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CEE262C Lecture 2: Nonlinear ODEs and Phase Diagrams

CEE262C Lecture 2: Nonlinear ODEs and Phase Diagrams. Overview. Nonlinear chaotic ODEs: the damped nonlinear forced pendulum 2 nd Order damped harmonic oscillator Systems of ODEs Phase diagrams Fixed points Isoclines/Nullclines

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CEE262C Lecture 2: Nonlinear ODEs and Phase Diagrams

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  1. CEE262C Lecture 2: Nonlinear ODEs and Phase Diagrams Overview • Nonlinear chaotic ODEs: the damped nonlinear forced pendulum • 2nd Order damped harmonic oscillator • Systems of ODEs • Phase diagrams • Fixed points • Isoclines/Nullclines References: Dym, Ch 7; Mooney & Swift, Ch 5.2-5.3; Kreyszig, Ch 4 CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  2. Forced pendulum Frictional effect m m g CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  3. Free-body diagram CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  4. Derivation of the governing ODE CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  5. m CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  6. CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  7. Reduce and nondimensionalize! CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  8. CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  9. Governing nondimensional ODE CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  10. Linearize CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  11. The damped harmonic oscillator CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  12. CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  13. CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  14. CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  15. The particular solution CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  16. CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  17. Simulating the nonlinear system pendulum.zip CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  18. Phase plane analysis CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  19. Direction field for a1=0.5 phasedirection.m CEE262C Lecture 2: Nonlinear ODEs and phase diagrams 24

  20. Computing phase lines analytically Solution in phase space Elliptic Integral! CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  21. Analytical Phase Lines for CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  22. Nullclines and fixed points CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  23. Plotting nullclines and fixed points q=0 (no acceleration) increasing friction p=0 (no velocity) Fixed points CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  24. where the point is a fixed point corresponding to 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 CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  25. Using the Taylor series expansion about the fixed point, we have Substitution into the ODEs gives Since the fixed points satisfy CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  26. and , then the perturbations satisfy In vector form, this is given by The Jacobian matrix is given by CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  27. 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 CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  28. complex pair, negative real part. two real negative roots. CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  29. complex pair, positive real part. two real positive roots. pure imaginary. CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  30. Phase plane analysis for the pendulum CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  31. CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  32. Underdamped Critical or overdamped CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  33. Spiral direction CW or CCW? Clockwise c<0 Counter- clockwise c>0 CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  34. Behavior around saddle point CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  35. CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

  36. CEE262C Lecture 2: Nonlinear ODEs and phase diagrams

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