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Computational Physics 5/18/2010

Computational Physics 5/18/2010. 黃信健. 10 Chaotic Oscillations. 10.1 The Oscillator 10.2 A Forced Nonlinear Oscillator 10.3 The Duffing equation 10.4 The Van der Pol Equation 10.5 Lorentz and R Ö ssler Systems. 10 Chaotic Oscillations. 10.1 A Forced Nonlinear Oscillator.

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Computational Physics 5/18/2010

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  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

  9. The driven, damped simple pendulum 10.2 Forced Oscillators

  10. 10.2.1 Driven Pendulum

  11. 10.2.2 Bifurcation in Forced Oscillators

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

  13. 10.3.1The Period 2 Case Period 2solution: the pattern repeats after 2 os.

  14. 10.3.2 The Chaotic Case A chaotic solution

  15. 10.3.3 Sensitive to IC F=0.325 F=0.40

  16. 10.4 An electronic oscillator • The Van derPol Equation • X > 1: damping, X < 1: - damping • Limit circle (not X = 1!) • Self-excited oscillations

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

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

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

  20. 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

  21. 10.5.4 The RÖsslerAttractor

  22. Coordinate transformation in 3D Graphics

  23. 眼睛座標和顯示座標

  24. Transformation Matrix

  25. OsLorentz.f90

  26. 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

  27. 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

  28. 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

  29. Conical Helix   

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