12
Nonlinear Mechanics and Chaos
Jen-Hao Yeh
Linear
Nonlinear
easy
hard
Uncommon
common
analytical
numerical
Superposition
principle
Chaos
Nonlinear
Chaotic
Linear
Driven, Damped Pendulum (DDP)
2
2



  2   sin( )   cos(t )
0
2  b / m
02  g / L
0
  F0 / mg
We expect something interesting to happen as  → 1,
i.e. the driving force becomes comparable to the weight
A Route to Chaos
NDSolve in Mathematica
Fig12p2
NDSolve
'' t
2
InterpolatingFunction
Plot Evaluate
t
' t
0 ^2
0., 6.
Sin
t
0 ^2
Cos
t ,
0
0, ' 0
,
. Fig12p2 , t, 0, 6 , Ticks
1, 2, 3, 4, 5, 6 ,
0.3, 0.3
0.3
1
2
3
4
0 , , t, 0, 6
5
6
0.3
Download Mathematica for free:
https://terpware.umd.edu/Windows/Package/2032
, PlotRange
All, AxesS
Driven, Damped Pendulum (DDP)
2
2



  2   sin( )   cos(t )
0
For all following plots:
  2
Period = 2/ = 1
0  1.5
  0 / 4
 (0)  (0)  0
0
Small Oscillations of the Driven, Damped Pendulum
 << 1 will give small oscillations
2
2
0
0
  2      cos(t )
 = 0.2
(t)
0.3
1
2
3
4
5
6
t
0.3
Calculated with Mathematica NDSolve
After the initial transient, the
solution looks like
 (t )  A cos(t   )
Periodic “attractor”
Fig. 12.2
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
 (t )  A cos(t   )
Moderate Driving Force Oscillations of the Driven, Damped Pendulum
 < 1 and the nonlinearity becomes significant…
1 3
2
2



  2  0      0 cos(t )
6 

Try
 (t )  A cos(t   )
This solution gives from the
3
3
term: cos x 
1
cos 3x  3 cos x 
4
 
Since there is no cos(3t) on the RHS, it must be that  ,  , 
all develop a cos(3t) time dependence. Hence we expect:
 (t )  A cos(t   )  B cos3(t   )
B  A
We expect to see a third harmonic as the driving force grows
Harmonics
Stronger Driving: The Nonlinearity Distorts the cos(t)
(t)
(t)
 = 0.9
cos(t)
2
2
1
2
3
4
5
6
t
t
5
6
-cos(3t)
2
2
The motion is periodic, but …
The third harmonic distorts the simple  (t ) 
A cos(t   )
Fig. 12.3
Even Stronger Driving: Complicated Transients – then Periodic!
After a wild initial transient, the motion becomes periodic
(t)
 = 1.06
4
2
t
3
6
9
12
15
After careful analysis of the long-term motion, it is found to be periodic
with the same period as the driving force
Fig. 12.4
Slightly Stronger Driving: Period Doubling
After a wilder initial transient, the motion becomes periodic, but period 2!
(t)
4
 = 1.073
2
t
5
2
10
15
20
25
8.
30
8.5
22
24
26
28
30
The long-term motion is TWICE the period of the driving force!
A SUB-Harmonic has appeared
Fig. 12.5
Harmonics
Sub-harmonic
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
 = 1.077
(t)
4
2
Period 3
2
4
6
8
10 12 14
t
Fig. 12.6
Multiple Attractors
The linear oscillator has a single attractor for a given set of initial conditions
For the driven damped pendulum:
Different initial conditions result in different long-term behavior (attractors)
 = 1.077
(t)
 (0)  0, (0)  0
4
Period 3
2
 (0)   / 2, (0)  0
2
4
6
8
10 12 14
t
Period 2
Fig. 12.7
Period Doubling Cascade
(t)
 = 1.06
(t)
t
2
4
6
8
10
2
 = 1.078
(0)  0
2
Close-up of steady-state
motion
Period 1
2.5
2
30
32
34
36
38
40
30
Period 2
32 34 36
38
40
t
2
2
4
6
8
10
2
2.5
2
2
 = 1.081
2
4
6
8
10
2
2.5
2
30
32
Period 4
34 36 38 40
2
 = 1.0826
2
2
 (0)   / 2
Early-time motion
2
4
6
8
10
Period 8
2.5
30
32
34
36
38
40
Fig. 12.8
Period Doubling Cascade
Period doubling continues in a sequence of ever-closer values of 
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(t)
/2
Period Doubling Cascade
 = 1.06
 = 1.078
 = 1.081
 = 1.0826
‘Bifurcation Points’ in the Period Doubling Cascade
 (0)   / 2
Driven Damped Pendulum
(0)  0
n
period
n
1
1→2
1.0663
2
3
4
2→4
4→8
8 → 16
1.0793
1.0821
1.0827
interval (n+1-n)
0.0130
0.0028
0.0006
The spacing between consecutive bifurcation points grows smaller at a steady rate:
( n 1   n ) 
1

