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

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 20 , 130 (1963) ˙ 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” Edward Lorenz [1917-2008] Michael Cross (Caltech) Chaos 29 May, 2008 2 / 25

Lorenz: a Pioneer Citations of Lorenz 1963 (4316 total) 250 200 150 Citations 100 50 0 1970 1980 1990 2000 Year Michael Cross (Caltech) Chaos 29 May, 2008 3 / 25

Rayleigh-Bénard Convection COLD HOT Michael Cross (Caltech) Chaos 29 May, 2008 4 / 25

Rayleigh-Bénard Convection COLD HOT Michael Cross (Caltech) Chaos 29 May, 2008 5 / 25

Rayleigh-Bénard Convection COLD HOT Michael Cross (Caltech) Chaos 29 May, 2008 6 / 25

Rayleigh-Bénard Convection COLD HOT Michael Cross (Caltech) Chaos 29 May, 2008 7 / 25

Lorenz Model (1963) COLD Z Y X HOT Michael Cross (Caltech) Chaos 29 May, 2008 8 / 25

Lorenz Equations ˙ X = − σ( X − Y ) ˙ = r X − Y − X Z Y ˙ = − bZ + XY Z (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 / R c , b = 8 / 3, and σ 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 only on 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

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 on 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

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 on 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

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 on 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

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 on 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

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

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

Sensitive Dependence on Initial Conditions Trajectories diverge exponentially Z δ u f X Y t f t 3 t 1 t 2 δ u 0 t 0 Lyapunov exponent: � � � � 1 δ u f � � λ = lim ln � � t f − t 0 δ u 0 t f →∞ � � Lyapunov eigenvector: δ 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 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

Phase Space Trajectory 40 30 Z 20 10 -10 0 0 10 10 Y 5 5 0 0 10 -5 -5 X X -10 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 dependence on 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

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

Phase Space Trajectory 40 30 Z 20 10 -10 0 0 10 10 Y 5 5 0 0 10 -5 -5 X X -10 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 qualitative trends. See Strogatz §9.5 for more details. Period Doubling : Successive period doubling bifurcations from a periodic orbit (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

Lorenz model does not describe Rayleigh-Bénard convection! Rayleigh-Benard Convection Thermosyphon However the ideas of low dimensional models do apply to fluids and other continuum systems. Michael Cross (Caltech) Chaos 29 May, 2008 18 / 25