[PPT] - The Chaotic Pendulum I Continuous Nonlinear Dynamics Rubin H Landau PowerPoint Presentation

SLIDE 1

ODE Free Pend Phase Space Implementation-Assessment

The Chaotic Pendulum I

Continuous Nonlinear Dynamics Rubin H Landau

Sally Haerer, Producer-Director

Based on A Survey of Computational Physics by Landau, Páez, & Bordeianu with Support from the National Science Foundation

Course: Computational Physics I

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SLIDE 2

ODE Free Pend Phase Space Implementation-Assessment

Problem: Realistic Single or Double Pendulum

Simulate Nonlinear, Chaotic System

m I θ α α f

l1 m1 θ1 l2 θ2 m2

Driven single pendulum Free, double pendulum loading TwoPend Large oscillations, even

ver-the-top

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SLIDE 3

ODE Free Pend Phase Space Implementation-Assessment

Chaotic Pendulum ODE

Newton’s Laws for Rotational Motion τ = I d2θ

dt2

m I θ α α f

Gravitation τ: −mgl sin θ Friction τ: −β ˙ θ External τ: τ0 cos ωt I d2θ dt2 = − mgl sin θ − β dθ dt + τ0 cos ωt (1) d2θ dt2 = − ω2

0 sin θ − α dθ

dt + f cos ωt (2)

ω0 = mgl I , α = β I , f = τ0 I

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SLIDE 4

ODE Free Pend Phase Space Implementation-Assessment

Chaotic Pendulum ODE

Newton’s Laws for Rotational Motion τ = I d2θ

dt2

m I θ α α f

Gravitation τ: −mgl sin θ Friction τ: −β ˙ θ External τ: τ0 cos ωt I d2θ dt2 = − mgl sin θ − β dθ dt + τ0 cos ωt (1) d2θ dt2 = − ω2

0 sin θ − α dθ

dt + f cos ωt (2)

ω0 = mgl I , α = β I , f = τ0 I

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SLIDE 5

ODE Free Pend Phase Space Implementation-Assessment

Chaotic Pendulum ODE

Newton’s Laws for Rotational Motion τ = I d2θ

dt2

m I θ α α f

Gravitation τ: −mgl sin θ Friction τ: −β ˙ θ External τ: τ0 cos ωt I d2θ dt2 = − mgl sin θ − β dθ dt + τ0 cos ωt (1) d2θ dt2 = − ω2

0 sin θ − α dθ

dt + f cos ωt (2)

ω0 = mgl I , α = β I , f = τ0 I

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SLIDE 6

ODE Free Pend Phase Space Implementation-Assessment

Chaotic Pendulum ODE

Newton’s Laws for Rotational Motion τ = I d2θ

dt2

m I θ α α f

Gravitation τ: −mgl sin θ Friction τ: −β ˙ θ External τ: τ0 cos ωt I d2θ dt2 = − mgl sin θ − β dθ dt + τ0 cos ωt (1) d2θ dt2 = − ω2

0 sin θ − α dθ

dt + f cos ωt (2)

ω0 = mgl I , α = β I , f = τ0 I

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SLIDE 7

ODE Free Pend Phase Space Implementation-Assessment

Chaotic Pendulum ODE

Newton’s Laws for Rotational Motion τ = I d2θ

dt2

m I θ α α f

Gravitation τ: −mgl sin θ Friction τ: −β ˙ θ External τ: τ0 cos ωt I d2θ dt2 = − mgl sin θ − β dθ dt + τ0 cos ωt (1) d2θ dt2 = − ω2

0 sin θ − α dθ

dt + f cos ωt (2)

ω0 = mgl I , α = β I , f = τ0 I

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SLIDE 8

ODE Free Pend Phase Space Implementation-Assessment

Chaotic Pendulum ODE

Standard ODE Form (rk4): ˙

y =

f( y, t)

m I θ α α f

d2θ dt2 = −ω2

0 sin θ − α dθ

dt + f cos ωt (1) 2nd O t-dependent nonlinear ODE Nonlinearity: sin θ ≃ θ − θ3/3! · · · y(0) = θ(t), y(1) = dθ(t)

dt

dy(0) dt = y(1) (2) dy(1) dt = − ω2

0 sin y(0) − αy(1) + f cos ωt

(3)

