Testing

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Class 26

• 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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Outline

• Origins and Definitions of chaos
• State Space
• Behavior of a driven damped pendulum (DDP)
• Non-linear behavior of a DDP
• Attractor
• Period doubling
• Sensitive dependence
• Bifurcation Plot

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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,
  unpredictability
• Technical 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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Weather and climate prediction

• Is important
• Can we go
• For a hike?
• Get married outdoors?
• Start a war?
• Will we be hurt by a
• Tornado?
• Hurricane?
• Lightning bolt?
• How much more fossil fuel can we burn before we
• Fry?
• Drown?
• Starve?

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Early work by Edward N. Lorenz

• 1960s 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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State Space / Phase Space
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Viscous Drag III Stokes Law
D

• Form-factor k becomes
• D is diameter of sphere
• Viscous drag on walls of sphere is responsible
  for retarding force.
• George Stokes 1819-1903 ?
• (Navier-Stokes equations/ Stokes theorem)

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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

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Damped Driven Pendulum (DDP)
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Conditions for chaos

• Dissipative Chaos
• Requires a differential equation with 3 or more
  independent variables.
• Requires a non-linear coupling between at least
  two of the variables.
• Requires a dissipative term (that will use up
  energy).
• Non-dissipative chaos
• Not in this course

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Sensitive dependence on initial conditions
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Sensitive dependence on initial conditions
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Worked problem

• 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-dot0
• 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