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Math 211Math 211
Lecture #42
The Pendulum
Predator-Prey
December 6, 2002
Return
2
The PendulumThe Pendulum
Return
2
The PendulumThe Pendulum
• The angle θ satisfies the nonlinear differential equation
mLθ′′ = −mg sin θ − D θ′,
Return
2
The PendulumThe Pendulum
• The angle θ satisfies the nonlinear differential equation
mLθ′′ = −mg sin θ − D θ′,
� We will write this as
θ′′ + d θ + b sin θ = 0.
Return
2
The PendulumThe Pendulum
• The angle θ satisfies the nonlinear differential equation
mLθ′′ = −mg sin θ − D θ′,
� We will write this as
θ′′ + d θ + b sin θ = 0.
• Introduce ω = θ′ to get the system
θ′ = ω
ω′ = −b sin θ − d ω
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3
AnalysisAnalysis
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3
AnalysisAnalysis
• The equilibrium points are (k π, 0)T where k is any
integer.
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3
AnalysisAnalysis
• The equilibrium points are (k π, 0)T where k is any
integer.
� If k is odd the equilibrium point is a saddle.
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3
AnalysisAnalysis
• The equilibrium points are (k π, 0)T where k is any
integer.
� If k is odd the equilibrium point is a saddle.
� If k is even the equilibrium point is a center if d = 0or a sink if d > 0.
Return Pendulum
4
The Inverted PendulumThe Inverted Pendulum
Return Pendulum
4
The Inverted PendulumThe Inverted Pendulum
• The angle θ measured from straight up satisfies the
nonlinear differential equation
mLθ′′ = mg sin θ − D θ′,
Return Pendulum
4
The Inverted PendulumThe Inverted Pendulum
• The angle θ measured from straight up satisfies the
nonlinear differential equation
mLθ′′ = mg sin θ − D θ′,
or
θ′′ +D
mLθ′ − g
Lsin θ = 0.
Return Pendulum
4
The Inverted PendulumThe Inverted Pendulum
• The angle θ measured from straight up satisfies the
nonlinear differential equation
mLθ′′ = mg sin θ − D θ′,
or
θ′′ +D
mLθ′ − g
Lsin θ = 0.
� We will write this as
θ′′ + d θ − b sin θ = 0.
Return Inverted pendulum Pendulum system
5
The Inverted Pendulum SystemThe Inverted Pendulum System
Return Inverted pendulum Pendulum system
5
The Inverted Pendulum SystemThe Inverted Pendulum System
• Introduce ω = θ′ to get the system
θ′ = ω
ω′ = b sin θ − d ω
Return Inverted pendulum Pendulum system
5
The Inverted Pendulum SystemThe Inverted Pendulum System
• Introduce ω = θ′ to get the system
θ′ = ω
ω′ = b sin θ − d ω
• The equilibrium point at (0, 0)T is a saddle point and
unstable.
Return Inverted pendulum Pendulum system
5
The Inverted Pendulum SystemThe Inverted Pendulum System
• Introduce ω = θ′ to get the system
θ′ = ω
ω′ = b sin θ − d ω
• The equilibrium point at (0, 0)T is a saddle point and
unstable.
• Can we find an automatic way of sensing the departure
of the system from (0, 0)T and moving the pivot to
bring the system back to the unstable point at (0, 0)T ?
Return Inverted pendulum Pendulum system
5
The Inverted Pendulum SystemThe Inverted Pendulum System
• Introduce ω = θ′ to get the system
θ′ = ω
ω′ = b sin θ − d ω
• The equilibrium point at (0, 0)T is a saddle point and
unstable.
• Can we find an automatic way of sensing the departure
of the system from (0, 0)T and moving the pivot to
bring the system back to the unstable point at (0, 0)T ?
� Experimentally the answer is yes.
Return Inverted pendulum Inverted pendulum system
6
The Control SystemThe Control System
• If we apply a force v moving the pivot to the right or
left, then θ satisfies
mLθ′′ = mg sin θ − D θ′ − v cos θ,
Return Inverted pendulum Inverted pendulum system
6
The Control SystemThe Control System
• If we apply a force v moving the pivot to the right or
left, then θ satisfies
mLθ′′ = mg sin θ − D θ′ − v cos θ,
• The system becomes
θ′ = ω
ω′ = b sin θ − d ω − u cos θ,
where u = v/mL.
