1 :60

ParameterSim.Time F0 F Damping Gamma Ovrd Theta-0 State

Comment (s) Hz Hz Nt-s/m Nt-s^2/m Degrees 0,1,2Lightly damped, undriven 15 3 1 0.4 0 1 90 0More damping, undriven (dies in 5 secs) 15 3 1 1 0 1 90 0Lightly damped, state space 15 3 1 1 0 1 90 1Same damping, driven 15 3 1 1 220 1 1 0Same damping, state space 15 3 1 1 220 1 0 1Different drive frequency 15 3 0.5 1 150 1 0 0Drive freq~res. Freq. 15 1 2 1 150 1 0 0Drive freq>res. Freq. 15 1 5 1 150 1 0 0Drive freq>>res. Freq. 15 1 10 1 150 1 0 0Ordinary driven 15 1 0.67 1 0.3 0 0 0Ordinary driven more damping 15 1 0.67 3.14 0.3 0 0 0Long Transient 22.5 1 0.67 3.14 1.06 0 0 0Long Transient, State space 100 1 0.67 3.14 1.06 0 0 1Period doubling 45 1 0.67 3.14 1.073 0 0 0Period tripling 45 1 0.67 3.14 1.077 0 0 0Different Attractors diff IC's 45 1 0.67 3.14 1.077 0 -90 0Period tripling, state space 45 1 0.67 3.14 1.077 0 0 0Different Attractors state space 45 1 0.67 3.14 1.077 0 -90 0Period doubling 45 1 0.67 3.14 1.078 0 -90 0Period quadrupling 45 1 0.67 3.14 1.081 0 -90 0Period quadrupling (state) 100 1 0.67 3.14 1.081 0 -90 2Period octupling 45 1 0.67 3.14 1.0826 0 -90 0True chaos 45 1 0.67 3.14 1.105 0 -90 0Chaos in state space 100 1 0.67 3.14 1.105 0 -90 2

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Parameter Time F0 F Damping Gamma Ovrd Theta-0 State

Comment (s) Hz Hz Nt-s/m Nt-s^2/m Degrees 0,1,2More damping, undriven - 3 1 1 0 1 90 -Lightly damp, state space - 3 1 1 0 1 90 1Same damping, driven - 3 1 1 220 1 0 0Same damping, state space - 3 1 1 220 1 - 1Different drive frequency - 3 0.5 1 150 1 - 0Drive freq~res. Freq. - 1 2 1 150 1 - -Drive freq>res. Freq. - - 5 1 150 1 - -Drive freq>>res. Freq. - - 10 1 150 1 - -Ordinary driven - 1 0.67 1 0.3 0 - -Driven more damped - 1 0.67 3.14 0.3 - - -Long Transient 22.5 - - - 1.06 - - -Long Transient, State space 100 - - - 1.06 - - 2Period doubling 45 - - - 1.073 - - 0Period tripling 45 - - - 1.077 - - 0Different Attractors diff IC's 45 - - - 1.077 - -90 0Period tripling, state space 45 - - - 1.077 - 0 2Different Attractors state space 45 - - - 1.077 - -90 2Period doubling 45 - - - 1.078 - -90 0Period quadrupling 45 - - - 1.081 - -90 0Period quadrupling (state) 100 - - - 1.081 - -90 2Period octupling 45 - - - 1.0826 - -90 0True chaos 45 - - - 1.105 - -90 0Chaos in state space 100 - - - 1.105 - -90 2

3

Handout #20

Nonlinear Systems and Chaos Most important concepts

Sensitive Dependence on Initial conditions Attractors

Other concepts State-space orbits Non-linear diff. eq. Driven oscillations Second Harmonic Generation Subharmonics Period-doubling cascade Bifurcation plot Poincare diagram Feigenbaum number Universality

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4

Outline

Origins and Definitions of chaosState SpaceBehavior of a driven damped pendulum (DDP)Non-linear behavior of a DDP Attractor Period doubling Sensitive dependence Bifurcation Plot

:02

5

Definition of chaos

The dynamical evolution that is aperiodic and sensitively dependent on initial conditions. In dissipative dynamical systems this involves trajectories that move on a strange attractor, a fractal subspace of the phase space. This term takes advantage of the colloquial meaning of chaos as random, unpredictable, and disorderly behavior, but the phenomena given the technical name chaos have an intrinsic feature of determinism and some characteristics of order. Colloquial meaning – disorder, randomness, unpredictabilityTechnical meaning – Fundamental unpredictability and apparent randomness from a system that is deterministic. Some real randomness may be included in real systems, but a model of the system without ANY added randomness should display the same behavior.

