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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 Sciences 20, 130 (1963)

Edward Lorenz [1917-2008]

X = (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-Bnard Convection

HOT

COLD

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

Rayleigh-Bnard Convection

HOT

COLD

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

Rayleigh-Bnard Convection

HOT

COLD

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

Rayleigh-Bnard 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 = (X Y )Y = r X Y X ZZ = bZ + XY

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

The equations give the flow f = (X , Y , Z) of the point X = (X, Y, Z) in thephase space

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

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

Properties of the Lorenz Equations

Autonomoustime 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-linearthe quadratic terms X Z and XY in the second and thirdequations;

Dissipativecrudely the diagonal terms such as X = X correspond todecaying motion. More systematically, volumes in phase space contractunder the flow (Strogatz 9.2)

f = X

[(X Y )] + Y

[r X Y X Z ] + Z

[bZ + XY ]= 1 b < 0

Solutions are boundedtrajectories 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

Autonomoustime 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-linearthe quadratic terms X Z and XY in the second and thirdequations;

Dissipativecrudely the diagonal terms such as X = X correspond todecaying motion. More systematically, volumes in phase space contractunder the flow (Strogatz 9.2)

f = X

[(X Y )] + Y

[r X Y X Z ] + Z

[bZ + XY ]= 1 b < 0

Solutions are boundedtrajectories 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

Autonomoustime 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-linearthe quadratic terms X Z and XY in the second and thirdequations;

Dissipativecrudely the diagonal terms such as X = X correspond todecaying motion. More systematically, volumes in phase space contractunder the flow (Strogatz 9.2)

f = X

[(X Y )] + Y

[r X Y X Z ] + Z

[bZ + XY ]= 1 b < 0

Solutions are boundedtrajectories 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

Autonomoustime does not explicitly appear on the right hand side;

Non-linearthe quadratic terms X Z and XY in the second and thirdequations;

f = X

[(X Y )] + Y

[r X Y X Z ] + Z

[bZ + XY ]= 1 b < 0

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

Properties of the Lorenz Equations

Autonomoustime does not explicitly appear on the right hand side;

Non-linearthe quadratic terms X Z and XY in the second and thirdequations;

f = X

[(X Y )] + Y

[r X Y X Z ] + Z

[bZ + XY ]= 1 b < 0

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 for r < (+b+3)

b1

r = (+b+3)b1 : subcritical Hopf bifurcation

Lorenz investigated the equations with b = 8/3, = 10 and r = 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 for r < (+b+3)

b1

r = (+b+3)b1 : subcritical Hopf bifurcation

Lorenz investigated the equations with b = 8/3, = 10 and r = 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

u0

ufX

Y

Z

Lyapunov exponent:

= limt f

[1

t f t0 lnu fu0

]

Lyapunov eigenvector: u f (t)

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

Butterfly Effect

The sensitive dependence on initial conditions found by Lorenz is often calledthe butterfly 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 seagulls wings would be enough to alter the course of the weather forever.

By the time of Lorenzs 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 Butterflys 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 a strange 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 a fractal ofnoninteger dimension.

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

Strange Attractor

Trajectories settle onto a strange 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 a fractal 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-Bnardconvection!

Thermosyphon Rayle igh-Benard Convect ion

However the ideas of low dimensional models do apply 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 aperiodi

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