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 James Gleick Vintage ISBN  £8.99

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 constant  Length and Time depend on observer Quantum Theory  Limits to measurement  Truly 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 equation  extremely sensitive to initial conditions  makes 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 velocity  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

Sensitivity to initial conditions Blue & Yellow differ in starting positions by 1 part in Evolution of system in phase space 

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

Onset of Chaos At low values of k (<3), the value of x t eventually stabilises to a single value - a fixed point 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. As k increases, further bifurcation events occur such that the periodicity of the attractor becomes 4, 8, 16 etc.

k<3 – Fixed Point Attractor At low values of k (<3), the value of x t 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 – 2 nd 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= – Onset of chaos When k is > 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 xtxt k = Onset of chaos

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

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