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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 x 1 -x 2 plane

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State space Produces a family of concentric closed curves as shown… How to compute? Integrate analytically!

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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 θ 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 Euler’s Method simple-minded, basis of many others Predictor-corrector methods can be useful Runge-Kutta (usually 4 th -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(h 2 ) Global accumulated) error = O(h) (Roughly: multiply by T/h )

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Euler Local error = O(h 2 ) Global (accumulated) error = O(h) Euler step

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Euler Local error = O(h 2 ) Global (accumulated) error = O(h) Taylor’s series with remainder Euler step

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Second-order Runge-Kutta (midpoint method) Local error = O(h 3 ) Global (accumulated) error = O(h 2 )

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Fourth-order Runge-Kutta Local error = O(h 5 ) Global (accumulated) error = O(h 4 )

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

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