1. 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

2. Example: RLC circuit

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

4. Population dynamics Malthus: Verhulst: Logistic growth  

5. Population dynamics Hudson Bay Company

6. Population dynamics V.Volterra, commercial fishing in the Adriatic

7. In the x 1 -x 2 plane

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

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

10. As functions of time

11. 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

12. Putting equations in state-space form 

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

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

15. Pendulum in the phase plane

16. Varieties of Behavior Stable focus Periodic Limit cycle

17. Varieties of Behavior Stable focus Periodic Limit cycle Chaos …Assignment

18. 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

19. 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

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

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

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

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

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

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