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SLIDE 9

ODE Free Pend Phase Space Implementation-Assessment

Chaotic Pendulum ODE

Standard ODE Form (rk4): ˙

y =

f( y, t)

m I θ α α f

d2θ dt2 = −ω2

0 sin θ − α dθ

dt + f cos ωt (1) 2nd O t-dependent nonlinear ODE Nonlinearity: sin θ ≃ θ − θ3/3! · · · y(0) = θ(t), y(1) = dθ(t)

dt

dy(0) dt = y(1) (2) dy(1) dt = − ω2

0 sin y(0) − αy(1) + f cos ωt

(3)

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SLIDE 10

ODE Free Pend Phase Space Implementation-Assessment

Chaotic Pendulum ODE

Standard ODE Form (rk4): ˙

y =

f( y, t)

m I θ α α f

d2θ dt2 = −ω2

0 sin θ − α dθ

dt + f cos ωt (1) 2nd O t-dependent nonlinear ODE Nonlinearity: sin θ ≃ θ − θ3/3! · · · y(0) = θ(t), y(1) = dθ(t)

dt

dy(0) dt = y(1) (2) dy(1) dt = − ω2

0 sin y(0) − αy(1) + f cos ωt

(3)

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SLIDE 11

ODE Free Pend Phase Space Implementation-Assessment

Chaotic Pendulum ODE

Standard ODE Form (rk4): ˙

y =

f( y, t)

m I θ α α f

d2θ dt2 = −ω2

0 sin θ − α dθ

dt + f cos ωt (1) 2nd O t-dependent nonlinear ODE Nonlinearity: sin θ ≃ θ − θ3/3! · · · y(0) = θ(t), y(1) = dθ(t)

dt

dy(0) dt = y(1) (2) dy(1) dt = − ω2

0 sin y(0) − αy(1) + f cos ωt

(3)

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SLIDE 12

ODE Free Pend Phase Space Implementation-Assessment

Chaotic Pendulum ODE

Standard ODE Form (rk4): ˙

y =

f( y, t)

m I θ α α f

d2θ dt2 = −ω2

0 sin θ − α dθ

dt + f cos ωt (1) 2nd O t-dependent nonlinear ODE Nonlinearity: sin θ ≃ θ − θ3/3! · · · y(0) = θ(t), y(1) = dθ(t)

dt

dy(0) dt = y(1) (2) dy(1) dt = − ω2

0 sin y(0) − αy(1) + f cos ωt

(3)

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SLIDE 13

ODE Free Pend Phase Space Implementation-Assessment

Start Simply: Free Oscillations (Test Algorithm & Physics)

Ignore Friction & External Torques (f = α = 0)

m I θ

¨ θ = − ω2

0 sin θ

(1) ¨ θ ≃ − ω2

0θ

(linear, θ ≃ 0) ⇒ θ(t) = θ0 sin(ω0t + φ) (2) (1): ”Analytic solution”; sort of:

T ∝ θm dθ

sin2(θm/2) − sin2(θ/2)

1/2 (3)

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SLIDE 14

ODE Free Pend Phase Space Implementation-Assessment

Start Simply: Free Oscillations (Test Algorithm & Physics)

Ignore Friction & External Torques (f = α = 0)

m I θ

¨ θ = − ω2

0 sin θ

(1) ¨ θ ≃ − ω2

0θ

(linear, θ ≃ 0) ⇒ θ(t) = θ0 sin(ω0t + φ) (2) (1): ”Analytic solution”; sort of:

T ∝ θm dθ

sin2(θm/2) − sin2(θ/2)

1/2 (3)

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SLIDE 15

ODE Free Pend Phase Space Implementation-Assessment

Start Simply: Free Oscillations (Test Algorithm & Physics)

Ignore Friction & External Torques (f = α = 0)

m I θ

¨ θ = − ω2

0 sin θ

(1) ¨ θ ≃ − ω2

0θ

(linear, θ ≃ 0) ⇒ θ(t) = θ0 sin(ω0t + φ) (2) (1): ”Analytic solution”; sort of:

