Introductory Concepts for Dynamical Systems: Chaos

Michael Cross

California Institute of Technology

29 May, 2008

Michael Cross (Caltech) Chaos 29 May, 2008 1 / 25

Lorenz ModelDeterministic Nonperiodic Flow, Journal of Atmospheric Sciences20, 130 (1963)

Edward Lorenz [1917-2008]

X = −σ(X − Y)

Y = r X − Y − X ZZ = −bZ + XY

“The feasibility of very long-rangeweather prediction is examined in thelight of these results”

Michael Cross (Caltech) Chaos 29 May, 2008 2 / 25

Lorenz: a Pioneer

0

50

100

150

200

250

1970 1980 1990 2000

Citations of Lorenz 1963 (4316 total)

Cita

tions

Year

Michael Cross (Caltech) Chaos 29 May, 2008 3 / 25

Rayleigh-Bénard Convection

HOT

COLD

Michael Cross (Caltech) Chaos 29 May, 2008 4 / 25

Rayleigh-Bénard Convection

HOT

COLD

Michael Cross (Caltech) Chaos 29 May, 2008 5 / 25

Rayleigh-Bénard Convection

HOT

COLD

Michael Cross (Caltech) Chaos 29 May, 2008 6 / 25

Rayleigh-Bénard Convection

HOT

COLD

Michael Cross (Caltech) Chaos 29 May, 2008 7 / 25

Lorenz Model (1963)

HOT

COLD

X

Z

Y

Michael Cross (Caltech) Chaos 29 May, 2008 8 / 25

Lorenz Equations

X = −σ(X − Y)

Y = r X − Y − X ZZ = −bZ + XY

(whereX = d X/dt, etc.).

The equations give the flowf = (X, Y, Z) of the pointX = (X, Y, Z) in thephase space

r = R/Rc, b = 8/3, andσ is the Prandtl number.

Michael Cross (Caltech) Chaos 29 May, 2008 9 / 25

Properties of the Lorenz Equations

Autonomous—time does not explicitly appear on the right hand side;

Involve only first order time derivatives so that the evolution depends onlyon the instantaneous value of(X, Y, Z);

Non-linear—the quadratic termsX Z andXY in the second and thirdequations;

Dissipative—crudely the diagonal terms such asX = −σ X correspond todecaying motion. More systematically,volumes in phase space contractunder the flow (Strogatz §9.2)

∇ · f = ∂

∂ X[−σ(X − Y)] + ∂

∂Y[r X − Y − X Z] + ∂

∂ Z[−bZ + XY]

= −σ − 1 − b < 0

Solutions are bounded—trajectories eventually enter and stay within anellipsoidal region (Strogatz Exercises 9.2.2-3)

Michael Cross (Caltech) Chaos 29 May, 2008 10 / 25

Properties of the Lorenz Equations

Autonomous—time does not explicitly appear on the right hand side;

Involve onlyfirst order time derivativesso that the evolution depends onlyon the instantaneous value of(X, Y, Z);

Non-linear—the quadratic termsX Z andXY in the second and thirdequations;

Dissipative—crudely the diagonal terms such asX = −σ X correspond todecaying motion. More systematically,volumes in phase space contractunder the flow (Strogatz §9.2)

∇ · f = ∂

∂ X[−σ(X − Y)] + ∂

∂Y[r X − Y − X Z] + ∂

∂ Z[−bZ + XY]

= −σ − 1 − b < 0

Solutions are bounded—trajectories eventually enter and stay within anellipsoidal region (Strogatz Exercises 9.2.2-3)

Michael Cross (Caltech) Chaos 29 May, 2008 10 / 25

Properties of the Lorenz Equations

Autonomous—time does not explicitly appear on the right hand side;

Involve only first order time derivatives so that the evolution depends onlyon the instantaneous value of(X, Y, Z);

Non-linear—the quadratic termsX Z andXY in the second and thirdequations;

Dissipative—crudely the diagonal terms such asX = −σ X correspond todecaying motion. More systematically,volumes in phase space contractunder the flow (Strogatz §9.2)

∇ · f = ∂

∂ X[−σ(X − Y)] + ∂

∂Y[r X − Y − X Z] + ∂

∂ Z[−bZ + XY]

= −σ − 1 − b < 0

Solutions are bounded—trajectories eventually enter and stay within anellipsoidal region (Strogatz Exercises 9.2.2-3)

