chaos, communication and consciousness module ph19510 lecture 16 chaos
TRANSCRIPT
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Chaos, Communication and ConsciousnessModule PH19510
Lecture 16
Chaos
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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
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Chaos – Making a New Science
James Gleick Vintage ISBN
0-749-38606-1
£8.99 http://www.around.com
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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
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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
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Modern Physics and the Deterministic Universe Relativity (Einstein)
Velocity of light constantLength and Time depend on observer
Quantum TheoryLimits to measurementTruly random processes
Chaos
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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
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What is Chaos ?
Not randomness Chaos is
deterministic – follows basic rule or equationextremely sensitive to initial conditionsmakes long term predictions useless
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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
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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
Damped Pendulum – Point Attractor
velo
cit
y
position
Undamped Pendulum – Limit Cycle Attractor
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The ‘Strange’ Attractor
Edward Lorentz From study of
weather patterns Simulation of
convection in 3D Simple as possible
with non-linear terms left in. The Lorenz Attractor
bzxydt
dz ,xzyrx
dt
dy ,xy
dt
dx
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Sensitivity to initial conditions
Blue & Yellow differ in starting positions by 1 part in 10-5
Evolution of system in phase space
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Simplest Chaotic System
Logistic equations Model populations in biological system
tt1t x1x kx
What happens as we change k ?
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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
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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.
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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.
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k=3.5699456 – Onset of chaos
When k is > 3.5699456 x becomes chaotic
Now have a Aperiodic Attractor
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Onset of chaos
Feigenbaum diagram
Shows bifurcation branches
Regions of order re-appear
Figure is ‘scale invariant’ k
xt
k = 3.5699456 Onset of chaos
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Instability in the Solar System
3 Body ProblemPossible to get exact, analytical solution for 2
bodies (planet+satellite)No exact solution for 3 body systemPossible to arrive at approximation by making
assumptionsSolutions show chaotic motion
The moon cannot have satellites
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Asteroid Orbits
Jupiter
Mars
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Asteroid Orbits
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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.
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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