( n   n 1 )
‘≈’ → ‘=’ as n → ∞
 = 4.6692016 is called the Feigenbaum number
The limiting value as n → ∞ is c = 1.0829. Beyond that is … chaos!
Period infinity
Chaos!
 = 1.105
(t)
5
10
15
20
25
30
t
The pendulum is “trying” to oscillate at the driving frequency, but
the motion remains erratic for all time
Fig. 12.10
Period Doubling Cascade
Period doubling continues in a sequence of ever-closer values of 
Such period-doubling cascades are seen in many nonlinear systems
Their 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
Chaos
• Nonperiodic
• Sensitivity to initial conditions
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 instead diverge quickly?
Two pendulums 1 (t ), 2 (t ) are given different initial conditions
Follow their evolution and calculate  (t )  2 (t )  1 (t )
For a linear oscillator  (t )  A cos(t   )  C1er1t  C2er2t
Long-term
attractor
Transient
behavior
r1, 2     i1
The initial conditions affect the transient behavior, the long-term attractor is the same
Hence
 (t )  De t cos(1t  1 )
Thus the trajectories will converge after the transients die out
Convergence of Trajectories in Linear Motion
 (t )  De t cos(1t  1 )
Take the logarithm to magnify small differences..
ln[|  (t ) |]  ln(D)  t  | cos(1t  1 ) |
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]
Convergence of Trajectories in Linear Motion
Log10[|(t)|]
2
2
4
4
6
8
10
t
 = 0.1
(0) = 0.1 Radians
6
8
10
12
The trajectories converge quickly for the small driving force (~ linear) case
This shows that the linear oscillator is essentially insensitive to its initial conditions!
Fig. 12.11
Convergence of Trajectories in Period-2 Motion
Log10[|(t)|]
5
2
10
15
20
25
30
35
40
t
 = 1.07
(0) = 0.1 Radians
4
6
8
The trajectories converge more slowly, but still converge
Fig. 12.12
Divergence of Trajectories in Chaotic Motion
 = 1.105
(0) = 0.0001 Radians
Log10[|(t)|]
1
2
4
6
8
10
12
14
16
t
1
2
3
4
5
If the motion remains bounded, as it does
in this case, then  can never exceed 2.
Hence this plot will saturate
6
The trajectories diverge, even when very close initially
(16) ~ , so there is essentially complete loss of correlation between the pendulums
Extreme Sensitivity to Initial Conditions
Fig. 12.13
The Lyapunov Exponent
 (t ) ~ Ke
t
K 0
 = Lyapunov exponent
 < 0: periodic motion in the long term
 > 0: chaotic motion
Chaos
• Nonperiodic
• Sensitivity to initial conditions
Linear
Nonlinear
Chaos
Drive period
Period
doubling
<0
Nonperiodic
<0
>0
What Happens if we Increase the Driving Force Further?
Does the chaos become more intense?
(0) = 0.001 Radians
(t)
 = 1.13
Log10[|(t)|]
5
10
15
20
t
2
3
t
5
10
6
15
9
Period 3 motion re-appears!
Fig. 12.14
What Happens if we Increase the Driving Force Further?
Does the chaos re-appear?
(0) = 0.001 Radians
 = 1.503
(t)
5
10
20
10
15
Log10[|(t)|]
t
20
25
1
1
2
3
4
5
10
15
t
Chaotic motion re-appears!
This is a kind of ‘rolling’ chaotic motion
Fig. 12.15
Divergence of Two Nearby Initial Conditions
for Rolling Chaotic Motion
 = 1.503
(t)
5
10
15
t
20
25
10
(0) = 0.001 Radians
20
Chaotic motion is always associated with extreme sensitivity to initial conditions
