1 Math 211

60
1 Math 211 Math 211 Lecture #42 The Pendulum Predator-Prey December 6, 2002

Transcript of 1 Math 211

Page 1: 1 Math 211

1

Math 211Math 211

Lecture #42

The Pendulum

Predator-Prey

December 6, 2002

Page 2: 1 Math 211

Return

2

The PendulumThe Pendulum

Page 3: 1 Math 211

Return

2

The PendulumThe Pendulum

• The angle θ satisfies the nonlinear differential equation

mLθ′′ = −mg sin θ − D θ′,

Page 4: 1 Math 211

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.

Page 5: 1 Math 211

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

Page 6: 1 Math 211

Return

3

AnalysisAnalysis

Page 7: 1 Math 211

Return

3

AnalysisAnalysis

• The equilibrium points are (k π, 0)T where k is any

integer.

Page 8: 1 Math 211

Return

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.

Page 9: 1 Math 211

Return

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.

Page 10: 1 Math 211

Return Pendulum

4

The Inverted PendulumThe Inverted Pendulum

Page 11: 1 Math 211

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′,

Page 12: 1 Math 211

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.

Page 13: 1 Math 211

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.

Page 14: 1 Math 211

Return Inverted pendulum Pendulum system

5

The Inverted Pendulum SystemThe Inverted Pendulum System

Page 15: 1 Math 211

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

Page 16: 1 Math 211

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.

Page 17: 1 Math 211

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 ?

Page 18: 1 Math 211

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.

Page 19: 1 Math 211

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,

Page 20: 1 Math 211

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.

Page 21: 1 Math 211

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.

Page 22: 1 Math 211

Return Inverted pendulum Inverted pendulum system Controls

7

The Controlled SystemThe Controlled System

Page 23: 1 Math 211

Return Inverted pendulum Inverted pendulum system Controls

7

The Controlled SystemThe Controlled System

• The Jacobian at the origin is

J =(

0 1b − c −d

)

Page 24: 1 Math 211

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.

Page 25: 1 Math 211

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.

Page 26: 1 Math 211

Return

8

Predator-PreyPredator-Prey

Lotka-Volterra system

x′ = (a − by)x (prey – fish)

y′ = (−c + dx)y (predator – sharks)

Page 27: 1 Math 211

Return

8

Predator-PreyPredator-Prey

Lotka-Volterra system

x′ = (a − by)x (prey – fish)

y′ = (−c + dx)y (predator – sharks)

• Equilbrium points:

Page 28: 1 Math 211

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

Page 29: 1 Math 211

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.

Page 30: 1 Math 211

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.

Page 31: 1 Math 211

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.

Page 32: 1 Math 211

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.

Page 33: 1 Math 211

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?

Page 34: 1 Math 211

Return System

9

Solutions are PeriodicSolutions are Periodic

Along the solution curve y = y(x) we have

Page 35: 1 Math 211

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)

.

Page 36: 1 Math 211

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

Page 37: 1 Math 211

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.

Page 38: 1 Math 211

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.

Page 39: 1 Math 211

System Return

10

Why Fishing Leads to More FishWhy Fishing Leads to More Fish

Page 40: 1 Math 211

System Return

10

Why Fishing Leads to More FishWhy Fishing Leads to More Fish

Compute the average of the fish & shark populations.

Page 41: 1 Math 211

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=

Page 42: 1 Math 211

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

Page 43: 1 Math 211

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

Page 44: 1 Math 211

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.

Page 45: 1 Math 211

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.

Page 46: 1 Math 211

System Averages

11

The effect of fishing that does not distinquish between fish

and sharks is the system

Page 47: 1 Math 211

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

Page 48: 1 Math 211

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.

Page 49: 1 Math 211

Averages

12

The average populations are

x1 =c + e

dand y1 =

a − e

b

Page 50: 1 Math 211

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.

Page 51: 1 Math 211

Return

13

Cottony Cushion Scale Insect & the

Ladybird Beetle

Cottony Cushion Scale Insect & the

Ladybird Beetle

Page 52: 1 Math 211

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.

Page 53: 1 Math 211

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.

Page 54: 1 Math 211

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

Page 55: 1 Math 211

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

Page 56: 1 Math 211

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.

Page 57: 1 Math 211

14

DDT kills the scale insect.

Page 58: 1 Math 211

14

DDT kills the scale insect.

• Massive spraying ordered.

Page 59: 1 Math 211

14

DDT kills the scale insect.

• Massive spraying ordered.

� Despite the warnings of mathematicians and

biologists.

Page 60: 1 Math 211

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.