12Nonlinear Mechanics and Chaos

Jen-Hao Yeh

Linear Nonlineareasy hard

Uncommon common

analytical numerical

Superposition principle

Chaos

Linear

Nonlinear

Chaotic

)cos()sin(2 20

20 t

Driven, Damped Pendulum (DDP)

mb /2 mgF /0Lg /20

We expect something interesting to happen as g → 1, i.e. the driving force becomes comparable to the weight

A Route to Chaos

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

Download Mathematica for free:https://terpware.umd.edu/Windows/Package/2032

)cos()sin(2 20

20 t

Driven, Damped Pendulum (DDP)

5.10

2Period = 2 /p w = 1

0)0()0(

4/0

For all following plots:

Small Oscillations of the Driven, Damped Pendulum

g << 1 will give small oscillations

)cos(2 20

20 t

1 2 3 4 5 6

0 .3

0 .3f(t)

t

g = 0.2

Fig. 12.2

Calculated with Mathematica NDSolve

)cos()( tAtAfter the initial transient, the solution looks like

Periodic “attractor”

Small Oscillations of the Driven, Damped Pendulum

g << 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

Moderate Driving Force Oscillations of the Driven, Damped Pendulum

g < 1 and the nonlinearity becomes significant…

)cos(6

12 2

032

0 t

)cos()( tAt

This solution gives from the f3 term: xxx cos33cos4

1cos3

Try

Since there is no cos(3wt) on the RHS, it must be thatall develop a cos(3wt) time dependence. Hence we expect:

,,

)(3cos)cos()( tBtAtAB

We expect to see a third harmonic as the driving force grows

Harmonics

5 6

2

2

Fig. 12.3

1 2 3 4 5 6

2

2

g = 0.9f(t)

t

f(t)

t

cos(wt)

-cos(3wt)

Stronger Driving: The Nonlinearity Distorts the cos(wt)

The third harmonic distorts the simple )cos()( tAtThe motion is periodic, but …

Even Stronger Driving: Complicated Transients – then Periodic!

Fig. 12.4

3 6 9 1 2 1 5

2

4

g = 1.06f(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

5 1 0 1 5 2 0 2 5 3 0 2

2

4

Slightly Stronger Driving: Period Doubling

Fig. 12.5

g = 1.073

f(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

HarmonicsSub-harmonic

Slightly Stronger Driving: Period 3

Fig. 12.6

2 4 6 8 1 0 1 2 1 4

2

4

f(t)

t

g = 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

Multiple Attractors

Fig. 12.7

For the driven damped pendulum: Different initial conditions result in different long-term behavior (attractors)

2 4 6 8 1 0 1 2 1 4

2

4

f(t)

t

g = 1.077

The linear oscillator has a single attractor for a given set of initial conditions

0)0(,0)0(

0)0(,2/)0(

Period 3

Period 2

Period Doubling Cascade

Fig. 12.8

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

g = 1.06

g = 1.078

g = 1.081

g = 1.0826

0)0(

2/)0(

Period 1

Period 2

Period 4

Period 8

Early-time motion

Close-up of steady-statemotion

f(t)

t

f(t)

t

Period Doubling Cascade

Period doubling continues in a sequence of ever-closer values of g

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(wt)

w/2

Period Doubling Cascade

g = 1.06

g = 1.078

g = 1.081

g = 1.0826

n period gn interval (gn+1-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

‘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

d = 4.6692016 is called the Feigenbaum number

‘≈’ → ‘=’ as n → ∞

The limiting value as n → ∞ is gc = 1.0829. Beyond that is … chaos!

Period infinity

5 1 0 1 5 2 0 2 5 3 0

Fig. 12.10

f(t)

t

g = 1.105

Chaos!

The pendulum is “trying” to oscillate at the driving frequency, butthe motion remains erratic for all time

Period Doubling Cascade

Period doubling continues in a sequence of ever-closer values of g

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

Chaos

• Nonperiodic• Sensitivity to initial conditions

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 instead 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

Convergence of Trajectories in Linear Motion

)cos()( 11 tDet t

|)cos(|)ln(|])(ln[| 11 ttDt

Take the logarithm to magnify small differences..

Plotting log10[|Df(t)|] vs. t should be a straight line of slope – ,bplus some wiggles from the cos(w1t – d1) term

Note that log10[x] = log10[e] ln[x]

2 4 6 8 1 0

1 2

1 0

8

6

4

2

Fig. 12.11

Log10[|Df(t)|]t

g = 0.1Df(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!

