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Chaos, Communication and Consciousness Module PH19510

Chaos, Communication and Consciousness Module PH19510. Lecture 16 Chaos. Overview of Lecture. The deterministic universe What is Chaos ? Examples of chaos Phase space Strange attractors Logistic differences – chaos in 1D Instability in the solar system. Chaos – Making a New Science.

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Chaos, Communication and Consciousness Module PH19510

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  1. Chaos, Communication and ConsciousnessModule PH19510 Lecture 16 Chaos

  2. Overview of Lecture • The deterministic universe • What is Chaos ? • Examples of chaos • Phase space • Strange attractors • Logistic differences – chaos in 1D • Instability in the solar system

  3. Chaos – Making a New Science • James Gleick • Vintage • ISBN • 0-749-38606-1 • £8.99 • http://www.around.com

  4. Before Chaos • A Newtonian Universe : • Fully deterministic with complete predictability of the universe. • Laplace thought that it would be possible to predict the future if we only knew the right equations. "Laplace's Demon." • Causal Determinism

  5. Weather Control in a deterministic universe • von Neumann (1946) • Identify ‘critical points’ in weather patterns using computer modelling • Modify weather by interventions at these points • Use as weapon to defeat communism

  6. Modern Physics and the Deterministic Universe • Relativity (Einstein) • Velocity of light constant • Length and Time depend on observer • Quantum Theory • Limits to measurement • Truly random processes • Chaos

  7. What is Chaos ? • Observed in non-linear dynamic systems • Linear systems • variables related by linear equations • equations solvable • behaviour predictable over time • Non-Linear systems • variables related by non-linear equations • equations not always solvable • behaviour not always predictable

  8. What is Chaos ? • Not randomness • Chaos is • deterministic – follows basic rule or equation • extremely sensitive to initial conditions • makes long term predictions useless

  9. Examples of Chaotic Behaviour • Dripping Tap • Weather patterns • Population • Turbulence in liquid or gas flow • Stock & commodity markets • Movement of Jupiter's red spot • Biology – many systems • Chemical reactions • Rhythms of heart or brain waves

  10. velocity  position  Damped Pendulum – Point Attractor Undamped Pendulum – Limit Cycle Attractor Phase Space • Mathematical map of all possibilities in a system • Eg Simple Pendulum • Plot x vs dx/dt • Damped Pendulum • Point Attractor • Undamped Pendulum • Limit cycle attractor

  11. The Lorenz Attractor The ‘Strange’ Attractor • Edward Lorentz • From study of weather patterns • Simulation of convection in 3D • Simple as possible with non-linear terms left in.

  12. Sensitivity to initial conditions • Blue & Yellow differ in starting positions by 1 part in 10-5 Evolution of system in phase space 

  13. Simplest Chaotic System • Logistic equations • Model populations in biological system What happens as we change k ?

  14. k<3 – Fixed Point Attractor • At low values of k (<3), the value of xt eventually stabilises to a single value - a fixed point attractor

  15. k=3 – Limit Cycle Attractor • When k is 3, the system changes to oscillate between two values. • This is called a bifurcation event. • Now have a limit cycle attractor of period 2.

  16. k=3.5 – 2nd Bifurcation event • When k is 3.5, the system changes to oscillate between four values. • Now have a limit cycle attractor of period 4.

  17. k=3.5699456 – Onset of chaos • When k is > 3.5699456 x becomes chaotic • Now have a Aperiodic Attractor

  18. Onset of chaos • Feigenbaum diagram • Shows bifurcation branches • Regions of order re-appear • Figure is ‘scale invariant’ xt k k = 3.5699456 Onset of chaos

  19. Instability in the Solar System • 3 Body Problem • Possible to get exact, analytical solution for 2 bodies (planet+satellite) • No exact solution for 3 body system • Possible to arrive at approximation by making assumptions • Solutions show chaotic motion • The moon cannot have satellites

  20. Asteroid Orbits Jupiter Mars

  21. Asteroid Orbits

  22. The Kirkwood gap • Daniel Kirkwood (1867) • No asteroids at 2.5 or 3.3 a.u. from sun • 2:1 & 3:1 resonance with Jupiter • Jack Wisdom (1981) solved three-body problem of Jupiter, the Sun and one asteroid at 3:1 resonance with Jupiter. • Showed that asteroids with such specifications will behave chaotically, and may undergo large and unpredictable changes in their orbits.

  23. Review of Lecture • The deterministic universe • What is Chaos ? • Examples of chaos • Phase space • Strange attractors • Logistic differences – chaos in 1D • Instability in the solar system

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