T ∝ θm dθ

sin2(θm/2) − sin2(θ/2)

1/2 (3)

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SLIDE 16

ODE Free Pend Phase Space Implementation-Assessment

Start Simply: Free Oscillations (Test Algorithm & Physics)

Ignore Friction & External Torques (f = α = 0)

m I θ

¨ θ = − ω2

0 sin θ

(1) ¨ θ ≃ − ω2

0θ

(linear, θ ≃ 0) ⇒ θ(t) = θ0 sin(ω0t + φ) (2) (1): ”Analytic solution”; sort of:

T ∝ θm dθ

sin2(θm/2) − sin2(θ/2)

1/2 (3)

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SLIDE 17

ODE Free Pend Phase Space Implementation-Assessment

Free Pendulum Implementation ¨ θ = −ω2

0 sin θ

Solve ODE with rk4

1

Initial conditions: {θ = 0, ˙ θ(0) = 0}; increase ˙ θ(0)

2

Verify SHM ¨ θ = −ω2

0θ ⇒ ω = ω0 = 2π/T = constant

3

Devise algorithm to determine period T ( 3 × θ = 0)

4

Determine T(θ) for realistic pendulum, compare

5

Verify as KE(0) ≤ 2mgl: non harmonic oscillations

6

Verify ⇒ separatrix (KE(0) → 2mgl), T → ∞

7

Listen harmonic & anharmonic motion (Hear now)

8

Hear Data applet

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SLIDE 18

ODE Free Pend Phase Space Implementation-Assessment

Free Pendulum Implementation ¨ θ = −ω2

0 sin θ

Solve ODE with rk4

1

Initial conditions: {θ = 0, ˙ θ(0) = 0}; increase ˙ θ(0)

2

Verify SHM ¨ θ = −ω2

0θ ⇒ ω = ω0 = 2π/T = constant

3

Devise algorithm to determine period T ( 3 × θ = 0)

4

Determine T(θ) for realistic pendulum, compare

5

Verify as KE(0) ≤ 2mgl: non harmonic oscillations

6

Verify ⇒ separatrix (KE(0) → 2mgl), T → ∞

7

Listen harmonic & anharmonic motion (Hear now)

8

Hear Data applet

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SLIDE 19

ODE Free Pend Phase Space Implementation-Assessment

Free Pendulum Implementation ¨ θ = −ω2

0 sin θ

Solve ODE with rk4

1

Initial conditions: {θ = 0, ˙ θ(0) = 0}; increase ˙ θ(0)

2

Verify SHM ¨ θ = −ω2

0θ ⇒ ω = ω0 = 2π/T = constant

3

Devise algorithm to determine period T ( 3 × θ = 0)

4

Determine T(θ) for realistic pendulum, compare

5

Verify as KE(0) ≤ 2mgl: non harmonic oscillations

6

Verify ⇒ separatrix (KE(0) → 2mgl), T → ∞

7

Listen harmonic & anharmonic motion (Hear now)

8

Hear Data applet

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SLIDE 20

ODE Free Pend Phase Space Implementation-Assessment

Free Pendulum Implementation ¨ θ = −ω2

0 sin θ

Solve ODE with rk4

1

Initial conditions: {θ = 0, ˙ θ(0) = 0}; increase ˙ θ(0)

2

Verify SHM ¨ θ = −ω2

0θ ⇒ ω = ω0 = 2π/T = constant

3

Devise algorithm to determine period T ( 3 × θ = 0)

4

Determine T(θ) for realistic pendulum, compare

5

Verify as KE(0) ≤ 2mgl: non harmonic oscillations

6

Verify ⇒ separatrix (KE(0) → 2mgl), T → ∞

7

Listen harmonic & anharmonic motion (Hear now)

8

Hear Data applet

20 / 58

SLIDE 21

ODE Free Pend Phase Space Implementation-Assessment

Free Pendulum Implementation ¨ θ = −ω2

0 sin θ

Solve ODE with rk4

1

Initial conditions: {θ = 0, ˙ θ(0) = 0}; increase ˙ θ(0)

