Computational Physics 5/18/2010
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Computational Physics 5/18/2010 黃黃黃
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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. - PowerPoint PPT Presentation
Transcript of Computational Physics 5/18/2010
Computational Physics 5/18/2010
黃信健
10 Chaotic Oscillations10.1 The Oscillator10.2 A Forced Nonlinear
Oscillator10.3 The Duffing equation10.4 The Van der Pol Equation10.5 Lorentz and RÖssler
Systems
10 Chaotic Oscillations10.1 A Forced Nonlinear Oscillator
A completely general spring not necessarily elastic/linear
10.1.1 Theory, Physics: Newton’s Laws
The equation of motion:
2
2
k ),()( dtxdmtxFxF ext
10.1.2 Damped Oscillations F v f bxmx bx kx b m
x x x k m
x Ae
Ar e A re A e
r r
r r
d d
rt
rt rt rt
/
= /
Assume
02
0 2
2 0
2 0
2 0
02
202
202
12
02
22
02
10.1.3 X(t) for Damped Oscillations
10.1.4 Anharmonic Oscillations
Nonlinear Differential Equations for realistic physical systems
1 ),321(
21)( 2 xxkxxV
10.1.5 Nonlinear OscillatorV x
pk x
F x dVdx
k x xx
p
p
( )
( )
1
1k
3p 2p
1p
)(,
3121)(
23
2
kxkxxxk
xF
kx
kxkx
xV
10.2 Forced Oscillators The driven, damped simple
pendulum sin cos( ) 2 0
2 F t
0 0 022t b F, / , / F
( ) , ( ) ( ) sin cos
ab F
10.2.1 Driven Pendulum
0
0
2 sin cos( )1, 2 / 3, 0.25,
(0) 0.09, (0) 0.01.0, 1.06, 1.25, 1.06 1.25
F t
F
vs. and vs. t
10.2.2 Bifurcation in Forced Oscillators
10.3 The Duffing equation The forced spring equation
>0, >0: hard spring <0: soft spring =0: nonharmonic =-1: inverted
cos( )X X X X F t 2 3
cos5.0
1 ,3 Fxxyy
yx
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=0.325F=0.40
( ) X X X X 1 02
10.4 An electronic oscillator The Van der Pol Equation
X > 1: damping, X < 1: - damping
Limit circle (not X = 1!) Self-excited oscillations
Use competition.f90X = (/0.0,3.5,0.0/), X =
(/0.0,1.5,0.0/)2
2
(1 ) 0
, (1 )5 (0) (0) 0.1 stepsize 0.01, 0..150.10 (0) (0) 1.5 stepsize 0.05, 0..30
X X X X
x y y x y xx y tx y t
10.4.1 Limit Cycle
10.5.1 Lorentz and RÖssler Systems
The Lorentz equation
= 10, b = 8/3, r: bifurcation parameter
10.5.2 The gossamer wings of a butterfly
1.r = 28, x(0) = 2, y(0) = 5, z(0) = 5
2.x(t)
10.5.3 The RÖssler System A simple artificial 3D system
a = 0.2, b = 0.2, c= 5.7, x(0) = -1, y(0) = 0, z(0) = 0
)(
)(
cxzbzayxyzyx
10.5.4 The RÖssler Attractor
Coordinate transformation in 3D Graphics
眼睛座標和顯示座標
Transformation Matrix
OsLorentz.f90
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) dta = 1 b = 0.9 c = 0.4 d = 6x0 , 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 ≤ 2http://technocosm.org/chaos/attractors.html
Tamari Attractorx' = ( 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 + g atan [ ( 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
Conical Helix
