12 nonlinear mechanics and chaos jen-hao yeh. linearnonlinear easyhard specialgeneral...
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12Nonlinear Mechanics and Chaos
Jen-Hao Yeh
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Linear Nonlineareasy hard
special general
analytical numerical
Superposition principle
Chaos
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Linear
Nonlinear
Chaotic
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)cos()sin(2 20
20 t
Driven, Damped Pendulum (DDP)
We expect something interesting to happen as → 1, i.e. the driving force becomes comparable to the weight
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A Route to Chaos
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Fig12p2 NDSolve ''t 2 't 0 ^2 Sint 0 ^2 Cos t, 0 0, '0 0, , t, 0, 6 InterpolatingFunction 0., 6., PlotEvaluatet . Fig12p2, t, 0, 6, Ticks 1, 2, 3, 4, 5, 6, 0.3, 0.3, PlotRange All, AxesStyle Thick, PlotStyle Thick, LabelStyle Bold, Medium
1 2 3 4 5 6
0 .3
0 .3
NDSolve in Mathematica
Fig12p2 NDSolve ''t 2 't 0 ^2 Sint 0 ^2 Cos t, 0 0, '0 0, , t, 0, 6 InterpolatingFunction 0., 6., PlotEvaluatet . Fig12p2, t, 0, 6, Ticks 1, 2, 3, 4, 5, 6, 0.3, 0.3, PlotRange All, AxesStyle Thick, PlotStyle Thick, LabelStyle Bold, Medium
1 2 3 4 5 6
0 . 3
0 . 3
Fig12p2 NDSolve ''t 2 't 0 ^2 Sint 0 ^2 Cos t, 0 0, '0 0, , t, 0, 6 InterpolatingFunction 0., 6., PlotEvaluate t . Fig12p2, t, 0, 6, Ticks 1, 2, 3, 4, 5, 6, 0.3, 0.3, PlotRange All, AxesStyle Thick, PlotStyle Thick, LabelStyle Bold, Medium
1 2 3 4 5 6
0 . 3
0 . 3
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)cos()sin(2 20
20 t
Driven, Damped Pendulum (DDP)
5.10
2Period = 2 = 1
0)0()0( 4/0
For all following plots:
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Small Oscillations of the Driven, Damped Pendulum
<< 1 will give small oscillations
)cos(2 20
20 t
1 2 3 4 5 6
0 .3
0 .3(t)
t
= 0.2
)cos()( tAt
After the initial transient dies out, the solution looks like
Periodic “attractor”
Linear
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Small Oscillations of the Driven, Damped Pendulum
<< 1 will give small oscillations
1) The motion approaches a unique periodic attractor independent of initial conditions
2) The motion is sinusoidal with the same frequency as the drive
)cos()( tAt
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Moderate Oscillations of the Driven, Damped Pendulum
< 1 and the nonlinearity becomes significant…
)cos(6
12 2
032
0 t
)cos()( tAt
This solution gives from the 3 term: xxx cos33cos4
1cos3
Try
Since there is no cos(3t) on the RHS, it must be thatall develop a cos(3t) time dependence. Hence we expect:
,,
)(3cos)cos()( tBtAtAB
We expect to see a third harmonic as the driving force grows
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Harmonics
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5 6
2
2
1 2 3 4 5 6
2
2
= 0.9(t)
t
(t)
t
cos(t)
-cos(3t)
Moderate Driving: The Nonlinearity Distorts the cos(t)
The third harmonic distorts the simple )cos()( tAtThe motion is periodic, but …
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Even Stronger Driving: Complicated Transients – then Periodic!
3 6 9 1 2 1 5
2
4
= 1.06(t)
t
After a wild initial transient, the motion becomes periodic
After careful analysis of the long-term motion, it is found to be periodicwith the same period as the driving force
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5 1 0 1 5 2 0 2 5 3 0 2
2
4
Slightly Stronger Driving: Period Doubling
= 1.073
(t)
t
After a wilder initial transient, the motion becomes periodic, but period 2!
The long-term motion is TWICE the period of the driving force!
