Introductory Concepts for Dynamical Systems: Chaos Michael Cross - - PowerPoint PPT Presentation

▶
introductory concepts for dynamical systems chaos
SMART_READER_LITE
LIVE PREVIEW

Introductory Concepts for Dynamical Systems: Chaos Michael Cross - - PowerPoint PPT Presentation

Nov 07, 2022 776 likes •1.1k views

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 Model Deterministic Nonperiodic Flow , Journal of Atmospheric Sciences

slide-1
SLIDE 1

Introductory Concepts for Dynamical Systems: Chaos

Michael Cross

California Institute of Technology

29 May, 2008

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

slide-2
SLIDE 2

Lorenz Model

Deterministic Nonperiodic Flow, Journal of Atmospheric Sciences 20, 130 (1963)

Edward Lorenz [1917-2008]

˙ X = −σ(X − Y) ˙ Y = r X − Y − X Z ˙ Z = −bZ + XY

“The feasibility of very long-range weather prediction is examined in the light of these results”

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

slide-3
SLIDE 3

Lorenz: a Pioneer

50 100 150 200 250 1970 1980 1990 2000 Citations of Lorenz 1963 (4316 total) Citations Year

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

slide-4
SLIDE 4

Rayleigh-Bénard Convection

HOT COLD

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

slide-5
SLIDE 5

Rayleigh-Bénard Convection

HOT COLD

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

slide-6
SLIDE 6

Rayleigh-Bénard Convection

HOT COLD

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

slide-7
SLIDE 7

Rayleigh-Bénard Convection

HOT COLD

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

slide-8
SLIDE 8

Lorenz Model (1963)

HOT COLD

X Z Y

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

slide-9
SLIDE 9

Lorenz Equations ˙ X = −σ(X − Y) ˙ Y = r X − Y − X Z ˙ Z = −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 the phase space r = R/Rc, b = 8/3, and σ is the Prandtl number.

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

slide-10
SLIDE 10

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 only

  • n the instantaneous value of (X, Y, Z);

Non-linear—the quadratic terms X Z and XY in the second and third equations; Dissipative—crudely the diagonal terms such as ˙ X = −σ X correspond to decaying motion. More systematically, volumes in phase space contract under 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 an ellipsoidal region (Strogatz Exercises 9.2.2-3)

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

slide-11
SLIDE 11

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 only

  • n the instantaneous value of (X, Y, Z);

Non-linear—the quadratic terms X Z and XY in the second and third equations; Dissipative—crudely the diagonal terms such as ˙ X = −σ X correspond to decaying motion. More systematically, volumes in phase space contract under 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 an ellipsoidal region (Strogatz Exercises 9.2.2-3)

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

slide-12
SLIDE 12

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 only

  • n the instantaneous value of (X, Y, Z);

Non-linear—the quadratic terms X Z and XY in the second and third equations; Dissipative—crudely the diagonal terms such as ˙ X = −σ X correspond to decaying motion. More systematically, volumes in phase space contract under 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 an ellipsoidal region (Strogatz Exercises 9.2.2-3)

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

slide-13
SLIDE 13

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 only

  • n the instantaneous value of (X, Y, Z);

Non-linear—the quadratic terms X Z and XY in the second and third equations; Dissipative—crudely the diagonal terms such as ˙ X = −σ X correspond to decaying motion. More systematically, volumes in phase space contract under 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 an ellipsoidal region (Strogatz Exercises 9.2.2-3)

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

slide-14
SLIDE 14

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 only

  • n the instantaneous value of (X, Y, Z);

Non-linear—the quadratic terms X Z and XY in the second and third equations; Dissipative—crudely the diagonal terms such as ˙ X = −σ X correspond to decaying motion. More systematically, volumes in phase space contract under 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 an ellipsoidal region (Strogatz Exercises 9.2.2-3)

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

slide-15
SLIDE 15

Solutions

r < 1: X = Y = Z = 0: stable fixed point r = 1: supercritical pitchfork bifurcation r > 1:

X = Y = Z = 0: unstable fixed point X = Y = ±√b(r − 1), Z = √ r − 1: fixed points, stable for r < σ(σ+b+3)

σ−b−1

r = σ(σ+b+3)

σ−b−1 : subcritical Hopf bifurcation

Lorenz investigated the equations with b = 8/3, σ = 10 and r = 27 and uncovered chaos!

