Nonlinear Mechanics and Chaos

Dr. Jen‐Hao YehProf. Anlage

This is a brief introduction to the ideas and concepts of nonlinear mechanics, and a discussion of various quantitativemethods for analyzing such problems

We will focus on the driven damped pendulum (DDP)

Linear Nonlineareasy hard

special general

analytical numerical

Superposition principle

Chaos

Linear

Nonlinear

Chaotic

)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

A Route to Chaos

Fig12p2 NDSolve''t 2 't 0 ^2 Sint 0 ^2 Cos t, 0 0, '0 0, , t, 0, 6

InterpolatingFunction0., 6.,

NDSolve in Mathematica

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

)cos()sin(2 20

20 t

Driven, Damped Pendulum (DDP)

5.10

2Period = 2 = 1

0)0()0(

4/0

For all following plots:

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()( tAtAfter the initial transient dies out, the solution looks like

Periodic “attractor”

Linear

Small Oscillations of the Driven, Damped Pendulum

<< 1 will give small oscillations

1) The motion approaches a unique periodic attractorindependent of initial conditions

2) The motion is sinusoidal with the same frequency as the drive

)cos()( tAt

Moderate Oscillations of the Driven, Damped Pendulum

< 1 and the nonlinearity becomes significant…

)cos(612 2

032

0 t

)cos()( tAtThis solution gives from the 3 term: xxx cos33cos

41cos3

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

Harmonics

5 6- p

2

p2

1 2 3 4 5 6

-p2

p2

= 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 …

Even Stronger Driving: Complicated Transients – then Periodic!

3 6 9 12 15

2 p

4 p = 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

5 10 15 20 25 30-2 p

2 p

4 p

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!

22 24 26 28 30-8.5

-8.

A SUB-Harmonic has appeared

Harmonics and SubharmonicsSubharmonic

Slightly Stronger Driving: Period 3

2 4 6 8 10 12 14

2 p

4 p

(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

Multiple Attractors

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

2 4 6 8 10 12 14

2 p

4 p

(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

Period Doubling Cascade

2 4 6 8 10

-2

2

30 32 34 36 38 40-2.5

-2

2 4 6 8 10

-2

2

30 32 34 36 38 40-2.5

-2

2 4 6 8 10

-2

2

30 32 34 36 38 40-2.5

-2

2 4 6 8 10

-2

2

30 32 34 36 38 40-2.5

-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

Period Doubling Cascade

= 1.06

= 1.078

= 1.081

= 1.0826

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!

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”

Period infinity 

5 10 15 20 25 30

-p

p

(t)

t

= 1.105

Chaos!

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

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 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(|])(ln[| 11 ttDt

Take the logarithm of |(t)| to magnify small differences.

Plotting log10[|(t)|] vs. t should be a straight line of slope –plus some wiggles from the ln[|cos(1t – 1)|] term

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

2 4 6 8 10

-12

-10

-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) caseThis shows that the linear oscillator is essentially insensitive to its initial conditions!

5 10 15 20 25 30 35 40

-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

2 4 6 8 10 12 14 16

-6

-5

-4

-3

-2

-1

1Log10[|(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

The Lyapunov Exponent

tKet ~)(

= Lyapunov exponent

< 0: periodic motion in the long term

> 0: chaotic motion

0K

Linear NonlinearChaos

Drive period Harmonics,Subharmonics,Period‐doubling

Nonperiodic,Extreme sensitivity

< 0 < 0 > 0

t = 1.13

(0) = 0.001 Radians

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

5 10 15

2 p

(t)

t

5 10 15 20

-9

-6

-3

Log10[|(t)|]

Period 3 motion re-appears!

With increasing the motion alternates between chaotic and periodic

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 10 15 20 25

-20 p

-10 p

t

5 10 15

-4-3-2-1

1

This is a kind of ‘rolling’ chaotic motion

5 10 15 20 25

-20 p

-10 p

= 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

Bifurcation Diagram

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

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 InsanityBy Jose Luis. Perez Velazquez

Fig. 12.9, Taylor

A period doubling cascade in convection of mercury in a smallconvection cell. The plots show the temperature at one fixedpoint in the cell as a function of time, for four successively larger temperature gradients as given by the parameter R/Rc

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 at a time after the attractor behavior has been achieved3) Move on to the next value of

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

Fig. 12.171.0663 1.07930)0(

2/)0(

30 32 34 36 38 40-2.5

-2

30 32 34 36 38 40-2.5

-2

30 32 34 36 38 40-2.5

-2

30 32 34 36 38 40-2.5

-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

5 10 15 20 25

-20 p

-10 p

= 1.503(t) t

(0) = 0.001 Radians

The Rolling Motion Renders the Bifurcation Diagram Useless

As an alternative, plot )(t

Prev

ious

dia

gram

ra

nge

Mos

tly c

haos

Peri

od-3

Mos

tly c

haos

Peri

od-1

follo

wed

by p

erio

d do

ublin

g bi

furc

atio

n

Mos

tly c

haos

Bifurcation Diagram Over a Broad Range of )(t

Rolling Motion(next slide)

Period-1 Rolling Motion at = 1.4

5 10

-20 p

-10 p 5 10

-20

-10

10

)(t = 1.4

t

t

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

(t)

An Alternative View: State Space Trajectory

)(tPlot vs. with time as a parameter)(t

1 2 3 4 5 6

-p2

p4

-p2 - p

4p4

-4

4

- p4

p4

-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(

- p4

p4

-4

4

- p4

p4

-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(

State Space Trajectory for Period Doubling Cascade

- p2

p2

-10

10

- p2

p2

-10

10

= 1.078 = 1.081

)(t

)(t

)(t

)(t

Period-2 Period-4Plotting cycles 20 to 60

Fig. 12.23

-1 1

-10

10

State Space Trajectory for Chaos

= 1.105

)(t

)(t

-1 1-10

10

)(t

)(t

Cycles 14 - 21

Cycles 14 - 94

The orbit has not repeated itself…

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 …

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 after the attractor behavior has been achieved

,...)2(),2(,)1(),1(,)(),( 000000 tttttt

= 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 moreelegant way to represent chaoticmotion

The Superconducting Josephson Junction as a Driven Damped Pendulum

= phase difference of SC wave-function across the junction

111

ie 222

ie

I1 2

(Tunnel barrier)

The JosephsonEquations

Radio Frequency (RF) Superconducting Quantum Interference Devices (SQUIDs)

L LJJ R C

FluxQuantization

in the loop

I(t)

Ic

0 ninducedapplied

Single rf-SQUID D. Zhang, et al., Phys. Rev. X (in press), arXiv:1504.08301

D. Zhang, et al., Phys. Rev. X (in press), arXiv:1504.08301

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, typeof material, width of mesa, etc.)

Many competing states do not showemission

Emission enhanced near cavity moderesonances → requires non-uniformcurrent injection, assisted by inhom. heating, -phase kinks, crystal defects

)V(VhefJJ THz/mV 483.02

L. Ozyuzer, et al., Science 318, 1291 (2007)

),(),(

1

1

nnn

nnn

sgsfs

Linear Maps for “Integrable” systems !!

),(),(

1

1

nnn

nnn

sgsfs

Non-Linear Maps for “Chaotic” systems !!

00s

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