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3 Definition Space M Homeomorphism f Trajectory … x -1 =f -1 (x), x 0 =x, x 1 = f(x), x 2 = f 2 (x), …
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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
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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
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