2

Verify SHM ¨ θ = −ω2

0θ ⇒ ω = ω0 = 2π/T = constant

3

Devise algorithm to determine period T ( 3 × θ = 0)

4

Determine T(θ) for realistic pendulum, compare

5

Verify as KE(0) ≤ 2mgl: non harmonic oscillations

6

Verify ⇒ separatrix (KE(0) → 2mgl), T → ∞

7

Listen harmonic & anharmonic motion (Hear now)

8

Hear Data applet

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SLIDE 22

ODE Free Pend Phase Space Implementation-Assessment

Free Pendulum Implementation ¨ θ = −ω2

0 sin θ

Solve ODE with rk4

1

Initial conditions: {θ = 0, ˙ θ(0) = 0}; increase ˙ θ(0)

2

Verify SHM ¨ θ = −ω2

0θ ⇒ ω = ω0 = 2π/T = constant

3

Devise algorithm to determine period T ( 3 × θ = 0)

4

Determine T(θ) for realistic pendulum, compare

5

Verify as KE(0) ≤ 2mgl: non harmonic oscillations

6

Verify ⇒ separatrix (KE(0) → 2mgl), T → ∞

7

Listen harmonic & anharmonic motion (Hear now)

8

Hear Data applet

22 / 58

SLIDE 23

ODE Free Pend Phase Space Implementation-Assessment

Free Pendulum Implementation ¨ θ = −ω2

0 sin θ

Solve ODE with rk4

1

Initial conditions: {θ = 0, ˙ θ(0) = 0}; increase ˙ θ(0)

2

Verify SHM ¨ θ = −ω2

0θ ⇒ ω = ω0 = 2π/T = constant

3

Devise algorithm to determine period T ( 3 × θ = 0)

4

Determine T(θ) for realistic pendulum, compare

5

Verify as KE(0) ≤ 2mgl: non harmonic oscillations

6

Verify ⇒ separatrix (KE(0) → 2mgl), T → ∞

7

Listen harmonic & anharmonic motion (Hear now)

8

Hear Data applet

23 / 58

SLIDE 24

ODE Free Pend Phase Space Implementation-Assessment

Free Pendulum Implementation ¨ θ = −ω2

0 sin θ

Solve ODE with rk4

1

Initial conditions: {θ = 0, ˙ θ(0) = 0}; increase ˙ θ(0)

2

Verify SHM ¨ θ = −ω2

0θ ⇒ ω = ω0 = 2π/T = constant

3

Devise algorithm to determine period T ( 3 × θ = 0)

4

Determine T(θ) for realistic pendulum, compare

5

Verify as KE(0) ≤ 2mgl: non harmonic oscillations

6

Verify ⇒ separatrix (KE(0) → 2mgl), T → ∞

7

Listen harmonic & anharmonic motion (Hear now)

8

Hear Data applet

24 / 58

SLIDE 25

ODE Free Pend Phase Space Implementation-Assessment

Free Pendulum Implementation ¨ θ = −ω2

0 sin θ

Solve ODE with rk4

1

Initial conditions: {θ = 0, ˙ θ(0) = 0}; increase ˙ θ(0)

2

Verify SHM ¨ θ = −ω2

0θ ⇒ ω = ω0 = 2π/T = constant

3

Devise algorithm to determine period T ( 3 × θ = 0)

4

Determine T(θ) for realistic pendulum, compare

5

Verify as KE(0) ≤ 2mgl: non harmonic oscillations

6

Verify ⇒ separatrix (KE(0) → 2mgl), T → ∞

7

Listen harmonic & anharmonic motion (Hear now)

8

Hear Data applet

25 / 58

SLIDE 26

ODE Free Pend Phase Space Implementation-Assessment

Visualization: Phase Space Orbits

Abstract Space: v(t) vs x(t) (x vs t, v vs t= Complicated)

V(x) x v(t) x(t)

E1 E1

x(t) v(t) V(x) x

E1 E2 E3

V(x) x v(t) x(t)

E1 E2 E3

Geometry easy to “see” SHM: Ellipse, E → size Anharmonic: + corners Ossc ⇒ CW Closed Non Ossc, repulse = open

x(t) =A sin(ωt), v(t) = ωA cos(ωt) (SHM) (1) E =KE + PE = mv 2/2 + ω2m2x2/2 = ellipse (2)

