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
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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