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 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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 James Gleick
 Vintage
 ISBN
 £8.99
 http://www.around.com

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 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

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 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

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 Relativity (Einstein)
 Velocity of light constant
 Length and Time depend on observer
 Quantum Theory
 Limits to measurement
 Truly random processes
 Chaos

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 Observed in nonlinear dynamic systems
 Linear systems
 variables related by linear equations
 equations solvable
 behaviour predictable over time
 NonLinear systems
 variables related by nonlinear equations
 equations not always solvable
 behaviour not always predictable

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

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 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

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 Mathematical map of all possibilities in a system
 Eg Simple Pendulum
 Plot x vs dx/dt
 Damped Pendulum
 Undamped Pendulum

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 Edward Lorentz
 From study of weather patterns
 Simulation of convection in 3D
 Simple as possible with nonlinear terms left in.

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 Blue & Yellow differ in starting positions by 1 part in 10^{5}

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 Logistic equations
 Model populations in biological system

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 At low values of k (<3), the value of x_{t} eventually
stabilises to a single value  a fixed point attractor

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 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.

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 When k is 3.5, the system changes to oscillate between four values.
 Now have a limit cycle attractor of period 4.

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 When k is > 3.5699456 x becomes chaotic
 Now have a Aperiodic Attractor

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 Feigenbaum diagram
 Shows bifurcation branches
 Regions of order reappear
 Figure is ‘scale invariant’

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 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

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 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 threebody 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.

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