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SLIDE 27

ODE Free Pend Phase Space Implementation-Assessment

Visualization: Phase Space Orbits

Abstract Space: v(t) vs x(t) (x vs t, v vs t= Complicated)

V(x) x v(t) x(t)

E1 E1

x(t) v(t) V(x) x

E1 E2 E3

V(x) x v(t) x(t)

E1 E2 E3

Geometry easy to “see” SHM: Ellipse, E → size Anharmonic: + corners Ossc ⇒ CW Closed Non Ossc, repulse = open

x(t) =A sin(ωt), v(t) = ωA cos(ωt) (SHM) (1) E =KE + PE = mv 2/2 + ω2m2x2/2 = ellipse (2)

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SLIDE 28

ODE Free Pend Phase Space Implementation-Assessment

Visualization: Phase Space Orbits

Abstract Space: v(t) vs x(t) (x vs t, v vs t= Complicated)

V(x) x v(t) x(t)

E1 E1

x(t) v(t) V(x) x

E1 E2 E3

V(x) x v(t) x(t)

E1 E2 E3

Geometry easy to “see” SHM: Ellipse, E → size Anharmonic: + corners Ossc ⇒ CW Closed Non Ossc, repulse = open

x(t) =A sin(ωt), v(t) = ωA cos(ωt) (SHM) (1) E =KE + PE = mv 2/2 + ω2m2x2/2 = ellipse (2)

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SLIDE 29

ODE Free Pend Phase Space Implementation-Assessment

Visualization: Phase Space Orbits

Abstract Space: v(t) vs x(t) (x vs t, v vs t= Complicated)

V(x) x v(t) x(t)

E1 E1

x(t) v(t) V(x) x

E1 E2 E3

V(x) x v(t) x(t)

E1 E2 E3

Geometry easy to “see” SHM: Ellipse, E → size Anharmonic: + corners Ossc ⇒ CW Closed Non Ossc, repulse = open

x(t) =A sin(ωt), v(t) = ωA cos(ωt) (SHM) (1) E =KE + PE = mv 2/2 + ω2m2x2/2 = ellipse (2)

29 / 58

SLIDE 30

ODE Free Pend Phase Space Implementation-Assessment

Visualization: Phase Space Orbits

Abstract Space: v(t) vs x(t) (x vs t, v vs t= Complicated)

V(x) x v(t) x(t)

E1 E1

x(t) v(t) V(x) x

E1 E2 E3

V(x) x v(t) x(t)

E1 E2 E3

Geometry easy to “see” SHM: Ellipse, E → size Anharmonic: + corners Ossc ⇒ CW Closed Non Ossc, repulse = open

x(t) =A sin(ωt), v(t) = ωA cos(ωt) (SHM) (1) E =KE + PE = mv 2/2 + ω2m2x2/2 = ellipse (2)

30 / 58

SLIDE 31

ODE Free Pend Phase Space Implementation-Assessment

Visualization: Phase Space Orbits

Abstract Space: v(t) vs x(t) (x vs t, v vs t= Complicated)

V(x) x v(t) x(t)

E1 E1

x(t) v(t) V(x) x

E1 E2 E3

V(x) x v(t) x(t)

E1 E2 E3

Geometry easy to “see” SHM: Ellipse, E → size Anharmonic: + corners Ossc ⇒ CW Closed Non Ossc, repulse = open

x(t) =A sin(ωt), v(t) = ωA cos(ωt) (SHM) (1) E =KE + PE = mv 2/2 + ω2m2x2/2 = ellipse (2)

31 / 58

SLIDE 32

ODE Free Pend Phase Space Implementation-Assessment

Undriven, Frictionless Pendulum in Phase Space

Separatrix Separates Open & Closed Orbits

rotating solutions

pendulum starts rotating pendulum falls back

4
2
2

2 2 4 6

.