2 2 2 4 2 6 2 8 3 0
8 .5
8 .
A SUB-Harmonic has appeared
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Harmonics and SubharmonicsSubharmonic
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Slightly Stronger Driving: Period 3
2 4 6 8 1 0 1 2 1 4
2
4
(t)
t
= 1.077
Period 3
The period-2 behavior still has a strong period-1 componentIncrease the driving force slightly and we have a very strong period-3 component
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Multiple Attractors
For the drive damped pendulum: Different initial conditions result in different long-term behavior (attractors)
2 4 6 8 1 0 1 2 1 4
2
4
(t)
t
= 1.077
The linear oscillator has a single attractor for a given set of initial conditions
0)0()0(
0)0(,2/)0(
Period 3
Period 2
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Period Doubling Cascade
2 4 6 8 1 0
2
2
3 0 3 2 3 4 3 6 3 8 4 0 2 .5
2
2 4 6 8 1 0
2
2
3 0 3 2 3 4 3 6 3 8 4 0 2 .5
2
2 4 6 8 1 0
2
2
3 0 3 2 3 4 3 6 3 8 4 0 2 .5
2
2 4 6 8 1 0
2
2
3 0 3 2 3 4 3 6 3 8 4 0 2 .5
2
= 1.06
= 1.078
= 1.081
= 1.0826
0)0(
2/)0(
Period 1
Period 2
Period 4
Period 8
Early-time motion
Close-up of steady-statemotion
(t)
t
(t)
t
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Period Doubling Cascade
= 1.06
= 1.078
= 1.081
= 1.0826
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n period n interval (n+1-n)
1 1 → 2 0.0130
2 2 → 4 0.0028
3 4 → 8 0.0006
4 8 → 16
‘Bifurcation Points’ in the Period Doubling Cascade
Driven Damped Pendulum0)0(
2/)0(
The spacing between consecutive bifurcation points grows smaller at a steady rate:
)(1
)( 11 nnnn
= 4.6692016 is called the Feigenbaum number
‘≈’ → ‘=’ as n → ∞
The limiting value as n → ∞ is c = 1.0829. Beyond that is … chaos!
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Period Doubling Cascade
Period doubling continues in a sequence of ever-closer values of
Such period-doubling cascades are seen in many nonlinear systemsTheir form is essentially the same in all systems – it is “universal”
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Period infinity
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5 1 0 1 5 2 0 2 5 3 0
(t)
t
= 1.105
Chaos!
The pendulum is “trying” to oscillate at the driving frequency, butthe motion remains erratic for all time
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Chaos
• Nonperiodic• Sensitivity to initial conditions
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Sensitivity of the Motion to Initial Conditions
Start the motion of two identical pendulums with slightly different initial conditionsDoes their motion converge to the same attractor?Does it diverge quickly?
)()()( 12 ttt
)(),( 21 tt Two pendulums are given different initial conditions
Follow their evolution and calculate
trtr eCeCtAt 2121)cos()( For a linear oscillator
Long-termattractor
Transientbehavior
12,1 ir
)cos()( 11 tDet t
The initial conditions affect the transient behavior, the long-term attractor is the same
Hence
Thus the trajectories will converge after the transients die out
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Convergence of Trajectories in Linear Motion
)cos()( 11 tDet t
|)cos(|)ln(|])(ln[| 11 ttDt
Take the logarithm to magnify small differences..
Plotting log10[|(t)|] vs. t should be a straight line of slope –plus some wiggles from the cos(1t – 1) term
Note that log10[x] = log10[e] ln[x]
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2 4 6 8 1 0
1 2
1 0
8
6
4
2
Log10[|(t)|]t
= 0.1(0) = 0.1 Radians
Convergence of Trajectories in Linear Motion
The trajectories converge quickly for the small driving force (~ linear) case
This shows that the linear oscillator is essentially insensitive to its initial conditions!
