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Symbolic Analysis of Dynamical systems

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Overview Definition an simple properties Symbolic Image Periodic points Entropy

Definition Calculation Example

Is this method important for us?

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Definition Space M Homeomorphism f Trajectory … x-1=f-1(x), x0=x, x1 =

f(x), x2 = f2(x), …

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Two mapsf(x, y) = (1-

1.4x2+0.3y, x)

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Types of trajectories Fixed points Periodic points All other

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

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Symbolic Image

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Background Measuring Errors Computation

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Construction Covering C = {M(i)} Corresponding vertex

«i» Cell’s Image

f(M(i)) ∩ M(j) ≠ 0 Graph construction

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Construction

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Path Sequence …, i0, … , in … is a path

if ik and ik+1 connected by an edge.i j

k

l

m

n

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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

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Periodic points

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

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

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Algorithm1. 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.

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AlgorithmInitialcovering

SymbolicImage

Findperiodic

Subdivide

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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

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Example

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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

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Conclusion What is the main stream

Formulating problem Translation into Symbolic Image

language Applying subdivision process

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Entropy

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

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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

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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

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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?

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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

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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…

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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

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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

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Difference Consider real line, its covering by

an intervals and identical map. All trajectories is a fixed points

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Difference. First definition All sequences from two neighbor

intervals is admissible coding N(K)≥n*2N

h≥1 But identical map is really

determenic

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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

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Let’s start a calculation!

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

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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)

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

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

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S and P Extend a map s to P2 and P1 Denote s(P2)=P1

2

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Proposition h(P1

2) ≤h(P1) h(P1

2) ≤h(P2)

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Inscribed coverings Let C0, C1, … , Ck, … be inscribed

coverings st(zt+1) = zt , for M(zt+1) M(zt)

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Paths

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What’s happened?

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Theorem Pl

k Plk+1 and h(Pl

k)≥h(Plk+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*

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Example

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

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Result

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Or in graphics

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Answer h* = 0.46 + eps Results of other methods h(f) =

0.4651 Quiet good result

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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

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Thank you for your attention

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Applause

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It is a question time