V ( )

pendulum starts rotating pendulum falls back

V ( )

rotating solutions

pendulum pendulum

Closed: oscillation Open: rotation Both: periodic Orbits do not cross Open orbits touch Hyperbolic points Unstable equilibrium

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SLIDE 33

ODE Free Pend Phase Space Implementation-Assessment

Undriven, Frictionless Pendulum in Phase Space

Separatrix Separates Open & Closed Orbits

rotating solutions

pendulum starts rotating pendulum falls back

4
2
2

2 2 4 6

.

V ( )

pendulum starts rotating pendulum falls back

V ( )

rotating solutions

pendulum pendulum

Closed: oscillation Open: rotation Both: periodic Orbits do not cross Open orbits touch Hyperbolic points Unstable equilibrium

33 / 58

SLIDE 34

ODE Free Pend Phase Space Implementation-Assessment

Undriven, Frictionless Pendulum in Phase Space

Separatrix Separates Open & Closed Orbits

rotating solutions

pendulum starts rotating pendulum falls back

4
2
2

2 2 4 6

.

V ( )

pendulum starts rotating pendulum falls back

V ( )

rotating solutions

pendulum pendulum

Closed: oscillation Open: rotation Both: periodic Orbits do not cross Open orbits touch Hyperbolic points Unstable equilibrium

34 / 58

SLIDE 35

ODE Free Pend Phase Space Implementation-Assessment

Undriven, Frictionless Pendulum in Phase Space

Separatrix Separates Open & Closed Orbits

rotating solutions

pendulum starts rotating pendulum falls back

4
2
2

2 2 4 6

.

V ( )

pendulum starts rotating pendulum falls back

V ( )

rotating solutions

pendulum pendulum

Closed: oscillation Open: rotation Both: periodic Orbits do not cross Open orbits touch Hyperbolic points Unstable equilibrium

35 / 58

SLIDE 36

ODE Free Pend Phase Space Implementation-Assessment

Undriven, Frictionless Pendulum in Phase Space

Separatrix Separates Open & Closed Orbits

rotating solutions

pendulum starts rotating pendulum falls back

4
2
2

2 2 4 6

.

V ( )

pendulum starts rotating pendulum falls back

V ( )

rotating solutions

pendulum pendulum

Closed: oscillation Open: rotation Both: periodic Orbits do not cross Open orbits touch Hyperbolic points Unstable equilibrium

36 / 58

SLIDE 37

ODE Free Pend Phase Space Implementation-Assessment

Undriven, Frictionless Pendulum in Phase Space

Separatrix Separates Open & Closed Orbits

rotating solutions

pendulum starts rotating pendulum falls back

4
2
2

2 2 4 6

.

V ( )

pendulum starts rotating pendulum falls back

V ( )

rotating solutions

pendulum pendulum

Closed: oscillation Open: rotation Both: periodic Orbits do not cross Open orbits touch Hyperbolic points Unstable equilibrium

37 / 58

SLIDE 38

ODE Free Pend Phase Space Implementation-Assessment

Undriven, Frictionless Pendulum in Phase Space

Separatrix Separates Open & Closed Orbits

rotating solutions

pendulum starts rotating pendulum falls back

4
2
2

2 2 4 6

.

V ( )

pendulum starts rotating pendulum falls back

V ( )

rotating solutions

pendulum pendulum

Closed: oscillation Open: rotation Both: periodic Orbits do not cross Open orbits touch Hyperbolic points Unstable equilibrium

38 / 58

SLIDE 39

ODE Free Pend Phase Space Implementation-Assessment

Undriven, Frictionless Pendulum in Phase Space

Separatrix Separates Open & Closed Orbits

rotating solutions

pendulum starts rotating pendulum falls back

4
2
2

2 2 4 6

.

