Quantifying Chaos

1. Introduction

2. Time Series of Dynamical Variables

3. Lyapunov Exponents

4. Universal Scaling of the Lyapunov Exponents

5. Invariant Measures

6. Kolmogorov-Sinai Entropy

7. Fractal Dimensions

8. Correlation Dimension & a Computational History

9. Comments & Conclusions

1. Introduction

Why quantify chaos?• To distinguish chaos from noise / complexities.

• To determine active degrees of freedom.

• To discover universality classes.

• To relate chaotic parameters to physical quantities.

2. Time Series of Dynamical Variables

• (Discrete) time series data:– x(t0), x(t1), …, x(tn) – Time-sampled (stroboscopic) measurements– Poincare section values

• Real measurements & calculations are always discrete.• Time series of 1 variable of n-D system :

– If properly chosen, essential features of system can be re-constructed:• Bifurcations• Chaos on-set

– Choice of sampling interval is crucial if noise is present (see Chap 10)• Quantification of chaos:

– Dynamical:• Lyapunov exponents• Kolmogorov-Sinai (K-S) Entropy

– Geometrical:• Fractal dimension• Correlation dimension

• Only 1-D dissipative systems are discussed in this chapter.

9.3. Lyapunov Exponents

0it t i i ix x tTime series:

Given i & j, let k j k i kd x x

System is chaotic if 0k

kd d e with 0

0

1ln nd

n d Lyapunov exponent

Technical Details:

• Check exponential dependence.

• λ is x dependent → λ = Σiλ(xi) / N .

• N can’t be too large for bounded systems.

• λ = 0 for periodic system.

• i & j shouldn’t be too close.

• Bit- version: dn = d0 2nλ Logistic Map

9.4. Universal Scaling of the Lyapunov Exponents

Period-doubling route to chaos:

Logistic map: A = 3.5699…LyapunovExponents.nb

λ < 0 in periodic regime.

λ = 0 at bifurcation point.(period-doubling)

λ > 0 in chaotic regime.

λ tends to increase with A

→ More chaotic as A increases.

Huberman & Rudnick: λ(A > A) is universal for periodic-doubling systems:

ln 2

ln0A A A

0.445

0 A A

4.669 = Feigenbaum δ

λ0 = 0.9

λ ~ order parameter

A A ~ T TC

Derivation of the Universal Law for λ

• Chaotic bands merge via “period-undoubling” for A > A.

• Ratio of convergence tends to Feigenbaum δ.

Logistic map

• Let 2m bands merge to 2m1 bands at A = Am .

• Reminder: 2m bands bifurcate to 2m+1 bands at A = Am .

Divergence of trajectories in 1 band : 0n

nd d e

Divergence of trajectories among 2m band :

202

m

md d e 0d e

λ = effective Lyapunov exponent denoting 2m iterations as one.

λ = Lyapunov exponent for 2m

f

If λ is the same for all bands, then 2m m

A

Ex.2.4-1: Assuming δn = δ gives 2

2 1 1n

nA A A A

Similarly: 2

1 2 1m

mA A A A

A

m

m

A

A A

ln

lnm

AA A

m

→

mmA A A

2m m

A

→

2

1

ln

ln mA A

A

1

ln

2ln 2 log mA A

A

ln 2

ln

2log mA A

A

ln 2

lnmA A

A

ln 2

ln0A A A i.e.,

ln 2

ln0 A

9.5. Invariant Measures

• Definition of Probability

• Invariant Measures

• Ergodic Behavior

For systems of large DoFs, geometric analysis becomes unwieldy.

Alternative approach: Statistical methods.

Basic quantity of interest: Probability of trajectory to pass through given region of state space.

Definition of Probability

Consider an experiment with N possible results (outcomes).

After M runs (trials) of the experiment, let there be mi occurrences of the ith outcome.

The probability pi of the ith outcome is defined as i

i

mp

M where

1

N

ii

m M

→1

1N

ii

p

( Normalization )

If the outcomes are described by a set of continuous parameters x, N = .

mi are finite → M = and pi = 0 i.

Remedy:

Divide range of x into cells/bins.

mi = number of outcomes belonging to the ith cell.

Invariant Measures

For an attractor in state space:

1. Divide attractor into cells.

2. 1-D case: pi mi / M.

