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

Example: RLC circuit

To illustrate: Population dynamics 1798 Malthusian catastrophe 1838 Verhulst, logistic growth Predator-prey systems, Volterra-Lotka

Population dynamics Malthus: Verhulst: Logistic growth  

Population dynamics Hudson Bay Company

Population dynamics V.Volterra, commercial fishing in the Adriatic

In the x 1 -x 2 plane

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

Population dynamics self-limiting term  stable focus Delay  limit cycle

As functions of time

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 Published by: The University of Chicago Press for The American Society of Naturalists Stable URL:

Putting equations in state-space form 

Traditional state space: Example: the (nonlinear) pendulum McMaster

Linear pendulum: small θ For simplicity, let g/l = 1 Circles!

Pendulum in the phase plane

Varieties of Behavior Stable focus Periodic Limit cycle

Varieties of Behavior Stable focus Periodic Limit cycle Chaos …Assignment

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

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

Euler Local error = O(h 2 ) Global accumulated) error = O(h) (Roughly: multiply by T/h )

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

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

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

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

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