V ( )

pendulum starts rotating pendulum falls back

V ( )

rotating solutions

pendulum pendulum

Closed: oscillation Open: rotation Both: periodic Orbits do not cross Open orbits touch Hyperbolic points Unstable equilibrium

39 / 58

SLIDE 40

ODE Free Pend Phase Space Implementation-Assessment

Include Friction, Driving Torque (small t steps)

Geometry Tends to Remain

t

.

t θ θ

Limit฀Cycles฀for฀Realistic฀Pendulum

m I θ α α f

Friction ⇒ ↓ E Inward Spiral τext can put E back Limit cycle = Balance < τext > = < friction >

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SLIDE 41

ODE Free Pend Phase Space Implementation-Assessment

Include Friction, Driving Torque (small t steps)

Geometry Tends to Remain

t

.

t θ θ

Limit฀Cycles฀for฀Realistic฀Pendulum

m I θ α α f

Friction ⇒ ↓ E Inward Spiral τext can put E back Limit cycle = Balance < τext > = < friction >

41 / 58

SLIDE 42

ODE Free Pend Phase Space Implementation-Assessment

Include Friction, Driving Torque (small t steps)

Geometry Tends to Remain

t

.

t θ θ

Limit฀Cycles฀for฀Realistic฀Pendulum

m I θ α α f

Friction ⇒ ↓ E Inward Spiral τext can put E back Limit cycle = Balance < τext > = < friction >

42 / 58

SLIDE 43

ODE Free Pend Phase Space Implementation-Assessment

Include Friction, Driving Torque (small t steps)

Geometry Tends to Remain

t

.

t θ θ

Limit฀Cycles฀for฀Realistic฀Pendulum

m I θ α α f

Friction ⇒ ↓ E Inward Spiral τext can put E back Limit cycle = Balance < τext > = < friction >

43 / 58

SLIDE 44

ODE Free Pend Phase Space Implementation-Assessment

Include Friction, Driving Torque (small t steps)

Geometry Tends to Remain

t

.

t θ θ

Limit฀Cycles฀for฀Realistic฀Pendulum

m I θ α α f

Friction ⇒ ↓ E Inward Spiral τext can put E back Limit cycle = Balance < τext > = < friction >

44 / 58

SLIDE 45

ODE Free Pend Phase Space Implementation-Assessment

Include Friction, Driving Torque (small t steps)

Geometry Tends to Remain

t

.

t θ θ

Limit฀Cycles฀for฀Realistic฀Pendulum

m I θ α α f

Friction ⇒ ↓ E Inward Spiral τext can put E back Limit cycle = Balance < τext > = < friction >

45 / 58

SLIDE 46

ODE Free Pend Phase Space Implementation-Assessment

Chaos As Viewed in Phase Space (Full Solution)

Look for in Your Simulations

. Complex ≤ Chaos ≤ Rand Fixed Params, all x0, ts: flows Chaos complex = mess Figs distort, remains Closed = periodic Simplicity in chaos [PS, = θ(t)] → attractors (return) Random = cloud fill E Bands ⇒ continuity, sequential ⇒ hypersensitive θ(t) Tools measure chaos

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SLIDE 47

ODE Free Pend Phase Space Implementation-Assessment

Chaos As Viewed in Phase Space (Full Solution)

Look for in Your Simulations

. Complex ≤ Chaos ≤ Rand Fixed Params, all x0, ts: flows Chaos complex = mess Figs distort, remains Closed = periodic Simplicity in chaos [PS, = θ(t)] → attractors (return) Random = cloud fill E Bands ⇒ continuity, sequential ⇒ hypersensitive θ(t) Tools measure chaos

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SLIDE 48

ODE Free Pend Phase Space Implementation-Assessment

Chaos As Viewed in Phase Space (Full Solution)

Look for in Your Simulations

. Complex ≤ Chaos ≤ Rand Fixed Params, all x0, ts: flows Chaos complex = mess Figs distort, remains Closed = periodic Simplicity in chaos [PS, = θ(t)] → attractors (return) Random = cloud fill E Bands ⇒ continuity, sequential ⇒ hypersensitive θ(t) Tools measure chaos

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SLIDE 49

ODE Free Pend Phase Space Implementation-Assessment

Chaos As Viewed in Phase Space (Full Solution)

Look for in Your Simulations

. Complex ≤ Chaos ≤ Rand Fixed Params, all x0, ts: flows Chaos complex = mess Figs distort, remains Closed = periodic Simplicity in chaos [PS, = θ(t)] → attractors (return) Random = cloud fill E Bands ⇒ continuity, sequential ⇒ hypersensitive θ(t) Tools measure chaos

