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Introduction to Chaos Clint Introduction to Chaos Clint

Introduction to Chaos Clint - PowerPoint Presentation

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Introduction to Chaos Clint - PPT Presentation

Sprott Department of Physics University of Wisconsin Madison USA Presented at the Utrecht Physics Challenge in Utrecht Netherlands on May 6 2017 Abbreviated History Kepler 1605 ID: 640307

chaos system object chaotic system chaos chaotic object attractor sprott conditions dependence sensitive physics problem effect laws butterfly circuit fractal orbits lorenz

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Slide1

Introduction to Chaos

Clint SprottDepartment of PhysicsUniversity of Wisconsin – Madison USAPresented at the Utrecht Physics Challengein Utrecht, Netherlands on May 6, 2017Slide2

Abbreviated History

Kepler (1605)Newton (1687)Poincare (1890)Lorenz (1963)Slide3

Johannes Kepler (1605)

Assistant to Tycho Brahe3 laws of planetary motionElliptical orbits

RepeatablePredictableSlide4

Isaac Newton (1687)

Invented calculusDerived 3 laws of motion F = maProposed law of gravity F = Gm1

m2/r 2Explained Kepler’s laws

Got headaches (3-body problem)Slide5

3-Body ProblemSlide6

Simplified Solar SystemSlide7

Henri Poincare (1890)

200 years later!King Oscar (Sweden, 1887)Prize won – 200 pagesNo analytic solution exists!Sensitive dependence on initial conditions (Lyapunov exponent)Chaos! (Li & Yorke, 1975) Slide8

Chaotic Double PendulumSlide9

Sensitive Dependence on I. C.Slide10

Double Pendulum SimulationSlide11

Equations for Double PendulumSlide12

Edward Lorenz (1963)

Meteorologist at MITHad his own personal computerRediscovered chaos in a simple system of equations:dx/dt = σ(y - x)dy/dt = -

xz + rx - ydz

/dt = xy - bz3 variables (x

,

y

,

z

)

2 nonlinearities (

xz

,

xy

)

3 parameters (

σ

,

r

,

b

)Slide13

Lorenz Attractor

Strange attractorA fractal object

Fractal dim ~ 2.05

Butterfly effectSlide14

Butterfly EffectSlide15

Sensitive Dependence on Init

CondInitial conditions differ by 0.01%Slide16

Conditions for Chaos

At least 3 variables (to keep the orbits from intersecting)At least one nonlinearity (to keep the orbits bounded)A source of energy (to keep the system going)Slide17

Usual Route to Chaos

Stable equilibrium (point attractor)

Limit cycle (periodic attractor)

Period doubling

Strange attractorSlide18

Period Doubling  Chaos

dx/dt = ydy/dt= zdz/dt= -a

z + y2 - x

The simplest chaotic flow!Sprott (1997)Slide19

Chaotic CircuitSlide20

Equations for Chaotic Circuit

dx

/d

t = ydy/

d

t

= z

d

z

/

d

t

= -

az

- by + c

(sgn

x

- x

)

Jerk system

Period doubling route to chaosSlide21

Bifurcation Diagram for Chaotic CircuitSlide22

Applications for Chaos

Secure communicationsMeteorologyEcologyEconomicsSociologyPsychologyPoliticsPhilosophySlide23

References

http://sprott.physics.wisc.edu/ lectures/utrecht.pptx

(this talk)

http://sprott.physics.wisc.edu/chaostsa/

(my chaos textbook)

sprott@physics.wisc.edu

(contact me)Slide24

Questions

Who won King Oscar’s Prize?Johannes KeplerIsaac NewtonHenri PoincareEdward LorenzWhat is the 3-body problem?Biological survival of the fittest

The motion of three bodies with mutual attractionThe social interactions of three friendsThe riddle of a multiple homicide

What is a nonlinear system?A system in which effects have multiple causesA system whose whole is not equal to the sum of its partsA system that exhibits chaosA system with many variables

What is a strange attractor?

An object whose shape is unpredictable

An object that attracts another dissimilar object

An object that attracts another similar object

A fractal produced by a chaotic process

What is the butterfly effect?

Turbulence produced by fluctuating organisms

Irregular oscillations of a dynamical system

Behavior of a complex system

Sensitive dependence on initial conditions