Ordinary Differential Equations (ODEs)

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Ordinary Differential Equations (ODEs)

• Differential equations are the ubiquitous, the
  lingua franca of the sciences many different
  fields are linked by having similar differential
  equations
• ODEs have one independent variable PDEs have
  more
• Examples electrical circuits
• Newtonian mechanics
• chemical reactions
• population dynamics
• economics and so on, ad
  infinitum

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Example RLC circuit
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To illustrate Population dynamics

• 1798 Malthusian catastrophe
• 1838 Verhulst, logistic growth
• Predator-prey systems, Volterra-Lotka

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Population dynamics

• Malthus
• Verhulst
• Logistic growth

?
?
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Population dynamics
Hudson Bay Company
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Population dynamics
V .Volterra, commercial fishing in the Adriatic
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In the x1-x2 plane
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State space
Integrate analytically!
Produces a family of concentric closed curves as
shown How to compute?
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Population dynamics
self-limiting term
? stable focus
Delay ? limit cycle
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As functions of time
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Do you believe this?

• Do hares eat lynx, Gilpin 1973

Do Hares Eat Lynx? Michael E. Gilpin The
American Naturalist, Vol. 107, No. 957 (Sep. -
Oct., 1973), pp. 727-730 Published by The
University of Chicago Press for The American
Society of Naturalists Stable URL
http//www.jstor.org/stable/2459670
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Putting equations in state-space form
?
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Traditional state space
Example the (nonlinear) pendulum
McMaster
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Linear pendulum small ?
For simplicity, let g/l 1
Circles!
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Pendulum in the phase plane
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Varieties of Behavior

• Stable focus
• Periodic
• Limit cycle

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Varieties of Behavior

• Stable focus
• Periodic
• Limit cycle
• Chaos
• Assignment

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Numerical integration of ODEs

• Eulers Method ? simple-minded, basis of
  many others
• Predictor-corrector methods ? can be useful
• Runge-Kutta (usually 4th-order) ?workhorse, good
  enough for our work, but not state-of-the-art

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Criteria for evaluating

• Accuracy ? use Taylor series, big-Oh, classical
  numerical analysis
• Efficiency ? running time may be hard to predict,
  sometimes step size is adaptive
• Stability ? some methods diverge on some problems

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Euler

• Local error O(h2)
• Global accumulated) error O(h)

(Roughly multiply by T/h )
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Euler

• Local error O(h2)
• Global (accumulated) error O(h)

Euler step
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Euler

• Local error O(h2)
• Global (accumulated) error O(h)

Taylors series with remainder
Euler step
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Second-order Runge-Kutta (midpoint method)

• Local error O(h3)
• Global (accumulated) error O(h2)

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Fourth-order Runge-Kutta

• Local error O(h5)
• Global (accumulated) error O(h4)

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Additional topics

• Stability, stiff systems
• Implicit methods
• Two-point boundary-value problems
• shooting methods
• relaxation methods