Dynamical Stability - maths.qmul.ac.ukfjw/goldsmiths/2009/WJ/stability.pdf · 3 Kapitza pendulum A...

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W. Just: Dynamical Stability 01234 Dynamical Stability Goldsmiths Teacher Course 20 - 24 July 2009 QMUL London Wolfram Just 0 Content 1 Stability 2 The pendulum 3 Kapitza pendulum 4 What else ? 1

Transcript of Dynamical Stability - maths.qmul.ac.ukfjw/goldsmiths/2009/WJ/stability.pdf · 3 Kapitza pendulum A...

Page 1: Dynamical Stability - maths.qmul.ac.ukfjw/goldsmiths/2009/WJ/stability.pdf · 3 Kapitza pendulum A cos( )Wt A ˛ 1 Ω ˛ 1 equations of motion d2ϕ(τ) d ...

W. Just: Dynamical Stability 0 1 2 3 4

Dynamical Stability

Goldsmiths Teacher Course

20 - 24 July 2009

QMUL London

Wolfram Just

0 Content

1 Stability

2 The pendulum

3 Kapitza pendulum

4 What else ?

1

Page 2: Dynamical Stability - maths.qmul.ac.ukfjw/goldsmiths/2009/WJ/stability.pdf · 3 Kapitza pendulum A cos( )Wt A ˛ 1 Ω ˛ 1 equations of motion d2ϕ(τ) d ...

W. Just: Dynamical Stability 0 1 2 3 4

1 Stability

• mathematical models for time evolution (in physics, chemistry, biology, medicine, engineering,social sciences, ...)

→ dynamical systems (differential equations, time discrete maps, cellular automata, ...)

• desired dynamical behaviour (e.g. aviation, heartbeat, economic cycles, ...)

→ stability and control

• quantitative approach

→ mathematical theory required

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Page 3: Dynamical Stability - maths.qmul.ac.ukfjw/goldsmiths/2009/WJ/stability.pdf · 3 Kapitza pendulum A cos( )Wt A ˛ 1 Ω ˛ 1 equations of motion d2ϕ(τ) d ...

W. Just: Dynamical Stability 0 1 2 3 4

2 The pendulum

ϕ (t)

mg sin( (t))ϕmg

l

m

equations of motion (physics)

mass × acceleration = force

m × `d2ϕ(t)

dt2= −mg sin(ϕ(t))| z

gravitation

− Γdϕ(t)

dt| z friction (Γ > 0)

• nondimensional units

τ = tq

g/`, γ = Γ/(mp

g`)

• equations of motion (mathematics)

d2ϕ(τ)

dτ 2= − sin(ϕ(τ))− γ

dϕ(τ)

• differential equation with given the initial values ϕ(0) and ϕ(0) → ϕ(τ) =?

• equilibrium solutions (fixed points): ϕ ≡ 0 and ϕ ≡ π → stability ?

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Page 4: Dynamical Stability - maths.qmul.ac.ukfjw/goldsmiths/2009/WJ/stability.pdf · 3 Kapitza pendulum A cos( )Wt A ˛ 1 Ω ˛ 1 equations of motion d2ϕ(τ) d ...

W. Just: Dynamical Stability 0 1 2 3 4

Energy

E =1

2

„dϕ(τ)

«2

| z kinetic energy

+ 1− cos(ϕ(τ))| z V (ϕ(τ)) potential energy

dE

dτ=

dϕ(τ)

d2ϕ(τ)

dτ 2+ sin(ϕ(τ))

dϕ(τ)

=dϕ(τ)

„− sin(ϕ(τ))− γ

dϕ(τ)

dτ+ sin(ϕ(τ))

«= −γ

„dϕ(τ)

«2

≤ 0

Energy decreases

• minimum at ϕ = 0 (and dϕ/dτ = 0) → stable

• “maximum” at ϕ = π (and dϕ/dτ = 0) → unstable 0

1

2

π0-πV

(ϕ)

ϕ

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Page 5: Dynamical Stability - maths.qmul.ac.ukfjw/goldsmiths/2009/WJ/stability.pdf · 3 Kapitza pendulum A cos( )Wt A ˛ 1 Ω ˛ 1 equations of motion d2ϕ(τ) d ...

W. Just: Dynamical Stability 0 1 2 3 4

numerical integration (maple)

ϕ(τ) for γ = 0.5 and initial condition ϕ(0) = 2, ϕ(0) = 1:

> deq1:=diff(x(t),t,t)+0.5*diff(x(t),t)+sin(x(t))=0:

> ic1:=x(0)=2.0,D(x)(0)=1.0:

> dsol1:=dsolve(deq1,ic1,numeric,range=0..20):

> with(plots):

> odeplot(dsol1,[t,x(t)],0..20);–0.5

0

0.5

1

1.5

2

2.5

x

5 10 15 20

t

Remark: The analytical stability considerations can be generalised to a large class of differentialequations and/or dynamical systems → Lyapunov method.

Main problem: how to find a suitable “energy” i.e. a Lyapunov function ?