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5 1 0 1 5 2 0 2 5 3 0 3 5 4 0
8
6
4
2
Log10[|(t)|]t
= 1.07(0) = 0.1 Radians
Convergence of Trajectories in Period-2 Motion
The trajectories converge more slowly, but still converge
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2 4 6 8 1 0 1 2 1 4 1 6
6
5
4
3
2
1
1
Log10[|(t)|]
t
= 1.105(0) = 0.0001 Radians
Divergence of Trajectories in Chaotic Motion
The trajectories diverge, even when very close initially(16) ~ , so there is essentially complete loss of correlation between the pendulums
If the motion remains bounded, as it doesin this case, then can never exceed 2.Hence this plot will saturate
Extreme Sensitivity to Initial ConditionsPractically impossible to predict the motion
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The Lyapunov Exponent
tKet ~)(
= Lyapunov exponent
< 0: periodic motion in the long term
> 0: chaotic motion
0K
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Linear NonlinearChaos
Drive period Harmonics,Subharmonics,
Period-doubling
Nonperiodic,Extreme sensitivity
< 0 < 0 > 0
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t = 1.13
(0) = 0.001 Radians
What Happens if we Increase the Driving Force Further?Does the chaos become more intense?
5 1 0 1 5
2
(t)
t
5 1 0 1 5 2 0
9
6
3
Log10[|(t)|]
Period 3 motion re-appears!
With increasing the motion alternates between chaotic and periodic
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t
= 1.503
(0) = 0.001 Radians
What Happens if we Increase the Driving Force Further?Does the chaos re-appear?
(t) Log10[|(t)|]
Chaotic motion re-appears!
5 1 0 1 5 2 0 2 5
2 0
1 0
t
5 1 0 1 5
4
3
2
1
1
This is a kind of ‘rolling’ chaotic motion
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5 1 0 1 5 2 0 2 5
2 0
1 0
= 1.503(t) t
(0) = 0.001 Radians
Divergence of Two Nearby Initial Conditions for Rolling Chaotic Motion
Chaotic motion is always associated with extreme sensitivity to initial conditions
Periodic and chaotic motion occur in narrow intervals of
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Bifurcation Diagram
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Period Doubling Cascade
Period doubling continues in a sequence of ever-closer values of
Such period-doubling cascades are seen in many nonlinear systemsTheir form is essentially the same in all systems – it is “universal”
Sub-harmonic frequency spectrumDriven Diode experiment
F0 cos(t)
/2
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Period Doubling Cascade
Period doubling continues in a sequence of ever-closer values of
Such period-doubling cascades are seen in many nonlinear systemsTheir form is essentially the same in all systems – it is “universal”
The Brain-behaviour Continuum: The Subtle Transition Between Sanity and Insanity By Jose Luis. Perez Velazquez
Fig. 12.9, Taylor
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Bifurcation Diagram
Used to visualize the behavior as a function of driving amplitude
1) Choose a value of 2) Solve for (t), and plot a periodic sampling of the function
t0 chosen so that the attractor behavior is achieved3) Move on to the next value of
),...4(),3(),2(),1(),( 00000 ttttt
Fig. 12.171.0663 1.07930)0(
2/)0(
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3 0 3 2 3 4 3 6 3 8 4 0 2 .5
2
3 0 3 2 3 4 3 6 3 8 4 0 2 .5
2
3 0 3 2 3 4 3 6 3 8 4 0 2 .5
2
3 0 3 2 3 4 3 6 3 8 4 0 2 .5
2
Period 1
Period 2
Period 4
Period 8
(t)
t
= 1.06
= 1.078
= 1.081
= 1.0826
Construction of the Bifurcation Diagram
Period 6window
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5 1 0 1 5 2 0 2 5
2 0
1 0
= 1.503(t) t
(0) = 0.001 Radians
The Rolling Motion Renders the Bifurcation Diagram Useless
As an alternative, plot )(t
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Pre
viou
s d
iagr
am
ran
ge
Mos
tly
chao
s
Per
iod
-3
Mos
tly
chao
s
Per
iod
-1 f
ollo
wed
by