Michael Cross (Caltech) Chaos 29 May, 2008 10 / 25

Properties of the Lorenz Equations

Autonomous—time does not explicitly appear on the right hand side;

Involve only first order time derivatives so that the evolution depends onlyon the instantaneous value of(X, Y, Z);

Non-linear—the quadratic termsX Z andXY in the second and thirdequations;

Dissipative—crudely the diagonal terms such asX = −σ X correspond todecaying motion. More systematically,volumes in phase space contractunder the flow (Strogatz §9.2)

∇ · f = ∂

∂ X[−σ(X − Y)] + ∂

∂Y[r X − Y − X Z] + ∂

∂ Z[−bZ + XY]

= −σ − 1 − b < 0

Solutions are bounded—trajectories eventually enter and stay within anellipsoidal region (Strogatz Exercises 9.2.2-3)

Michael Cross (Caltech) Chaos 29 May, 2008 10 / 25

Properties of the Lorenz Equations

Autonomous—time does not explicitly appear on the right hand side;

Involve only first order time derivatives so that the evolution depends onlyon the instantaneous value of(X, Y, Z);

Non-linear—the quadratic termsX Z andXY in the second and thirdequations;

Dissipative—crudely the diagonal terms such asX = −σ X correspond todecaying motion. More systematically,volumes in phase space contractunder the flow (Strogatz §9.2)

∇ · f = ∂

∂ X[−σ(X − Y)] + ∂

∂Y[r X − Y − X Z] + ∂

∂ Z[−bZ + XY]

= −σ − 1 − b < 0

Solutions are bounded—trajectories eventually enter and stay within anellipsoidal region (Strogatz Exercises 9.2.2-3)

Michael Cross (Caltech) Chaos 29 May, 2008 10 / 25

Solutions

r < 1: X = Y = Z = 0: stable fixed point

r = 1: supercritical pitchfork bifurcation

r > 1:

X = Y = Z = 0: unstable fixed pointX = Y = ±√

b(r − 1), Z = √r − 1: fixed points, stable forr <

σ(σ+b+3)σ−b−1

r = σ(σ+b+3)σ−b−1 : subcritical Hopf bifurcation

Lorenz investigated the equations withb = 8/3, σ = 10 andr = 27 anduncovered chaos!

Michael Cross (Caltech) Chaos 29 May, 2008 11 / 25

Solutions

r < 1: X = Y = Z = 0: stable fixed point

r = 1: supercritical pitchfork bifurcation

r > 1:

X = Y = Z = 0: unstable fixed pointX = Y = ±√

b(r − 1), Z = √r − 1: fixed points, stable forr <

σ(σ+b+3)σ−b−1

r = σ(σ+b+3)σ−b−1 : subcritical Hopf bifurcation

Lorenz investigated the equations withb = 8/3, σ = 10 andr = 27 anduncovered chaos!

Michael Cross (Caltech) Chaos 29 May, 2008 11 / 25

Sensitive Dependence on Initial Conditions

Trajectories diverge exponentially

t0

t1 t2

t3

tf

δu0

δufX

Y

Z

Lyapunov exponent:

λ = limt f →∞

[1

t f − t0ln

∣∣∣∣δu f

δu0

∣∣∣∣]

Lyapunov eigenvector:δu f (t)

Michael Cross (Caltech) Chaos 29 May, 2008 12 / 25

Butterfly Effect

Thesensitive dependence on initial conditionsfound by Lorenz is often calledthebutterfly effect, and is the essential feature of chaos.

In fact Lorenz first said (Transactions of the New York Academy of Sciences,1963):

One meteorologist remarked that if the theory were correct, one flap of the seagull’s wings would be enough to alter the course of the weather forever.

By the time of Lorenz’s talk at the December 1972 meeting of the AmericanAssociation for the Advancement of Science in Washington D.C., the sea gullhad evolved into the more poetic butterfly — the title of his talk was:

Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornadoin Texas?

Michael Cross (Caltech) Chaos 29 May, 2008 13 / 25

Phase Space Trajectory

-10-5

05

10

X

-10

0

10

Y

0

10

20

30

40

Z

-50

510

X

Michael Cross (Caltech) Chaos 29 May, 2008 14 / 25

Strange Attractor

Trajectories settle onto astrange attractor:

Definition: strange attractor— an attractor that exhibits sensitive dependenceon initial conditions (Ruelle and Takens).