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6

Weather and climate prediction

Is important Can we go:

a) For a hike?b) Get married outdoors?c) Start a war?

Will we be hurt by a:a) Tornado?b) Hurricane?c) Lightning bolt?

How much more fossil fuel can we burn before we:a) Fry?b) Drown?c) Starve?

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7

Early work by Edward N. Lorenz

1960’s at MIT Early computer models of the atmosphere

Were very simple (Computers were stupid)

Were very helpful Results were not reproducible!!

Lorenz ultimately noticed that For 7-10 days of prediction, all of his models reproduced very well. After 7-10 days, the same model could be run twice and give the

same result – BUT!! Changes that he thought were trivial (e.g. changing the density of air by rounding it out at the 3th

decimal place, or slightly modifying the initial conditions at beginning of model)

Produced COMPLETELY DIFFERENT results This came to be called “The Butterfly effect”

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8

State Space

:02

sin 2 ;

si

sin 0.35s

sin

n 2 2co

in 2 ;

cos 0.7c

;

sin

os 2

o

s

s

2

c

x t

dy x t

x t t

x t

dy x t t

dt

td

y

t

x t t

9

Viscous Drag III – Stokes Law

Form-factor k becomes “D” is diameter of sphereViscous drag on walls of

sphere is responsible for retarding force.

George Stokes [1819-1903] (Navier-Stokes equations/ Stokes’

theorem)

xu ˆD

xuDFdrag ˆ3 dragF

3

rbFdrag

:45

10

Damped Driven Pendulum (DDP)

Damping Pendulum immersed in fluid

with Newtonian viscosity Damping proportional to

velocity (and angular velocity)

Driving Constant amplitude drive

bar Connected to pendulum via

torsion spring Torque on Pendulum is

:02

( )FL k mgbv

( )F t

2 2 sin ( )mL bL mgL LF t

11

Damped Driven Pendulum (DDP)

:02

2 2

2 20 0

2 20 0

2 20 0

sin ( )

( )sin

( )2 sin

( )2 sin

2 sin cos

( )

mL bL mgL LF t

b g F t

m L mLL F t

g mL

F t

mg

t

F t

mg

0 0 / 2 1

0.667

2 3.14 /

f Hz

Hz

N s m

Force

pendulum weight

12

Conditions for chaos

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Dissipative Chaos

1. Requires a differential equation with 3 or more independent variables.

2. Requires a non-linear coupling between at least two of the variables.

3. Requires a dissipative term (that will use up energy).

Non-dissipative chaos

Not in this course

13

Chaos on the ski-slope

:607 “Ideal skiers” follow the fall-line and end up very different

places

14

Insensitive dependence on initial conditions

:60

0: 90Plot A

0: 175Plot B

: ( )

0.1

Plot A B Log of

15

Sensitive dependence on initial conditions

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0: 90Plot A

0: 90.0001Plot B

: ( )

1.105

Plot A B Log of

16

Worked problem

:60

Sketch a state-space plot for the magnetic pendulum

Indicate the attractors and repellers

Show some representative trajectories

First explore the trajectories beginning with theta-dot=0

Then explore trajectories that begin with theta in some state near an attractor but through proper choice of theta-dot move to the other attractor.

Sketch the basins of attraction

17

Handout #20 Windup

:60

Review class – Can be held Friday afternoon 12/12 or Saturday 12/13

TestIs due last day of class … midnight

Requires certified original work

You may use any scripts that already exist in computer lab. Do not trade scripts on the PC’s in the computer lab. Keep your work on a floppy – submit floppy along with written work.