering C with diameter d0.

2. Construct covering’s symbolic image.3. Find all his periodic points. Consider

union of cells. Name it Pk4. Subdivide this cells. New diameter

d0/2. Go to step 2.

17

Algorithm

Initialcovering

SymbolicImage

Findperiodic

Subdivide

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

1

kk

P

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) = ANα 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(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 Pl

k+1 and h(Plk)≥h(Pl

k+1)

Set of coded trajectories Codl = ∩k>lPl

k

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

51

Thank you for your attention

52

Applause

53

It is a question time