(See Strogatz §9.3 for complete definitions.)

The Lorenz attractor has no volume but is not a sheet: it is afractal ofnoninteger dimension.

Michael Cross (Caltech) Chaos 29 May, 2008 15 / 25

Strange Attractor

Trajectories settle onto astrange attractor:

Definition: strange attractor— an attractor that exhibits sensitive dependenceon initial conditions (Ruelle and Takens).

(See Strogatz §9.3 for complete definitions.)

The Lorenz attractor has no volume but is not a sheet: it is afractal ofnoninteger dimension.

Michael Cross (Caltech) Chaos 29 May, 2008 15 / 25

Phase Space Trajectory

-10-5

05

10

X

-10

0

10

Y

0

10

20

30

40

Z

-50

510

X

Michael Cross (Caltech) Chaos 29 May, 2008 16 / 25

Routes to Chaos

How does chaos develop from simpler dynamics (fixed points, limit cycles, etc.)as a parameter of the system is changed?

Lorenz Model: Complicated — I will just show you some qualitativetrends. See Strogatz §9.5 for more details.

Period Doubling: Successive period doubling bifurcations from a periodicorbit (period 2,4,…∞ → chaos). Feigenbaum showed there are universalfeatures of this route to chaos.

Breakdown of Quasiperiodicity: Ruelle and Takens discussed thestructural instability of quasiperiodic motion with many frequencies.

…

Michael Cross (Caltech) Chaos 29 May, 2008 17 / 25

Lorenz model does not describe Rayleigh-Bénardconvection!

Thermosyphon Rayle igh-Benard Convect ion

However the ideas of low dimensional modelsdoapply to fluids and othercontinuum systems.

Michael Cross (Caltech) Chaos 29 May, 2008 18 / 25

Experimental Chaos in Fluids

Some highlights:

Ahlers (1974)Transition from time independent flow to aperiodic flow atR/Rc ∼ 2 (cylinder with aspect ratio 5)

Gollub and Swinney (1975)Onset of aperiodic flow from time-periodic flow inTaylor-Couette

Maurer and Libchaber, Ahlers and Behringer (1978)Transition fromquasiperiodic flow to aperiodic flow in small aspect ratioconvection

Lichaber, Laroche, and Fauve (1982)Quantitative demonstration of theFiegenbaum period doubling route to chaos

Michael Cross (Caltech) Chaos 29 May, 2008 19 / 25

Rayleigh-Bénard Convection: aspect ratio 4.7 cylinderJ. Scheel: Caltech PhD Thesis

R = 3127 R = 6949

Michael Cross (Caltech) Chaos 29 May, 2008 20 / 25

Time Series (Heat Flow)Paul, MCC, Fischer, and Greenside

0 500 1000 1500 2000time

1.4

1.5

1.6

1.7

1.8

1.9

2N

u

R = 6949R = 4343R = 3474R = 3127R = 2804R = 2606

50 τh

Γ = 4.72σ = 0.78 (Helium)Random Initial Conditions

Michael Cross (Caltech) Chaos 29 May, 2008 21 / 25

Power Spectrum

10-2 10-1 100 101 102

ω10-12

10-11

10-10

10-9

10-8

10-7

10-6

P(ω

)

R = 6949ω-4

Γ = 4.72, σ = 0.78, R = 6949Conducting sidewalls

R/Rc = 4.0

Michael Cross (Caltech) Chaos 29 May, 2008 22 / 25

Lyapunov ExponentScheel and MCC

10 20 30 40 500

10

20

30

t

log

|Nor

m|

dataλ = 0.6

Michael Cross (Caltech) Chaos 29 May, 2008 23 / 25

Lyapunov Eigenvector

Temperature Temperature Perturbation

Michael Cross (Caltech) Chaos 29 May, 2008 24 / 25

Further Discussion

Quantifying Chaos

Multiple Lyapunov exponentsDimensions of the strange attractorInformation and entropy

Universal aspects of some routes to chaos (period doubling, onset from2-torus)

Predicting chaos (Homoclinic tangles, Melnikov …)

Applications

Control

Hamiltonian chaos

Quantum chaos

See Strogatz, and my website …

Michael Cross (Caltech) Chaos 29 May, 2008 25 / 25