Return Inverted pendulum Inverted pendulum system
6
The Control SystemThe Control System
• If we apply a force v moving the pivot to the right or
left, then θ satisfies
mLθ′′ = mg sin θ − D θ′ − v cos θ,
• The system becomes
θ′ = ω
ω′ = b sin θ − d ω − u cos θ,
where u = v/mL.
• Assume the force is a linear response to the detected
value of θ, so u = cθ, where c is a constant.
Return Inverted pendulum Inverted pendulum system Controls
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The Controlled SystemThe Controlled System
Return Inverted pendulum Inverted pendulum system Controls
7
The Controlled SystemThe Controlled System
• The Jacobian at the origin is
J =(
0 1b − c −d
)
Return Inverted pendulum Inverted pendulum system Controls
7
The Controlled SystemThe Controlled System
• The Jacobian at the origin is
J =(
0 1b − c −d
)
• The origin is asymptotically stable if T = −d < 0 and
D = c − b > 0.
Return Inverted pendulum Inverted pendulum system Controls
7
The Controlled SystemThe Controlled System
• The Jacobian at the origin is
J =(
0 1b − c −d
)
• The origin is asymptotically stable if T = −d < 0 and
D = c − b > 0. Therefore require
c > b =g
L.
Return
8
Predator-PreyPredator-Prey
Lotka-Volterra system
x′ = (a − by)x (prey – fish)
y′ = (−c + dx)y (predator – sharks)
Return
8
Predator-PreyPredator-Prey
Lotka-Volterra system
x′ = (a − by)x (prey – fish)
y′ = (−c + dx)y (predator – sharks)
• Equilbrium points:
Return
8
Predator-PreyPredator-Prey
Lotka-Volterra system
x′ = (a − by)x (prey – fish)
y′ = (−c + dx)y (predator – sharks)
• Equilbrium points: (0, 0) is a saddle
Return
8
Predator-PreyPredator-Prey
Lotka-Volterra system
x′ = (a − by)x (prey – fish)
y′ = (−c + dx)y (predator – sharks)
• Equilbrium points: (0, 0) is a saddle,
(x0, y0) = (c/d, a/b) is a linear center.
Return
8
Predator-PreyPredator-Prey
Lotka-Volterra system
x′ = (a − by)x (prey – fish)
y′ = (−c + dx)y (predator – sharks)
• Equilbrium points: (0, 0) is a saddle,
(x0, y0) = (c/d, a/b) is a linear center.
• The axes are invariant.
Return
8
Predator-PreyPredator-Prey
Lotka-Volterra system
x′ = (a − by)x (prey – fish)
y′ = (−c + dx)y (predator – sharks)
• Equilbrium points: (0, 0) is a saddle,
(x0, y0) = (c/d, a/b) is a linear center.
• The axes are invariant.
• The positive quadrant is invariant.
Return
8
Predator-PreyPredator-Prey
Lotka-Volterra system
x′ = (a − by)x (prey – fish)
y′ = (−c + dx)y (predator – sharks)
• Equilbrium points: (0, 0) is a saddle,
(x0, y0) = (c/d, a/b) is a linear center.
• The axes are invariant.
• The positive quadrant is invariant.
• The solution curves appear to be closed.
Return
8
Predator-PreyPredator-Prey
Lotka-Volterra system
x′ = (a − by)x (prey – fish)
y′ = (−c + dx)y (predator – sharks)
• Equilbrium points: (0, 0) is a saddle,
(x0, y0) = (c/d, a/b) is a linear center.
• The axes are invariant.
• The positive quadrant is invariant.
• The solution curves appear to be closed. Is this
actually true?
Return System
9
Solutions are PeriodicSolutions are Periodic
Along the solution curve y = y(x) we have
Return System
9
Solutions are PeriodicSolutions are Periodic
Along the solution curve y = y(x) we have
dy
dx=
y(−c + dx)x(a − by)
.
Return System
9
Solutions are PeriodicSolutions are Periodic
Along the solution curve y = y(x) we have
dy
dx=
y(−c + dx)x(a − by)
.
The solution is
H(x, y) = by − a ln y + dx − c lnx = C
Return System
9
Solutions are PeriodicSolutions are Periodic
Along the solution curve y = y(x) we have
dy
dx=
y(−c + dx)x(a − by)
.
The solution is
H(x, y) = by − a ln y + dx − c lnx = C
• This is an implicit equation for the solution curve.
