dynamical stability - qmul mathsfjw/goldsmiths/2008/wj/gtc_ds.pdf · remark: the analytical...
TRANSCRIPT
W. Just: Dynamical Stability 0 1 2 3 4
Dynamical Stability
Goldsmiths Teachers Course
20 - 25 July 2008
QMUL London
Wolfram Just
0 Content
1 Stability
2 The pendulum
3 Kapitza pendulum
4 What else ?
1
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
2
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)
dτ
• differential equation with given the initial values ϕ(0) and ϕ(0) → ϕ(τ) =?
3
W. Just: Dynamical Stability 0 1 2 3 4
Equilibrium solutions (fixed points): ϕ ≡ 0 and ϕ ≡ π → stability ?
Energy
E =1
2
„dϕ(τ)
dτ
«2
| z kinetic energy
+ 1− cos(ϕ(τ))| z V (ϕ(τ)) potential energy
dE
dτ=
dϕ(τ)
dτ
d2ϕ(τ)
dτ 2+ sin(ϕ(τ))
dϕ(τ)
dτ
=dϕ(τ)
dτ
„− sin(ϕ(τ))− γ
dϕ(τ)
dτ+ sin(ϕ(τ))
«= −γ
„dϕ(τ)
dτ
«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-π
V(ϕ
)
ϕ
4
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 ?
5
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)
dτ
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
6
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
dτ
˛ 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)
dτ
' − sin(ϕsl(τ))−A cos(Ωτ)ϕft(τ) cos(ϕsl(τ))− γdϕsl(τ)
dτ
' − sin(ϕsl(τ))−A2
4Ω2sin(2ϕsl(τ))− γ
dϕsl(τ)
dτ
7
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.
8
W. Just: Dynamical Stability 0 1 2 3 4
4 What else ?
• Dependence on initial conditions (I)
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”
9
W. Just: Dynamical Stability 0 1 2 3 4
• Dependence on initial conditions (II)
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”
10