Computational Physics 5/18/2010

1. Computational Physics5/18/2010 黃信健

2. 10 Chaotic Oscillations 10.1 The Oscillator 10.2 A Forced Nonlinear Oscillator 10.3 The Duffing equation 10.4 The Van derPol Equation 10.5 Lorentz and RÖssler Systems

3. 10 Chaotic Oscillations 10.1 A Forced Nonlinear Oscillator • A completely general spring not necessarily elastic/linear

4. 10.1.1 Theory, Physics: Newton’s Laws The equation of motion:

5. 10.1.2 Damped Oscillations

6. 10.1.3 X(t) for Damped Oscillations

7. 10.1.4 Anharmonic Oscillations • Nonlinear Differential Equations for realistic physical systems

8. 10.1.5 Nonlinear Oscillator

10. The driven, damped simple pendulum 10.2 Forced Oscillators

11. 10.2.1 Driven Pendulum

12. 10.2.2 Bifurcation in Forced Oscillators

13. The forced spring equation >0, >0: hard spring <0: soft spring =0: nonharmonic =-1: inverted 10.3 The Duffing equation

14. 10.3.1The Period 2 Case Period 2solution: the pattern repeats after 2 os.

15. 10.3.2 The Chaotic Case A chaotic solution

16. 10.3.3 Sensitive to IC F=0.325 F=0.40

17. 10.4 An electronic oscillator • The Van derPol Equation • X > 1: damping, X < 1: - damping • Limit circle (not X = 1!) • Self-excited oscillations

18. Use competition.f90 X = (/0.0,3.5,0.0/), X = (/0.0,1.5,0.0/) 10.4.1 Limit Cycle

19. The Lorentz equation  = 10, b = 8/3, r: bifurcation parameter 10.5.1 Lorentz and RÖssler Systems

20. r = 28, x(0) = 2, y(0) = 5, z(0) = 5 x(t) 10.5.2 The gossamer wings of a butterfly

21. A simple artificial 3D system a = 0.2, b = 0.2, c= 5.7, x(0) = -1, y(0) = 0, z(0) = 0 10.5.3 The RÖssler System

22. 10.5.4 The RÖsslerAttractor

23. Coordinate transformation in 3D Graphics

24. 眼睛座標和顯示座標

25. Transformation Matrix

26. OsLorentz.f90

27. Ikeda - Laser x' = a + b ( x cos z - y sin z) y' = b (x sin z + y cos z) z' = c - d / (1+x2+y2) dt a = 1 b = 0.9 c = 0.4 d = 6 x0 , y0 , z0 = 0 -2 ≤ x , y 𕟄 2

28. Pickover x' =  sin (ay) - z cos( bx) y' = z sin (cx) - cos (dy)z' = e sin (x) a = 2.0 b = 0.5 c= - 1.0 d = - 1.0 e = 2.0x0 , y0 , z0 = 0 -2 ≤ x , y ≤ 2 http://technocosm.org/chaos/attractors.html

29. Tamari Attractor x' =  ( x - a y ) cos( z ) - b y sin ( z )    "x" the  outputy' =  ( x + c y ) sin ( z ) + d y cos( z )   "y" the moneyz' =  e + fz + gatan[ ( 1 - u) y  / ( 1 - i) x ] "z" the pricinga ≡ Inertia = 1.013 b ≡ Productivity = -0.011c ≡ Printing = 0.02 d ≡ Adaptation = 0.96e ≡ Exchange = 0 f ≡ Indexation 0.01g ≡ Expectations  = 1 u ≡ Unemployment = 0.05i  ≡ Interest= 0.05x0 , y0 , z0 = 1 1 ≤ x , y ≤ 4

30. Conical Helix