2. The Universality of Chaos Some common features found in non-linear systems: Sequences of...
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Transcript of 2. The Universality of Chaos Some common features found in non-linear systems: Sequences of...
2. The Universality of Chaos
Some common features found in non-linear systems:
• Sequences of bifurcations (routes to chaos).
• Feigenbaum numbers.
Ref: P.Cvitanovic,”Universality in Chaos”, 2nd ed.,Adam Hilger (89)
Logistic Map:
Xn+1 = A xn
(1-xn)
Sine Map:
Xn+1 = B sin(Πxn)
1 2 3 4
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
2.2. The Feigenbaum Numbers See Appendix
F
1
1
n nn
n n
A A
A A
lim
4.66920161...
nn
1
lim 2.5029...n
nn
d
d
Rate of convergence
Size scaling:
A1S
3.236
Numerical Determination of δ
SnA
1
1
S SS n nn S S
n n
A A
A A
?
S
Value of A for the period 2n supercycle.
Supercycles: Orbits that contain xmax
limS Sn
n
Real Period-Doubling Systems
Examples: diode circuit, fluid convection, modulated laser, acoustic waves, chemical reactions, mechanical oscillations, etc.
Problem: δn measurable only for small n’s ( < 4 or 5 )
Lucky break: δ for logistic map converges very rapidly.
Logistic-map-like region
δ~ 3.57(10) δ~ 4.7(1)
All disagreement are within 20%
Traditional physics:
Common behavior common physical cause
eg. Harmonic oscillations in low-excited systems
→ potential ~ quadratic around its minimaChaos / complexity :
Common behavior universality
( Common features in state space )
2.4. Using δ (see Chap 5)
1
1
n nn
n n
A A
A A
1 1
1n n n nA A A A
1 1
1n n n nA A A A
Quantitative predictions on system with unsolvable or unknown dynamical equations.
Period –doubling systems:
(if exists )
1 1 2 1 2 12
1 1n n n n n nA A A A A A
1 2 12
1 1n n nA A A
2 3 23 2
1 1 1n n nA A A
2 1 2
1
1A A A
2 1 21
1 1
1A A A
1
11
1k
n k n k n kmm
A A A
2 1 21
1m
m
A AA A
Ex 2.4-1, Show that
2
2 11n
nA A A A
Logistic map :
1 23.00000 3.4931A A
13.49310 3.00000 3.49310
4.6693.
209
16274
A
1 23.23607 3.49856S SA A 1
(3.49856 3.23607) 3.498564.66920 1
3.5701
A
3.569946Actual A
Values taken from tables 2.1-2
2 1 2
1
1A A A A
Diode experiment:
1 230 3.470 2.505kHz V V
12.505 3.470 2.505
4.66920 12.242
V
2.26Actual V
1 285 0.438 0.946kHz V V
10.438 0.946 0.438
4.66920 10.29955
V
0.286Actual V
2 1 2
1
1A A A A
2.5. Feigenbaum Size Scaling
1
lim 2.5029...n
nn
d
d
α and δ are about the right size for experimental observations
Ratio of corresponding branches:
Also:
lim 2.5029...n
nn
d
d
2.6. Self-similarity
Fractals
No inherent size-scale
2.8. Models & Universality
Non-chaotic system :• Model: retain only relevant features.• Justification: prediction ~ observation.• Uniqueness: assumed.• Causality.
Chaotic systems :• Universality → models not unique.• Common features ( not physical )• No insight to microscopic structure gained.
• Complexity
2.9. Computers & Chaos
• Computers & graphics are crucial to study of chaos.
• Divergence of nearby trajectories + runoff errors / noise → chaos
• Question: Can any numerical computation be “meaningful” ?
• Partial answer : Calculated result is always a possible evolution of the system, even though it may not be the one you wish to investigate.
• Characteristics of system can still be studied in a statistical sense.