5 1 0 1 5 2 0 2 5 3 0 3 5 4 0

8

6

4

2

Log10[|Df(t)|]t

g = 1.07Df(0) = 0.1 Radians

Convergence of Trajectories in Period-2 Motion

The trajectories converge more slowly, but still converge

Fig. 12.12

2 4 6 8 1 0 1 2 1 4 1 6

6

5

4

3

2

1

1

Log10[|Df(t)|]

t

g = 1.105Df(0) = 0.0001 Radians

Divergence of Trajectories in Chaotic Motion

The trajectories diverge, even when very close initiallyDf(16) ~ p, so there is essentially complete loss of correlation between the pendulums

Fig. 12.13

If the motion remains bounded, as it doesin this case, then Df can never exceed 2p.Hence this plot will saturate

Extreme Sensitivity to Initial Conditions

The Lyapunov Exponent

tKet ~)(

l = Lyapunov exponent

l < 0: periodic motion in the long term

l > 0: chaotic motion

0K

Chaos

• Nonperiodic• Sensitivity to initial conditions

Linear Nonlinear ChaosDrive period Period

doublingNonperiodic

l < 0 l < 0 l > 0

tg = 1.13

Df(0) = 0.001 Radians

What Happens if we Increase the Driving Force Further?Does the chaos become more intense?

Fig. 12.14

5 1 0 1 5

2

f(t)

t

5 1 0 1 5 2 0

9

6

3

Log10[|Df(t)|]

Period 3 motion re-appears!

t

g = 1.503

Df(0) = 0.001 Radians

What Happens if we Increase the Driving Force Further?Does the chaos re-appear?

Fig. 12.15

f(t) Log10[|Df(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

5 1 0 1 5 2 0 2 5

2 0

1 0

g = 1.503f(t) t

Df(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 g

Fig. 12.16

Bifurcation Diagram

Used to visualize the behavior as a function of driving amplitude g

1) Choose a value of g2) Solve for f(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 g

),...4(),3(),2(),1(),( 00000 ttttt

Fig. 12.171.0663 1.07930)0(

2/)0(

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

f(t)

t

g = 1.06

g = 1.078

g = 1.081

g = 1.0826

Construction of the Bifurcation Diagram

Period 6window

5 1 0 1 5 2 0 2 5

2 0

1 0

g = 1.503f(t) t

Df(0) = 0.001 Radians

The Rolling Motion Renders the Bifurcation Diagram Useless

As an alternative, plot )(t

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 g)(t

Rolling Motion(next slide)

Period-1 Rolling Motion at g = 1.4

5 1 0

2 0

1 0 5 1 0

2 0

1 0

1 0

)(tg = 1.4

t

t

Even though the pendulum is “rolling”, is periodic)(t

f(t)

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

f(t)

t

g = 0.6

)(t

)(t )(t

)(t

First 20 cycles Cycles 5 -20

start

periodicattractor

0)0(

2/)0(

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

g = 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(

State Space Trajectory for Period Doubling Cascade

2

2

1 0

1 0

2

2

1 0

1 0

g = 1.078 g = 1.081

)(t

)(t

)(t

)(t

Period-2 Period-4Plotting cycles 20 to 60

Fig. 12.23

1 1

1 0

1 0

State Space Trajectory for Chaos

g = 1.105

)(t

)(t

1 1

1 0

1 0

)(t

)(t

Cycles 14 - 21

Cycles 14 - 94

The orbit has not repeated itself…

State Space Trajectory for Chaos

g = 1.5b = w0/8

Cycles 10 – 200

Chaotic rolling motionMapped into the interval –p < f < p

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 f(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

g = 1.5b = w0/8 Samples 10 – 60,000

Enlarged on the next slide

The Poincaré Section is a Fractal

The Poincaré section is a much moreelegant way to represent chaoticmotion

The Superconducting Josephson Junction as a Driven Damped Pendulum

=f f2-f1 = phase difference of SC wave-function across the junction

111

ie 222

ie

I

1 2

(Tunnel barrier)

The JosephsonEquations

),(

),(

1

1

nnn

nnn

sg

sfs

Linear Maps for “Integrable” systems !!

),(

),(

1

1

nnn

nnn

sg

sfs

Non-Linear Maps for “Chaotic” systems !!

0

0s

1s

0

1

• 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