Nonlinear Mechanics and Chaos

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Nonlinear Mechanics and Chaos
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• Dr. Jen-Hao Yeh
• Prof. Anlage

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This is a brief introduction to the ideas and
concepts of nonlinear mechanics, and a
discussion of various quantitative methods for
analyzing such problems We will focus on the
driven damped pendulum (DDP)
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Linear Nonlinear
easy hard
special general
analytical numerical
Superposition principle
Chaos
 
 
 
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Nonlinear
Chaotic
Linear
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Driven, Damped Pendulum (DDP)
 
 
We expect something interesting to happen as g ?
1, i.e. the driving force becomes comparable to
the weight
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A Route to Chaos
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NDSolve in Mathematica
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Driven, Damped Pendulum (DDP)
For all following plots
Period 2p/w 1
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Small Oscillations of the Driven, Damped Pendulum
g ltlt 1 will give small oscillations
Linear
g 0.2
f(t)
t
After the initial transient dies out, the
solution looks like
Periodic attractor
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Small Oscillations of the Driven, Damped Pendulum
g ltlt 1 will give small oscillations

• The motion approaches a unique periodic attractor
  
• independent of initial conditions
• The motion is sinusoidal with the same frequency
  as the drive

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Moderate Oscillations of the Driven, Damped
Pendulum
g lt 1 and the nonlinearity becomes significant
Try
This solution gives from the f3 term
Since there is no cos(3wt) on the RHS, it must be
that all develop a cos(3wt) time dependence.
Hence we expect
We expect to see a third harmonic as the driving
force grows
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Harmonics

•  

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Moderate Driving The Nonlinearity Distorts the
cos(wt)
cos(wt)
f(t)
f(t)
g 0.9
t
t
-cos(3wt)
The motion is periodic, but
The third harmonic distorts the simple
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Even Stronger Driving Complicated Transients
then Periodic!
After a wild initial transient, the motion
becomes periodic
f(t)
g 1.06
t
After careful analysis of the long-term motion,
it is found to be periodic with the same period
as the driving force
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Slightly Stronger Driving Period Doubling
After a wilder initial transient, the motion
becomes periodic, but period 2!
f(t)
g 1.073
t
The long-term motion is TWICE the period of the
driving force!
A SUB-Harmonic has appeared
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Harmonics and Subharmonics
 
 
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Slightly Stronger Driving Period 3
The period-2 behavior still has a strong period-1
component Increase the driving force slightly and
we have a very strong period-3 component
g 1.077
f(t)
Period 3
t
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Multiple Attractors
The linear oscillator has a single attractor for
a given set of initial conditions
For the drive damped pendulum Different
initial conditions result in different long-term
behavior (attractors)
g 1.077
f(t)
Period 3
t
Period 2
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Period Doubling Cascade
f(t)
f(t)
Early-time motion
g 1.06
t
Close-up of steady-state motion
Period 1
t
g 1.078
Period 2
g 1.081
Period 4
g 1.0826
Period 8
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Period Doubling Cascade
g 1.06
g 1.078
g 1.081
g 1.0826
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Bifurcation Points in the Period Doubling
Cascade
Driven Damped Pendulum
n period gn interval
(gn1-gn)
1 1 ? 2 1.0663 0.0130 2
2 ? 4 1.0793 0.0028 3
4 ? 8 1.0821 0.0006 4
8 ? 16 1.0827
The spacing between consecutive bifurcation
points grows smaller at a steady rate
? as n ? 8
d 4.6692016 is called the Feigenbaum number
The limiting value as n ? 8 is gc 1.0829.
Beyond that is chaos!
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Period Doubling Cascade
Period doubling continues in a sequence of
ever-closer values of g
Such period-doubling cascades are seen in many
nonlinear systems Their form is essentially the
same in all systems it is universal
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Period infinity
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Chaos!
g 1.105
f(t)
t
The pendulum is trying to oscillate at the
driving frequency, but the 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 conditions Does their
motion converge to the same attractor? Does it
diverge quickly?
 
