Bifurcations
Chaos and TimeSeries Analysis
10/17/00 Lecture #7 in Physics 505
Comments on Homework
#5 (Hénon Map)

Everyone did fine

Many noted the large number of iterates required when zoming
in on the attractor

Only a couple of people had a good plot of the basin
of attraction
Review (last
week)  Strange Attractors

KaplanYorke (Lyapunov) Dimension

D_{KY} = D  (l_{1}
+ l_{2} + ... + l_{D})
/ l_{D+1}

where D is the largest integer for which l_{1}
+ l_{2} + ... + l_{D}
> 0

D_{KY} = 1.258 for Hénon map (B = 0.3,
C
= 1.4)

D_{KY} = 2.062 for Lorenz attractor (p = 10,
r
= 28, b = 8/3)

Chaotic flows always have D_{KY} > 2

Hence we need visualization techniques

Chaotic maps always have D_{KY} >
1

Why not use a multipoint interpolation?

Is chaos the rule or the exception?

Polynomial maps and flows

Artificial neural networks

Examples of strange attractors

Properties of strange attractors

Dimension of strange attractors

Strange attractors as general approximators

Strange attractors as objects of art
What is the "Most Chaotic"
2D Quadratic Map?

This work is unpublished

Use genetic algorithm to maximize l_{1}
for 12 parameters

Mate 2 chaotic cases (arbitrarily chosen)

Kill inferior offspring (eugenics)

Introduce occasional mutations

Replace parents with superior children

The answer?  2 decoupled logistic maps!

X_{n}_{+1} = 4X_{n}(1  X_{n})

Y_{n}_{+1} = 4Y_{n}(1  Y_{n})

This system has l_{1} = l_{2}
= log(2) as expected

It is area expanding but folded in both directions

Its KaplanYorke dimension is D_{KY} = 2

Largest Lyapunov exponent generally decreases
with D

Implications

Complex systems evolve at the "edge of chaos"

Allows exploration of new regions of state space

But retains good shortterm memory
Shift Map (1D Nonlinear)

Start with logistic map: X_{n}_{+1} = 4X_{n}(1
 X_{n})

Let X = sin^{2} pY

Then sin^{2} pY_{n}_{+1}
= 4 sin^{2} pY_{n} (1
 sin^{2} pY_{n})
= 4 sin^{2} pY_{n}
cos^{2} pY_{n}

But 2 sin q cos q
= sin 2q (from trigonometry)

Hence sin^{2} pY_{n}_{+1}
= sin^{2} 2pY_{n}

Thus Y_{n}_{+1} = 2Y_{n}
(mod 1) (shift map)

"mod 1" means take only the fractional part of 2Y_{n}

Caution: mod only works for integers on some compilers
In this case, use instead:
IF X >= 1 THEN X = X  INT(X)
IF X < 0 THEN X = INT(X)  X

Shift map is conjugate to the logistic map
(equivalent except for a nonlinear change of variables)

More specifically, this is a piecewise linear map

Maps the unit interval (0, 1) back onto itself twice:

Involves a stretching and tearing

Lyapunov exponent: l = log(2)

Invariant measure (probability density) is uniform

Generates apparently random numbers in (0, 1)

But these numbers are strongly correlated (obviously)

Solution: Y_{n} = 2^{n}Y_{0}
mod 1

Why is it called a "shift map"?

Represent initial condition in binary: 0.1011010011...

Or in (left/right) symbols: RLRRLRLLRR...

Each iteration leftshifts by 1: 1.011010011...

Mod 1 discards the leading 1: 0.011010011...