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Page 6: Dynamical Stability - maths.qmul.ac.ukfjw/goldsmiths/2009/WJ/stability.pdf · 3 Kapitza pendulum A cos( )Wt A ˛ 1 Ω ˛ 1 equations of motion d2ϕ(τ) d ...

W. Just: Dynamical Stability 0 1 2 3 4

3 Kapitza pendulum

A cos( )Ωτ

A 1

Ω 1

equations of motion

d2ϕ(τ)

dτ 2= − (1 + A cos(Ωτ)) sin(ϕ(τ))− γ

dϕ(τ)

numerical integration (maple)

ϕ(τ) for γ = 0.5, A = 20, Ω = 10, and initial conditionϕ(0) = 2, ϕ(0) = 1:

> deq2:=diff(x(t),t,t)+0.5*diff(x(t),t)+

(1+20*cos(10*t))*sin(x(t))=0:

> ic2:=x(0)=2.0,D(x)(0)=1.0:

> dsol2:=dsolve(deq2,ic2,numeric,range=0..20):

> odeplot(dsol2,[t,x(t)],0..20,refine=3);2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

x

0 5 10 15 20

t

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Page 7: Dynamical Stability - maths.qmul.ac.ukfjw/goldsmiths/2009/WJ/stability.pdf · 3 Kapitza pendulum A cos( )Wt A ˛ 1 Ω ˛ 1 equations of motion d2ϕ(τ) d ...

W. Just: Dynamical Stability 0 1 2 3 4

qualitative analysis

fast and slow modulations

ϕ(τ) = ϕsl(τ) + ϕft(τ), |ϕft| 1,

˛d2ϕft

dτ 2

˛

˛dϕft

˛ 1

average over the fast period

ϕft(τ) = 0, ϕ(τ) = ϕsl(τ)

• fast part of the dynamics: largest terms in the equations of motion

d2ϕft(τ)

dτ 2' −A cos(Ωτ) sin(ϕsl(τ)) ⇒ ϕft(τ) '

A

Ω2cos(Ωτ) sin(ϕsl(τ))

• slow part of the dynamics: averaged equations of motion

d2ϕsl(τ)

dτ 2= −sin(ϕsl(τ) + ϕft(τ))−A cos(Ωτ) sin(ϕsl(τ) + ϕft(τ))| z

' sin(ϕsl) + ϕft cos(ϕsl)

− γdϕsl(τ)

' − sin(ϕsl(τ))−A cos(Ωτ)ϕft(τ) cos(ϕsl(τ))− γdϕsl(τ)

' − sin(ϕsl(τ))−A2

4Ω2sin(2ϕsl(τ))− γ

dϕsl(τ)

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Page 8: Dynamical Stability - maths.qmul.ac.ukfjw/goldsmiths/2009/WJ/stability.pdf · 3 Kapitza pendulum A cos( )Wt A ˛ 1 Ω ˛ 1 equations of motion d2ϕ(τ) d ...

W. Just: Dynamical Stability 0 1 2 3 4

• effective equation of motion with potential energy

V (ϕsl) = 1− cos(ϕsl)−A2

8Ω2cos(2ϕsl)

• stability of the inverted equilibrium position (ϕ = π)

unstable (A = 0)

0

1

2

π0-π

V(ϕ

)

ϕ

A2/Ω2=0

critical (A =√

2Ω)

0

1

2

π0-π

V(ϕ

)

ϕ

A2/Ω2=2

stable (A = 2Ω)

0

1

2

π0-π

V(ϕ

)

ϕ

A2/Ω2=4

Remark: the qualitative analysis can be rephrased as a mathematically rigorous perturbationexpansion.

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Page 9: Dynamical Stability - maths.qmul.ac.ukfjw/goldsmiths/2009/WJ/stability.pdf · 3 Kapitza pendulum A cos( )Wt A ˛ 1 Ω ˛ 1 equations of motion d2ϕ(τ) d ...

W. Just: Dynamical Stability 0 1 2 3 4

4 What else ?

• Dependence on initial conditions (I)

Dynamics of the pendulum for γ = 1/10, A = 10, and Ω = 2

random, “chaotic” motion

–80

–60

–40

–20

0

20

40

60

x

500 1000 1500 2000 2500t

sensitive dependence on two initialconditions which differ by ∆ϕ = 0.0001

–35

–30

–25

–20

–15

–10

–5

0

y, x

10 20 30 40t

→ “Chaos”

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Page 10: Dynamical Stability - maths.qmul.ac.ukfjw/goldsmiths/2009/WJ/stability.pdf · 3 Kapitza pendulum A cos( )Wt A ˛ 1 Ω ˛ 1 equations of motion d2ϕ(τ) d ...

W. Just: Dynamical Stability 0 1 2 3 4

• Dependence on initial conditions (II)

Initial conditions ϕ(0),v(0) = ϕ(0) which lead to:

– normal equilibriumposition ϕ ≡ 0

– inverted equilibriumposition ϕ ≡ π

– permanent rotation

for γ = 1/2, A = 12, andΩ = 6.

-20

0

20

π0-π

v

ϕSet with complicated geometry → “Fractals”

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