Class 26
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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

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Class #26

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Class 26

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


Outline

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


Definition of chaos

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.


Weather and climate prediction

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?


Early work by edward n lorenz

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”


State space phase space

State Space / Phase Space


Viscous drag iii stokes law

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)


Damped driven pendulum ddp

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


Damped driven pendulum ddp1

Damped Driven Pendulum (DDP)


Conditions for chaos

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


Sensitive dependence on initial conditions

Sensitive dependence on initial conditions


Sensitive dependence on initial conditions1

Sensitive dependence on initial conditions


Worked problem

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-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


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