The sequence is determined by the initial condition

Only irrational initial conditions give chaos

Any sequence of RL's can be generated

Computer hint: Use X_{n}_{+1} = 1.999999X_{n}
mod 1

Dynamics are similar in the tent map:
Computer Random Number
Generators

A generalization of the shift map: Y_{n}_{+1}
= (AY_{n} + B) mod C

A, B, and C must be chosen optimally (large
integers)

A is the number of cycles

B is the "phase" (horizontal shift)

C is the number of distinct values

Example: A = 1366, B = 150889, C = 714025

There are many other choices (see Knuth)

Must also choose an initial "seed" Y_{0}

This is called a "linear congruential generator"

Lyapunov exponent: l = log(A)
(very large)

Numbers produced this way are pseudorandom

The sequence will repeat after at most C steps

In QuickBASIC the numbers repeat after 16,777,216 = 2^{14}
steps

The repetition time is much longer in PowerBASIC

Cycle time can be increased with shuffling
Intermittency  Logistic
Map at A = 3.8284

This is just to the left of the period3
window

Dynamics change abruptly from period3 to chaos

Time series (X_{n} versus n):

This is a result of a tangent bifurcation(X_{n}_{+3}
versus X_{n})

Can be understood by the cobweb diagram

Orbit gets caught for many iterations in a narrow channel

This is the intermittency route to chaos (cf: transient
chaos)
Bifurcations  General

A qualitative change in behavior at a critical parameter
value

Observation of a bifurcation verifies determinism

Flows are often analyzed using their maps (Poincaré
section)

Classifications:

Local  involves one or more equilibrium points

Global  equilibrium points appear or vanish

Continuous (subtle)  eigenvalues cross unit circle

Discontinuous (catastrophic)  eigenvalues appear or vanish

Explosive  like catastrophic but no hysteresis
(occur when attractor touches the basin boundary)

There are dozens of bifurcations, many not discovered

Terminology is not precise or universal (still evolving)

Transcritical Bifurcation

A simple form where a stable fixed point becomes unstable

Or an unstable point becomes stable

Example: Fixed points of logistic map

X* = 0, 1  1/A

At A = 1, stability of points switch

Exchange of stability between two fixed points

Pitchfork Bifurcation

This is a local bifurcation

Stable branch becomes unstable

Two new stable branches are born

Happens when eigenvalue of fixed point reaches +1

This usually occurs when there is a symmetry in the problem

Flip Bifurcation

As above but solution oscillates between the branches

This is the common perioddoubling route to chaos

As occurs in the logistic map at 3 < A < 3.5699

Happens when eigenvalue of fixed point reaches 1

Can double and then halve without reaching chaos

Can occur only in maps (not flows)

Tangent (or SaddleNode or Blue Sky) Bifurcation

This was previously discussed under intermittency

Provides a new route to chaos

This is also a local bifurcation

It is sometimes called an interior crisis

Basic mechanism for creating and destroying fixed points

Catastrophe (1D example)

Cubic map: X_{n}_{+1} = AX_{n}(1
 X_{n}^{2})

Antisymmetric about X_{n} = 0 (allows negative
solutions)

Catastrophe occurs at A = 27^{1/2}/2 = 2.59807...
where attractors collide

This system is exhibits hysteresis (decrease in A can leave
X
< 0)

Also occurs in two backtoback logistic maps

Can also have infinitely many attractors

Process equation: X_{n}_{+1} = X_{n}
+ A sin(X_{n})

Fixed points: X* = n pi for n = 0, +
1, +2, ...

Attractors collide at A = 4.669201...

Orbits diffuse in X for A > 4.669201...

Hopf Bifurcation

A stable focus becomes unstable and a limit cycle is born

Example: Van der Pol equation at b = 0

This bifurcation is local and continuous

It occurs when complex eigenvalues touch the unit circle

Niemark (or Secondary Hopf) Bifurcation

A stable limit cycle becomes unstable and a 2torus is born

The Poincaré section exhibits a Hopf bifurcation

Main sequence (quasiperiodic route to chaos)

fixed point > limit cycle > 2torus > chaos

Ntorus with N > 2 not usually seen (Piexito's Theorem)
(3torus and higher are structurally unstable)

This contradicts the Landau theory of turbulence
(turbulence is a sum of very many periodic modes)

Also called the NewhouseRuelleTakens route

Probably the most common route to chaos at highD
J. C. Sprott  Physics 505
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