Set {pi} is a natural probability measure if it is independent of (almost all) IC.

i ip x cell i

p x dx

Let 1n nx f x then μ is an invariant probability measure if x f x

p(x) dx = probability of trajectory visiting interval [ x, x+dx ] or [ xdx/2 , x+dx/2 ].

= probability of trajectory visiting cell i.

Treating M as total mass → p(x) = ρ(x)

Example: Logistic Map, A = 4

From § 4.8: For A = 4, logistic map is equivalent to Bernoulli shift.

1 4 1n n nx x x → 1 2 mod 1n n with 11 cos

2x

1 1

0 0

1 dx p x d P 1

0

ddx P

dx

→ dp x P

dx

11cos 1 2x

2

1 2

1 1 2p x

x

1P 1

1x x

Numerical:

1024 iterations into 20 bins

→

Ergodic Behavior

Time average of B(x): 0

0

1t T

tt

B dt B x tT

1

1 N

ii

B x tN

0i

Tt t i

N

Bt should be independent of t0 as T → .

Ensemble average of B(x):

p

B dx p x B x 1

N

i ii

B x p

System is ergodic if Bt = Bp .

Comments:

• Bp is meaningful only for invariant probability measures.

• p(x) may not exist, e.g., strange attractors.

Example: Logistic Map, A = 4

Local values of the Lyapunov exponent:

lnx f x ln 4 1 2x

Ensemble average value of the Lyapunov exponent:

lndx p x f x

1

0

1ln 4 1 2

1dx x

x x

11 cos

2x

1

0

ln 4cosd

ln 2 ( same as the Bernoulli shift )

Same as that calculated by time average (c.f. §5.4):

1

ln1

i

N

i

f xN

9.6. Kolmogorov-Sinai Entropy

Brief Review of Entropy:

• Microcanonical ensemble (closed, isolated system in thermal equilibrium):

S = k ln N = k ln p p = 1/N

• Canonical ensemble (small closed subsystem):

S = k Σi pi ln pi Σi pi = 1

• 2nd law: ΔS 0 for spontaneous processes in closed isolated system.

→ S is maximum at thermodynamic equilibrium

• Issue: No natural way to count states in classical mechanics.

→ S is defined only up to an constant ( only ΔS physically meaningful )

Quantum mechanics: phase space volume of each state = hn , n = DoF.

Entropy for State Space Dynamics

1. Divide state space into cells (e.g., hypercubes of volume LDof ).

2. For dissipative systems, replace state space with attractors.

3. Start evolution for an ensemble of I.C.s (usually all located in 1 cell).

4. After n time steps, count number of states in each cell.

lnnr r

r

M M

MS k

M lnr r

r

p pk

Note: 10 0

lnlim ln limp p

pp p

p

1

20limp

p

p

0limp

p

0

• Non-chaotic motion:

• Number of cells visited (& hence S ) is independent of t & M on the macroscopic time-scale.

• Chaotic motion:

• Number of cells visited (& hence S ) increases with t but independent of M.

• Random motion:

• Number of cells visited (& hence S ) increases with both t & M

k = Boltzmann constant

Only ΔS is physically significant.

Kolmogorov-Sinai entropy rate = K-S entropy = K is defined as

dSK

dt 0

0 0lim lim lim N

L N

S S

N

1

10 0

0

1lim lim lim

N

n nL N

n

S SN

For iterated maps or Poincare sections, τ= 1 so that

1

10

0

1lim lim

N

n nL N

n

K S SN

0

0lim lim N

L N

S S

N

E.g., if the number of occupied cells Nn is given by0

nnN N e

and all occupied cells have the same probability1

rn

pN

then1 1

lnn

nN cells n n

S kN N

ln nk N 0lnk N n

0 0ln lnlim

N

k N N k NK

N

k

Pesin identity:i

i

K k λi = positive average Lyapunov exponents

Alternative Definition of the K-S Entropy See Schuster

1. Map out attractor by running a single trajectory for a long time.

2. Divide attractor into cells.

3. Start a trajectory of N steps & mark the cell it’s in at t = nτas b(n).

4. Do the same for a series of other slightly different trajectories starting from the same initial cell.

5. Calculate the fraction p(i) of trajectories described by the ith cell sequence.

0

lnNi b

S k p i p i 0lim N

N

S SK

N

Then where

Exercise: Show that both definitions of K give roughly the same result for all 3 types of motions discussed earlier.