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Convergence of Trajectories in Linear Motion
Take the logarithm of Df(t) to magnify small
differences.
Plotting log10Df(t) vs. t should be a
straight line of slope b, plus some wiggles from
the lncos(w1t d1) term
Note that log10x log10e lnx
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Convergence of Trajectories in Linear Motion
Log10Df(t)
t
g 0.1
Df(0) 0.1 Radians
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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Convergence of Trajectories in Period-2 Motion
Log10Df(t)
t
g 1.07
Df(0) 0.1 Radians
The trajectories converge more slowly, but still
converge
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Divergence of Trajectories in Chaotic Motion
g 1.105
Log10Df(t)
Df(0) 0.0001 Radians
t
If the motion remains bounded, as it does in this
case, then Df can never exceed 2p. Hence this
plot will saturate
The trajectories diverge, even when very close
initially Df(16) p, so there is essentially
complete loss of correlation between the pendulums
Extreme Sensitivity to Initial Conditions Practica
lly impossible to predict the motion
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The Lyapunov Exponent
l Lyapunov exponent
l lt 0 periodic motion in the long term
l gt 0 chaotic motion
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Linear Nonlinear Nonlinear
Linear Chaos
Drive period Harmonics, Subharmonics, Period-doubling Nonperiodic, Extreme sensitivity
l lt 0 l lt 0 l gt 0
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What Happens if we Increase the Driving Force
Further? Does the chaos become more intense?
Df(0) 0.001 Radians
g 1.13
f(t)
Log10Df(t)
t
t
Period 3 motion re-appears!
With increasing g the motion alternates between
chaotic and periodic
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What Happens if we Increase the Driving Force
Further? Does the chaos re-appear?
Df(0) 0.001 Radians
g 1.503
f(t)
Log10Df(t)
t
t
Chaotic motion re-appears!
This is a kind of rolling chaotic motion
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Divergence of Two Nearby Initial Conditions for
Rolling Chaotic Motion
g 1.503
f(t)
t
Df(0) 0.001 Radians
Chaotic motion is always associated with extreme
sensitivity to initial conditions
Periodic and chaotic motion occur in narrow
intervals of g
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Bifurcation Diagram
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Period Doubling Cascade
Period doubling continues in a sequence of
ever-closer values of g
Such period-doubling cascades are seen in many
nonlinear systems Their form is essentially the
same in all systems it is universal
Sub-harmonic frequency spectrum
Driven Diode experiment
F0 cos(wt)
w/2
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Period Doubling Cascade
Period doubling continues in a sequence of
ever-closer values of g
Such period-doubling cascades are seen in many
nonlinear systems Their form is essentially the
same in all systems it is universal
A period doubling cascade in convection of
mercury in a small convection cell. The plots
show the temperature at one fixed point in the
cell as a function of time, for four successively
larger temperature gradients as given by the
parameter R/Rc
Fig. 12.9, Taylor
The Brain-behaviour Continuum The Subtle
Transition Between Sanity and Insanity  By Jose
Luis. Perez Velazquez
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Bifurcation Diagram
Used to visualize the behavior as a function of
driving amplitude g

• Choose a value of g
• Solve for f(t), and plot a periodic sampling of
  the function
• t0 chosen at a time after the attractor behavior
  has been achieved
• Move on to the next value of g

1.0793
1.0663
Fig. 12.17
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Construction of the Bifurcation Diagram
f(t)
g 1.06
Period 1
t
g 1.078
Period 2
g 1.081
Period 4
Period 6 window
g 1.0826
Period 8
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The Rolling Motion Renders the Bifurcation
Diagram Useless
g 1.503
f(t)
t
Df(0) 0.001 Radians
As an alternative, plot
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Bifurcation Diagram Over a Broad Range of g
Period-1 followed by period doubling bifurcation
Previous diagram range
Mostly chaos
Mostly chaos
Mostly chaos
Period-3
Rolling Motion (next slide)
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Period-1 Rolling Motion at g 1.4
g 1.4
t
f(t)
t
Even though the pendulum is rolling, is
periodic
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An Alternative View State Space Trajectory
Plot vs. with time as a parameter
f(t)
t
g 0.6
periodic attractor
start
First 20 cycles
Cycles 5 -20
Fig. 12.20, 12.21
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An Alternative View State Space Trajectory
Plot vs. with time as a parameter
g 0.6
start
periodic attractor
Cycles 5 -20
First 20 cycles
The periodic attractor , is an
ellipse
The state space point moves clockwise on the orbit
Fig. 12.22
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State Space Trajectory for Period Doubling Cascade
g 1.078
g 1.081
Period-2
Period-4
Plotting cycles 20 to 60
Fig. 12.23
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State Space Trajectory for Chaos
g 1.105
Cycles 14 - 21
Cycles 14 - 94
The orbit has not repeated itself
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State Space Trajectory for Chaos

• 1.5
• b w0/8

Chaotic rolling motion Mapped into the interval
p lt f lt p
Cycles 10 200
This plot is still quite messy. Theres 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

• Solve for f(t), and construct the state-space
  orbit
• Plot a periodic sampling of the orbit
• with t0 chosen after the attractor behavior has
  been achieved

• 1.5
• b w0/8

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 more elegant way
to represent chaotic motion
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The Superconducting Josephson Junction as a
Driven Damped Pendulum
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1
I
(Tunnel barrier)
The Josephson Equations
f f2-f1 phase difference of SC wave-function
across the junction
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Radio Frequency (RF) Superconducting Quantum
Interference Devices (SQUIDs)
Flux Quantization in the loop
Ic
F
I(t)
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Single rf-SQUID
D. Zhang, et al., Phys. Rev. X (in press),
arXiv1504.08301
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D. Zhang, et al., Phys. Rev. X (in press),
arXiv1504.08301
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THz Emission from the Intrinsic Josephson
Effect A classic problem in nonlinear physics
 
 
F0 h/2e 2.07 x 10-15 Tm2
DC voltage on junction creates an oscillating
f(t), which in turn creates an AC current
that radiates
L. Ozyuzer, et al., Science 318, 1291 (2007)
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
emission Emission enhanced near cavity mode
resonances ? requires non-uniform current
injection, assisted by inhom. heating, p-phase
kinks, crystal defects
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Chaos in Newtonian Billiards
Imagine a point-particle trapped in a 2D
enclosure and making elastic collisions with the
walls
Describe the successive wall-collisions with a
mapping function

• The Chaos arises due to the shape of the
  boundaries enclosing the system.

Computer animation of extreme sensitivity to
initial conditions for the stadium billiard