Periodic and chaotic motion occur in narrow intervals of 
Fig. 12.16
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 ),  (t0  1),  (t0  2),  (t0  3),  (t0  4),...
t0 chosen so that the attractor behavior is achieved
3) Move on to the next value of 
 (0)   / 2
(0)  0
1.0663
1.0793
Fig. 12.17
Construction of the Bifurcation Diagram
(t)
 = 1.06
2
2.5
 = 1.078
36
38
40
Period 2
32 34 36
38
40
t
2
2.5
 = 1.081
Period 1
30 32 34
30
2
2.5
30
32
Period 4
34 36 38 40
Period 6
window
 = 1.0826
2
Period 8
2.5
30
32
34
36
38
40
The Rolling Motion Renders the Bifurcation Diagram Useless
 = 1.503
(t)
5
10
15
t
20
25
10
(0) = 0.001 Radians
20
As an alternative, plot (t )
Rolling Motion
(next slide)
Mostly chaos
Period-1 followed
by period doubling
bifurcation
Mostly chaos
Period-3
Mostly chaos
Previous diagram
range
(t ) Bifurcation Diagram Over a Broad Range of 
Period-1 Rolling Motion at  = 1.4
 = 1.4
(t )
(t)
5
10
t
10
5
10
10
20
20
Even though the pendulum is “rolling”, (t ) is periodic
10
t
An Alternative View: State Space Trajectory
Plot (t ) vs.  (t ) with time as a parameter
(t)
4
 (0)   / 2
(0)  0
1
2
(t )
t
(t )
 (t )
4
First 20 cycles
6
4
start
4
5
periodic
attractor
4
4
4
 = 0.6
2
2
3
 (t )
4
4
4
Cycles 5 -20
Fig. 12.20, 12.21
An Alternative View: State Space Trajectory
Plot (t ) vs.  (t ) with time as a parameter
 = 0.6
(t )
 (0)  0
(t )
(0)  0
4
4
 (t )
start
4
4
4
 (t )
4
4
4
periodic
attractor
First 20 cycles
The periodic attractor:
[ (t ) ,(t ) ] is an ellipse
Cycles 5 -20
 (t )  A cos(t   )
(t )   A sin(t   )
The state space point moves clockwise on the orbit
Fig. 12.22
State Space Trajectory for Period Doubling Cascade
 = 1.078
 = 1.081
(t )
(t )
10
10
 (t )
2
2
 (t )
2
10
2
10
Period-2
Period-4
Plotting cycles 20 to 60
Fig. 12.23
State Space Trajectory for Chaos
(t )
 = 1.105
Cycles 14 - 21
10
1
1
 (t )
10
(t )
10
1
Cycles 14 - 94
The orbit has not repeated itself…
1
10
 (t )
State Space Trajectory for Chaos
 = 1.5
 = 0/8
Chaotic rolling motion
Mapped into the interval – <  < 
Cycles 10 – 200
This plot is still quite messy. There’s got to be a better way to visualize the motion …
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 orbit
2) Plot a periodic sampling of the orbit
 (t ),(t ),  (t
0
0
0


 1), (t0  1) ,  (t0  2), (t0  2) ,...
with t0 chosen so that the attractor behavior is achieved
 = 1.5
 = 0/8
Samples 10 – 60,000
Enlarged on the next slide
The Poincaré Section is a Fractal
The Poincaré section is a much more
elegant way to represent chaotic
motion
The Superconducting Josephson Junction as a Driven Damped Pendulum
1  1 ei1
2  2 ei2
2
1
I
(Tunnel barrier)
The Josephson
Equations
 21 = phase difference of SC wave-function across the junction
Chaos in Newtonian Billiards
Imagine a point-particle trapped in a 2D enclosure and making
elastic collisions with the walls
s1
1
Describe the successive wall-collisions
with a “mapping function”
sn1  f (sn , n )
0
 n1  g (sn , n )
s0
Linear Maps for “Integrable” systems !!
0
sn1  f (sn , n )
 n1  g (sn , n )
Non-Linear Maps for “Chaotic” systems !!
• 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