per
iod
dou
bli
ng
bif
urc
atio
n
Mos
tly
chao
s
Bifurcation Diagram Over a Broad Range of )(t
Rolling Motion(next slide)
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Period-1 Rolling Motion at = 1.4
5 1 0
2 0
1 0 5 1 0
2 0
1 0
1 0
)(t = 1.4
t
t
Even though the pendulum is “rolling”, is periodic)(t
(t)
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An Alternative View: State Space Trajectory
)(tPlot vs. with time as a parameter)(t
1 2 3 4 5 6
2
4
2
4
4
4
4
4
4
4
4
Fig. 12.20, 12.21
(t)
t
= 0.6
)(t
)(t )(t
)(t
First 20 cycles Cycles 5 -20
start
periodicattractor
0)0(
2/)0(
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4
4
4
4
4
4
4
4
An Alternative View: State Space Trajectory
)(tPlot vs. with time as a parameter)(t
Fig. 12.22
= 0.6
)(t
)(t)(t
)(t
First 20 cycles Cycles 5 -20
start
The state space point moves clockwise on the orbit
)cos()( tAt)sin()( tAt
The periodic attractor:[ , ] is an ellipse)(t )(t
periodicattractor
0)0(
0)0(
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State Space Trajectory for Period Doubling Cascade
2
2
1 0
1 0
2
2
1 0
1 0
= 1.078 = 1.081
)(t
)(t
)(t
)(t
Period-2 Period-4Plotting cycles 20 to 60
Fig. 12.23
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1 1
1 0
1 0
State Space Trajectory for Chaos
= 1.105
)(t
)(t
1 1
1 0
1 0
)(t
)(t
Cycles 14 - 21
Cycles 14 - 94
The orbit has not repeated itself…
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State Space Trajectory for Chaos
= 1.5 = 0/8
Cycles 10 – 200
Chaotic rolling motionMapped into the interval – < <
This plot is still quite messy. There’s got to be a better way to visualize the motion …
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The Poincaré Section
Similar to the bifurcation diagram, look at a sub-set of the data
1) Solve for (t), and construct the state-space orbit2) Plot a periodic sampling of the orbit
with t0 chosen so that the attractor behavior is achieved
,...)2(),2(,)1(),1(,)(),( 000000 tttttt
= 1.5 = 0/8 Samples 10 – 60,000
Enlarged on the next slide
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The Poincaré Section is a Fractal
The Poincaré section is a much moreelegant way to represent chaoticmotion
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The Superconducting Josephson Junction as a Driven Damped Pendulum
= phase difference of SC wave-function across the junction
111
ie 222
ie
I
1 2
(Tunnel barrier)
The JosephsonEquations
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Radio Frequency (RF) Superconducting Quantum Interference Devices (SQUIDs)
L LJJ R C
FluxQuantization
in the loop
I(t)insulator ||eiq
||eiq
1
2 Ic
0 ninducedapplied
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THz Emission from the Intrinsic Josephson EffectA classic problem in nonlinear physics
DC voltage on junction createsan oscillating (t), which in turncreates an AC current thatradiates
0 = h/2e = 2.07 x 10-15 Tm2
Best emission is seen when the crystal is partially heated above Tc!Results are extremely sensitive to details (number of layers, edge properties, type of material, width of mesa, etc.)Many competing states do not show emissionEmission enhanced near cavity mode resonances → requires non-uniform current injection, assisted by inhom. heating, -phase kinks, crystal defects
)V(Vh
ef JJ THz/mV 483.0
2
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),(
),(
1
1
nnn
nnn
sg
sfs
qqq
Linear Maps for “Integrable” systems !!
),(
),(
1
1
nnn
nnn
sg
sfs
qqq
Non-Linear Maps for “Chaotic” systems !!
0
0s
1s
0q
1q
• The “Chaos” arises due to the shape of the boundaries enclosing the system.
Chaos in Newtonian BilliardsImagine a point-particle trapped in a 2D enclosure and making
elastic collisions with the walls
Describe the successive wall-collisionswith a “mapping function”
Computer animation of extreme sensitivity to initial conditions for the stadium billiard