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Chaos, Communication and ConsciousnessModule 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
James Gleick Vintage ISBN
0-749-38606-1
£8.99 http://www.around.com
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
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
Modern Physics and the Deterministic Universe Relativity (Einstein)
Velocity of light constantLength and Time depend on observer
Quantum TheoryLimits to measurementTruly random processes
Chaos
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
What is Chaos ?
Not randomness Chaos is
deterministic – follows basic rule or equationextremely sensitive to initial conditionsmakes long term predictions useless
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
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
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
Sensitivity to initial conditions
Blue & Yellow differ in starting positions by 1 part in 10-5
Evolution of system in phase space
Simplest Chaotic System
Logistic equations Model populations in biological system
tt1t x1x kx
What happens as we change k ?
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
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.
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.
k=3.5699456 – Onset of chaos
When k is > 3.5699456 x becomes chaotic
Now have a Aperiodic Attractor
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
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
Asteroid Orbits
Jupiter
Mars
Asteroid Orbits
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.
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