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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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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
10 Chaotic Oscillations 10.1 A Forced Nonlinear Oscillator • A completely general spring not necessarily elastic/linear
10.1.1 Theory, Physics: Newton’s Laws The equation of motion:
10.1.4 Anharmonic Oscillations • Nonlinear Differential Equations for realistic physical systems
The driven, damped simple pendulum 10.2 Forced Oscillators
The forced spring equation >0, >0: hard spring <0: soft spring =0: nonharmonic =-1: inverted 10.3 The Duffing equation
10.3.1The Period 2 Case Period 2solution: the pattern repeats after 2 os.
10.3.2 The Chaotic Case A chaotic solution
10.3.3 Sensitive to IC F=0.325 F=0.40
10.4 An electronic oscillator • The Van derPol Equation • X > 1: damping, X < 1: - damping • Limit circle (not X = 1!) • Self-excited oscillations
Use competition.f90 X = (/0.0,3.5,0.0/), X = (/0.0,1.5,0.0/) 10.4.1 Limit Cycle
The Lorentz equation = 10, b = 8/3, r: bifurcation parameter 10.5.1 Lorentz and RÖssler Systems
r = 28, x(0) = 2, y(0) = 5, z(0) = 5 x(t) 10.5.2 The gossamer wings of a butterfly
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
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
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
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