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

slide-16
SLIDE 16

Solutions

r < 1: X = Y = Z = 0: stable fixed point r = 1: supercritical pitchfork bifurcation r > 1:

X = Y = Z = 0: unstable fixed point X = Y = ±√b(r − 1), Z = √ r − 1: fixed points, stable for r < σ(σ+b+3)

σ−b−1

r = σ(σ+b+3)

σ−b−1 : subcritical Hopf bifurcation

Lorenz investigated the equations with b = 8/3, σ = 10 and r = 27 and uncovered chaos!

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

slide-17
SLIDE 17

Sensitive Dependence on Initial Conditions

Trajectories diverge exponentially

t0 t1 t2 t3 tf δu0 δuf X Y Z

Lyapunov exponent: λ = lim

t f →∞

  • 1

t f − t0 ln

  • δu f

δu0

  • Lyapunov eigenvector: δu f (t)

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

slide-18
SLIDE 18

Butterfly Effect

The sensitive dependence on initial conditions found by Lorenz is often called the 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 sea gull’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 American Association for the Advancement of Science in Washington D.C., the sea gull had 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 Tornado in Texas?

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

slide-19
SLIDE 19

Phase Space Trajectory

  • 10
  • 5

5 10 X

  • 10

10 Y 10 20 30 40 Z

  • 5

5 10 X Michael Cross (Caltech) Chaos 29 May, 2008 14 / 25

slide-20
SLIDE 20

Strange Attractor

Trajectories settle onto a strange attractor: Definition: strange attractor — an attractor that exhibits sensitive dependence

  • n 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 of noninteger dimension.

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

slide-21
SLIDE 21

Strange Attractor

Trajectories settle onto a strange attractor: Definition: strange attractor — an attractor that exhibits sensitive dependence

  • n 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 of noninteger dimension.

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

slide-22
SLIDE 22

Phase Space Trajectory

  • 10
  • 5

5 10 X

  • 10

10 Y 10 20 30 40 Z

  • 5

5 10 X Michael Cross (Caltech) Chaos 29 May, 2008 16 / 25

slide-23
SLIDE 23

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 qualitative

  • trends. See Strogatz §9.5 for more details.

Period Doubling: Successive period doubling bifurcations from a periodic

  • rbit (period 2,4,…∞ → chaos). Feigenbaum showed there are universal

features of this route to chaos. Breakdown of Quasiperiodicity: Ruelle and Takens discussed the structural instability of quasiperiodic motion with many frequencies. …

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

slide-24
SLIDE 24

Lorenz model does not describe Rayleigh-Bénard convection!

Thermosyphon Rayleigh-Benard Convection

However the ideas of low dimensional models do apply to fluids and other continuum systems.

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

slide-25
SLIDE 25

Experimental Chaos in Fluids

Some highlights: Ahlers (1974) Transition from time independent flow to aperiodic flow at R/Rc ∼ 2 (cylinder with aspect ratio 5) Gollub and Swinney (1975) Onset of aperiodic flow from time-periodic flow in Taylor-Couette Maurer and Libchaber, Ahlers and Behringer (1978) Transition from quasiperiodic flow to aperiodic flow in small aspect ratio convection Lichaber, Laroche, and Fauve (1982) Quantitative demonstration of the Fiegenbaum period doubling route to chaos

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

slide-26
SLIDE 26

Rayleigh-Bénard Convection: aspect ratio 4.7 cylinder

  • J. Scheel: Caltech PhD Thesis

R = 3127 R = 6949

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

slide-27
SLIDE 27

Time Series (Heat Flow)

Paul, MCC, Fischer, and Greenside

500 1000 1500 2000

time

1.4 1.5 1.6 1.7 1.8 1.9 2

Nu

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

50 τh

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

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

slide-28
SLIDE 28

Power Spectrum

10

  • 2

10

  • 1

10 10

1

10

2

ω

10

  • 12

10

  • 11

10

  • 10

10

  • 9

10

  • 8

10

  • 7

10

  • 6

P(ω)

R = 6949 ω

  • 4

Γ = 4.72, σ = 0.78, R = 6949 Conducting sidewalls R/Rc = 4.0

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

slide-29
SLIDE 29

Lyapunov Exponent

Scheel and MCC

10 20 30 40 50 10 20 30 t log |Norm| data λ = 0.6

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

slide-30
SLIDE 30

Lyapunov Eigenvector

Temperature Temperature Perturbation

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

slide-31
SLIDE 31

Further Discussion

Quantifying Chaos

Multiple Lyapunov exponents Dimensions of the strange attractor Information and entropy

Universal aspects of some routes to chaos (period doubling, onset from 2-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