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SLIDE 50

ODE Free Pend Phase Space Implementation-Assessment

Chaos As Viewed in Phase Space (Full Solution)

Look for in Your Simulations

. Complex ≤ Chaos ≤ Rand Fixed Params, all x0, ts: flows Chaos complex = mess Figs distort, remains Closed = periodic Simplicity in chaos [PS, = θ(t)] → attractors (return) Random = cloud fill E Bands ⇒ continuity, sequential ⇒ hypersensitive θ(t) Tools measure chaos

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SLIDE 51

ODE Free Pend Phase Space Implementation-Assessment

Chaos As Viewed in Phase Space (Full Solution)

Look for in Your Simulations

. Complex ≤ Chaos ≤ Rand Fixed Params, all x0, ts: flows Chaos complex = mess Figs distort, remains Closed = periodic Simplicity in chaos [PS, = θ(t)] → attractors (return) Random = cloud fill E Bands ⇒ continuity, sequential ⇒ hypersensitive θ(t) Tools measure chaos

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SLIDE 52

ODE Free Pend Phase Space Implementation-Assessment

Chaos As Viewed in Phase Space (Full Solution)

Look for in Your Simulations

. Complex ≤ Chaos ≤ Rand Fixed Params, all x0, ts: flows Chaos complex = mess Figs distort, remains Closed = periodic Simplicity in chaos [PS, = θ(t)] → attractors (return) Random = cloud fill E Bands ⇒ continuity, sequential ⇒ hypersensitive θ(t) Tools measure chaos

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SLIDE 53

ODE Free Pend Phase Space Implementation-Assessment

Chaos As Viewed in Phase Space (Full Solution)

Look for in Your Simulations

. Complex ≤ Chaos ≤ Rand Fixed Params, all x0, ts: flows Chaos complex = mess Figs distort, remains Closed = periodic Simplicity in chaos [PS, = θ(t)] → attractors (return) Random = cloud fill E Bands ⇒ continuity, sequential ⇒ hypersensitive θ(t) Tools measure chaos

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SLIDE 54

ODE Free Pend Phase Space Implementation-Assessment

Chaos As Viewed in Phase Space (Full Solution)

Look for in Your Simulations

. Complex ≤ Chaos ≤ Rand Fixed Params, all x0, ts: flows Chaos complex = mess Figs distort, remains Closed = periodic Simplicity in chaos [PS, = θ(t)] → attractors (return) Random = cloud fill E Bands ⇒ continuity, sequential ⇒ hypersensitive θ(t) Tools measure chaos

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SLIDE 55

ODE Free Pend Phase Space Implementation-Assessment

Chaos As Viewed in Phase Space (Full Solution)

Look for in Your Simulations

. Complex ≤ Chaos ≤ Rand Fixed Params, all x0, ts: flows Chaos complex = mess Figs distort, remains Closed = periodic Simplicity in chaos [PS, = θ(t)] → attractors (return) Random = cloud fill E Bands ⇒ continuity, sequential ⇒ hypersensitive θ(t) Tools measure chaos

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SLIDE 56

ODE Free Pend Phase Space Implementation-Assessment

Chaos As Viewed in Phase Space (Full Solution)

Look for in Your Simulations

. Complex ≤ Chaos ≤ Rand Fixed Params, all x0, ts: flows Chaos complex = mess Figs distort, remains Closed = periodic Simplicity in chaos [PS, = θ(t)] → attractors (return) Random = cloud fill E Bands ⇒ continuity, sequential ⇒ hypersensitive θ(t) Tools measure chaos

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SLIDE 57

ODE Free Pend Phase Space Implementation-Assessment

Chaos As Viewed in Phase Space (Full Solution)

Look for in Your Simulations

. Complex ≤ Chaos ≤ Rand Fixed Params, all x0, ts: flows Chaos complex = mess Figs distort, remains Closed = periodic Simplicity in chaos [PS, = θ(t)] → attractors (return) Random = cloud fill E Bands ⇒ continuity, sequential ⇒ hypersensitive θ(t) Tools measure chaos

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SLIDE 58

ODE Free Pend Phase Space Implementation-Assessment

Implementation: Let’s Get Down to Work

Good Time for a Break!

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