2. The Universality of Chaos Some common features found in non-linear systems: Sequences of...

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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)
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Page 1: 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.

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)

Page 2: 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.

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

Page 3: 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.

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

Page 4: 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.

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

Page 5: 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.

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)

Page 6: 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.

All disagreement are within 20%

Page 7: 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.

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 )

Page 8: 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.

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

Page 9: 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.

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

Page 10: 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.

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

Page 11: 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.

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

Page 12: 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.

2.6. Self-similarity

Fractals

No inherent size-scale

Page 13: 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.

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

Page 14: 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.

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.