Math 211 Math 211 Lecture #42 The Pendulum Predator-Prey - - PowerPoint PPT Presentation

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Math 211 Math 211

Lecture #42 The Pendulum Predator-Prey December 6, 2002

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The Pendulum The Pendulum

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The Pendulum The Pendulum

• The angle θ satisfies the nonlinear differential equation

mLθ′′ = −mg sin θ − D θ′,

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The Pendulum The 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.

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The Pendulum The 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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Analysis Analysis

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Analysis Analysis

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

integer.

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Analysis Analysis

• 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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Analysis Analysis

• 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 = 0

• r a sink if d > 0.

Return Pendulum

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The Inverted Pendulum The Inverted Pendulum

Return Pendulum

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The Inverted Pendulum The Inverted Pendulum

• The angle θ measured from straight up satisfies the

nonlinear differential equation mLθ′′ = mg sin θ − D θ′,

Return Pendulum

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The Inverted Pendulum The Inverted Pendulum

• The angle θ measured from straight up satisfies the

nonlinear differential equation mLθ′′ = mg sin θ − D θ′,

• r

θ′′ + D mLθ′ − g L sin θ = 0.

Return Pendulum

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The Inverted Pendulum The Inverted Pendulum

• The angle θ measured from straight up satisfies the

nonlinear differential equation mLθ′′ = mg sin θ − D θ′,

• r

θ′′ + D mLθ′ − g L sin θ = 0.

We will write this as

θ′′ + d θ − b sin θ = 0.

Return Inverted pendulum Pendulum system

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The Inverted Pendulum System The Inverted Pendulum System

Return Inverted pendulum Pendulum system

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The Inverted Pendulum System The Inverted Pendulum System

• Introduce ω = θ′ to get the system

θ′ = ω ω′ = b sin θ − d ω

Return Inverted pendulum Pendulum system

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The Inverted Pendulum System The 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

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The Inverted Pendulum System The 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
• f 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

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The Inverted Pendulum System The 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
• f 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

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The Control System The 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

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The Control System The 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

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The Control System The 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 System The Controlled System

Return Inverted pendulum Inverted pendulum system Controls

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The Controlled System The Controlled System

• The Jacobian at the origin is

J =

• 1

b − c −d

Return Inverted pendulum Inverted pendulum system Controls

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The Controlled System The Controlled System

• The Jacobian at the origin is

J =

• 1

b − c −d

• The origin is asymptotically stable if T = −d < 0 and

D = c − b > 0.

Return Inverted pendulum Inverted pendulum system Controls

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The Controlled System The Controlled System

• The Jacobian at the origin is

J =

• 1

b − c −d

• The origin is asymptotically stable if T = −d < 0 and

D = c − b > 0. Therefore require c > b = g L.

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Predator-Prey Predator-Prey

Lotka-Volterra system x′ = (a − by)x (prey – fish) y′ = (−c + dx)y (predator – sharks)

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Predator-Prey Predator-Prey

Lotka-Volterra system x′ = (a − by)x (prey – fish) y′ = (−c + dx)y (predator – sharks)

• Equilbrium points:

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Predator-Prey Predator-Prey

Lotka-Volterra system x′ = (a − by)x (prey – fish) y′ = (−c + dx)y (predator – sharks)

• Equilbrium points: (0, 0) is a saddle

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Predator-Prey Predator-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.

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Predator-Prey Predator-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.

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Predator-Prey Predator-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.

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Predator-Prey Predator-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.

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Predator-Prey Predator-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

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Solutions are Periodic Solutions are Periodic

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

Return System

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Solutions are Periodic Solutions are Periodic

Along the solution curve y = y(x) we have dy dx = y(−c + dx) x(a − by) .

Return System

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Solutions are Periodic Solutions 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 ln x = C

Return System

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Solutions are Periodic Solutions 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 ln x = C

• This is an implicit equation for the solution curve.

Return System

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Solutions are Periodic Solutions 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 ln x = C

• This is an implicit equation for the solution curve. ⇒

All solution curves are closed, and represent periodic solutions.

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Why Fishing Leads to More Fish Why Fishing Leads to More Fish

System Return

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Why Fishing Leads to More Fish Why Fishing Leads to More Fish

Compute the average of the fish & shark populations.

System Return

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Why Fishing Leads to More Fish Why Fishing Leads to More Fish

Compute the average of the fish & shark populations. d dt ln x(t) = x′ x =

System Return

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Why Fishing Leads to More Fish Why Fishing Leads to More Fish

Compute the average of the fish & shark populations. d dt ln x(t) = x′ x = a − by

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Why Fishing Leads to More Fish Why Fishing Leads to More Fish

Compute the average of the fish & shark populations. d dt ln x(t) = x′ x = a − by 0 = 1 T T d dt ln x(t) dt

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Why Fishing Leads to More Fish Why Fishing Leads to More Fish

Compute the average of the fish & shark populations. d dt ln x(t) = x′ x = a − by 0 = 1 T T d dt ln x(t) dt = a − by. So y = a/b = y0.

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Why Fishing Leads to More Fish Why Fishing Leads to More Fish

Compute the average of the fish & shark populations. d dt ln x(t) = x′ x = a − by 0 = 1 T T d dt ln x(t) dt = a − by. So y = a/b = y0. Similarly x = x0 = c/d.

System Averages

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The effect of fishing that does not distinquish between fish and sharks is the system

System Averages

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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

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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

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The average populations are x1 = c + e d and y1 = a − e b

Averages

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The average populations are x1 = c + e d and y1 = a − e b Fishing causes the average fish population to increase and the average shark population to decrease.

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Cottony Cushion Scale Insect & the Ladybird Beetle Cottony Cushion Scale Insect & the Ladybird Beetle

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Cottony Cushion Scale Insect & the Ladybird Beetle Cottony Cushion Scale Insect & the Ladybird Beetle

• Cottony cushion scale insect accidentally introduced

from Australia in 1868.

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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.

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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

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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

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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.

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DDT kills the scale insect.

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DDT kills the scale insect.

• Massive spraying ordered.

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DDT kills the scale insect.

• Massive spraying ordered.

Despite the warnings of mathematicians and

biologists.

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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.