1. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams1 CEE162 Lecture 2: Nonlinear ODEs and phase diagrams; predator-prey Overview Nonlinear chaotic ODEs: the damped nonlinear forced pendulum 2 nd Order damped harmonic oscillator Systems of ODEs Phase diagrams –Fixed points –Isoclines/Nullclines Predator-prey model References: Dym, Ch 7; Mooney & Swift, Ch 5.2-5.3; Kreyszig, Ch 4

2. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams2 Forced pendulum g m m Frictional effect

3. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams3 Free-body diagram

4. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams4 Derivation of the governing ODE

5. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams5 m

6. 6

7. 7 Reduce and nondimensionalize!

8. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams8

9. 9 Governing nondimensional ODE

10. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams10 Linearize

11. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams11 The damped harmonic oscillator

12. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams12

13. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams13

14. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams14

15. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams15 The particular solution

16. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams16

17. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams17 Simulating the nonlinear system pendulum.zip

18. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams18 Phase plane analysis

19. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams19 Direction field for  1 =0.5 24 phasedirection.m

20. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams20 Computing phase lines analytically Elliptic Integral! Solution in phase space

21. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams21 Analytical Phase Lines for

22. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams22 Nullclines and fixed points

23. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams23 Plotting nullclines and fixed points p=0 (no velocity) q=0 (no acceleration) increasing friction Fixed points

24. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams24 Behavior in the vicinity of fixed points Suppose we have a nonlinear coupled set of ODEs in the form We can determine the behavior of this ODE in the vicinity of the fixed points by analyzing the behavior of disturbances applied to the fixed points such that where the point is a fixed point corresponding to

25. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams25 Using the Taylor series expansion about the fixed point, we have Substitution into the ODEs gives Since the fixed points satisfy

26. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams26 and, then the perturbations satisfy In vector form, this is given by The Jacobian matrix is given by

27. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams27 The behavior of the solution in the phase plane in the vicinity of the fixed points is determined by the behavior of the eigenvalues of the Jacobian. If then the eigenvalues of J are given by

28. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams28 two real negative roots. complex pair, negative real part.

29. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams29 two real positive roots. complex pair, positive real part. pure imaginary.

30. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams30 Phase plane analysis for the pendulum

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32. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams32 Underdamped Critical or overdamped

33. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams33 Spiral direction CW or CCW? Clockwise Counter- clockwise c>0 c<0

34. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams34 Behavior around saddle point

35. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams35

36. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams36

37. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams37 The predator-prey problem Overview Lotka-Volterra predator-prey model –Phase-plane analysis –Analytical solutions –Numerical solutions References: Mooney & Swift, Ch 5.2-5.3;

38. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams38 Compartmental Analysis Tool to graphically set up an ODE-based model –Example: Population Immigration: ix Births: bx Deaths: dx Emigration: ex Population: x

39. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams39 Logistic equation Population: x Can flow both directions but the direction shown is defined as positive

40. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams40 Income class model Lower x Middle y Upper z

41. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams41 For a system the fixed points are given by the Null space of the matrix A. For the income class model:

42. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams42 Classical Predator-Prey Model Predator yPrey x Die-off in absence of prey dy Growth in absence of predators ax bxy cxy Lotka-Volterra predator-prey equations

43. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams43 Assumptions about the interaction term xy xy = interaction; bxy: b = likelihood that it results in a prey death; cxy: c = likelihood that it leads to predator success. An "interaction" results when prey moves into predator territory. Animals reside in a fixed region (an infinite region would not affect number of interactions). Predators never become satiated.

44. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams44 Phase-plane analysis

45. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams45

46. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams46 Analytical solution

47. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams47 Solution with Matlab % Initial condition is a low predator population with % a fixed-point prey population. X0 = [x0,.25*y0]'; % Decrease the relative tolerance opts = odeset('reltol',1e-4); [t,X]=ode23(@pprey,[0 tmax],X0,opts); lvdemo.m function Xdot = pprey(t,X) % Constants are set in lvdemo.m (the calling function) global a b c d % Must return a column vector Xdot = zeros(2,1); % dx/dt=Xdot(1), dy/dt=Xdot(2) Xdot(1) = a*X(1)-b*X(1)*X(2); Xdot(2) = c*X(1)*X(2)-d*X(2); pprey.m

48. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams48 at t=0, x=20 y=19.25

49. CEE162/262c Lecture 2: Nonlinear ODEs and phase diagrams49 Nonlinear Linear