9.7. Fractal Dimensions

Geometric aspects of attractorsDistribution of state space points of a long time series

→ Dimension of attractor

Importance of dimensionality:• Determines range of possible dynamical behavior.• Dictates long-term dynamics.• Reveals active degrees of freedom.

For a dissipative system :• D < d,

D dimension of attractor, d dimension of state space.

• D* < D, D* = dimension of attractor on Poincare section.

For a Hamiltonian system, • D d 1,

D = dimension of points generated by one trajectory( trajectory is confined on constant energy surface )

• D* < D, D* = dimension of points on Poincare section.

• Dimension is further reduced if there are other constants of motion.

Example: 3-D state space x f x

0 f x → attractor must shrink to a point or a curve x

→ system can’t be quasi-periodic ( no torus )

→ no q.p. solutions for the Lorenz system.

Dissipative system:Strange attractor = Attractor with fractional dimensions (fractals)Caution: There’re many inequivalent definitions of fractal dimension.

See J.D.Farmer, E.Ott, J.A.Yorke, Physica D7, 153-80 (1983)

Capacity ( Box-Counting ) Dimension Db

• Easy to understand.• Not good for high d systems.

1st used by Komogorov

0

lim bD

RN R k R

N(R) = Number of boxes of side R that covers the object

0

ln lim ln lnbR

N R k D R

0

ln lnlim

ln lnb R

N R kD

R R

0

lnlim

lnb R

N RD

R

Example 1: Points in 2-D space

A single point: Box = square of sides R.

1N R →0

ln1lim 0

lnbR

DR

Set of N isolated points: Box = square of sides R. R = ½ (minimal distance between points).

N R N →0

lnlim 0

lnbR

ND

R

Example 2: Line segment of length L in 2-D space

LN R

R →

0

lnlim

lnbR

LRDR

Box = square of sides R.

0

lnlim 1

lnR

L

R

1

Example 3: Cantor Set

Starting with a line segment of length 1, take out repeatedly the middle third of each remaining segment.

Caution:Given M finite, set consists of 2M line segments → Db =

1.Given M infinite, set consists of discrete points → Db = 0.

Limits M → and R → 0 must be taken simultaneously.

ln 2lim

1ln

3

M

b MMD

ln 2

ln 3 0.63

At step M, there remain 2M segments, each of length 1/3M.

Measure of the Cantor set:

1lim 2

3

MM

M

Length of set 0

1

1

11 2

3

MM

M

0

1 11 2

3 3

MM

M

11

3

1

3

12

0

Ex. 9.7-5: Fat Fractal

Example 4: Koch Curve

Start with a line segment of length 1. a) Construct an equilateral triangle with the middle third segment as base.b) Discard base segment.Repeat a) and b) for each remaining segment.

At step M, there exists 4M segments of length 1/3M each.

ln 4lim

1ln

3

M

b MMD

ln 4

ln 3 1.26

Types of Fractals

Fractals with self-similarity:

small section of object, when magnified, is identical with the whole.

• Fractals with self-affinity:

same as self-similarity, but with anisotropic magnification.

• Deterministic fractals:

Fixed construction rules.

• Random fractals:

Stochastic construction rules (see Chap 11).

Fractal Dimensions of State Space Attractors

Difficulty: R → 0 not achievable due to finite precision of data.

Remedy: Alternate definition of fractal dimension (see §9.8)

Logistic map at A , renormalization method: Db = 0.5388… (universal)

Elementary estimates:Consider A → A

+ ( from above ).

Sarkovskii’s theorem → chaotic bands undergo doubling-splits as A → A+

.Feigenbaum universality → splitted bands are narrower by 1/α and 1/α2 .Assume points in each band distributed uniformly → splitting is Cantor-set like.

1st estimate: R decreases by factor 1/α at each splitting.

11

ln 2lim

ln /

n

b nnD

R ln 2

ln 0.4498...

2nd estimate:

1nn

RR

12

1 1 1

2n nR R

11n

R

→

→ 1

1 2

ln 2lim

1 1 1ln

2

n

b nnD

R

2

ln 2

1 1 1ln

2

0.543...

Db procedure dependent.

An infinity of dimensional measures needed to characterize object (see Chap 10)

The Similarity Dimensions for Nonuniform Fractals

9.8. Correlation Dimension & a Computational History

9.9. Comments & Conclusions