Return System
9
Solutions are PeriodicSolutions are Periodic
Along the solution curve y = y(x) we have
dy
dx=
y(−c + dx)x(a − by)
.
The solution is
H(x, y) = by − a ln y + dx − c lnx = C
• This is an implicit equation for the solution curve. ⇒All solution curves are closed, and represent periodic
solutions.
System Return
10
Why Fishing Leads to More FishWhy Fishing Leads to More Fish
System Return
10
Why Fishing Leads to More FishWhy Fishing Leads to More Fish
Compute the average of the fish & shark populations.
System Return
10
Why Fishing Leads to More FishWhy Fishing Leads to More Fish
Compute the average of the fish & shark populations.
d
dtlnx(t) =
x′
x=
System Return
10
Why Fishing Leads to More FishWhy Fishing Leads to More Fish
Compute the average of the fish & shark populations.
d
dtlnx(t) =
x′
x= a − by
System Return
10
Why Fishing Leads to More FishWhy Fishing Leads to More Fish
Compute the average of the fish & shark populations.
d
dtlnx(t) =
x′
x= a − by
0 =1T
∫ T
0
d
dtlnx(t) dt
System Return
10
Why Fishing Leads to More FishWhy Fishing Leads to More Fish
Compute the average of the fish & shark populations.
d
dtlnx(t) =
x′
x= a − by
0 =1T
∫ T
0
d
dtlnx(t) dt = a − by.
So y = a/b = y0.
System Return
10
Why Fishing Leads to More FishWhy Fishing Leads to More Fish
Compute the average of the fish & shark populations.
d
dtlnx(t) =
x′
x= a − by
0 =1T
∫ T
0
d
dtlnx(t) dt = a − by.
So y = a/b = y0. Similarly x = x0 = c/d.
System Averages
11
The effect of fishing that does not distinquish between fish
and sharks is the system
System Averages
11
The effect of fishing that does not distinquish between fish
and sharks is the system
x′ = (a − by)x − ex
y′ = (−c + dx)y − ey
System Averages
11
The effect of fishing that does not distinquish between fish
and sharks is the system
x′ = (a − by)x − ex
y′ = (−c + dx)y − ey
This is the same system with a replaced by a − e and c
replaced by c + e.
Averages
12
The average populations are
x1 =c + e
dand y1 =
a − e
b
Averages
12
The average populations are
x1 =c + e
dand y1 =
a − e
b
Fishing causes the average fish population to increase and
the average shark population to decrease.
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13
Cottony Cushion Scale Insect & the
Ladybird Beetle
Cottony Cushion Scale Insect & the
Ladybird Beetle
Return
13
Cottony Cushion Scale Insect & the
Ladybird Beetle
Cottony Cushion Scale Insect & the
Ladybird Beetle• Cottony cushion scale insect accidentally introduced
from Australia in 1868.
Return
13
Cottony Cushion Scale Insect & the
Ladybird Beetle
Cottony Cushion Scale Insect & the
Ladybird Beetle• Cottony cushion scale insect accidentally introduced
from Australia in 1868.
� Threatened the citrus industry.
Return
13
Cottony Cushion Scale Insect & the
Ladybird Beetle
Cottony Cushion Scale Insect & the
Ladybird Beetle• Cottony cushion scale insect accidentally introduced
from Australia in 1868.
� Threatened the citrus industry.
• Ladybird beetle imported from Australia
Return
13
Cottony Cushion Scale Insect & the
Ladybird Beetle
Cottony Cushion Scale Insect & the
Ladybird Beetle• Cottony cushion scale insect accidentally introduced
from Australia in 1868.
� Threatened the citrus industry.
• Ladybird beetle imported from Australia
� Natural predator
Return
13
Cottony Cushion Scale Insect & the
Ladybird Beetle
Cottony Cushion Scale Insect & the
Ladybird Beetle• Cottony cushion scale insect accidentally introduced
from Australia in 1868.
� Threatened the citrus industry.
• Ladybird beetle imported from Australia
� Natural predator – reduced the insects to
manageable low.
14
DDT kills the scale insect.
14
DDT kills the scale insect.
• Massive spraying ordered.
14
DDT kills the scale insect.
• Massive spraying ordered.
� Despite the warnings of mathematicians and
biologists.
14
DDT kills the scale insect.
• Massive spraying ordered.
� Despite the warnings of mathematicians and
biologists.
• The scale insect increased in numbers, as predicted by
Volterra.