In the early 1960's Edward Lorenz, a MIT meteorologist was working on developing a System to simplify
the convection rolls in upper atmosphere for long-range weather prediction (5+ days).
However, the weather is complicated!
A theoretical simplification was necessary.
In 1963 derived a three dimensional system in efforts to model long range predictions
for the weather.
Using this system ran heading into "sensitivity to initial conditions".
In the process, he sketched the outlines of one of the first recognized chaotic attractors.
And he came to the conclusion that model equation are inaccurate in their representation of
some aspect of the weather OR the model may be accurate but there is some anomalous property
of the equations that makes our prediction difficult.
The Lorenz systems describes the motion of a fluid between two layers at different temperature.
Specifically, the fluid is heated uniformly from below and cooled uniformly from above.
By rising the temperature difference between the two surfaces, we observe initially a linear
temperature gradient, and then the formation of Rayleigh-Benard convection cells.
After convection, turbulent regime is also observed:
The Lorenz attractor defines a 3-dimensional trajectory by the differential equations:
σ, r, b are parameters
When calculating this model, Lorenz encountered a strange phenomenon.
After entering slightly different input values for two successive attempts he obtained completely
different outputs.
This effect was later named a butterfly effect; the "Butterfly Effect" is the propensity of
a system to be sensitive to initial conditions.
Such systems over time become unpredictable, this idea gave rise to the notion of a butterfly
flapping it's wings in one area of the world, causing a tornado or some such weather event
to occur in another remote area of the world.
The following graphs show time dependence of functions x(t) and z(t) in the Lorenz attractor
for the recommended parameter values, while blue curves are related to initial conditions
x(0)=1; y(0)=1; z(0)=10 and red curves are related to initial conditions x(0)=1; y(0)=1;
z(0)=10.01
Very slight change of initial condition results in large change in solution of the function.
If we calculate the Jacobian of the system at S1 then we have:
And Eigenvalues of J are:
If you do the same for the other two points then for the recommended parameter values
all three fixed points are unstable, because they all include an eigenvalue with positive
real part.
A chaotic system is roughly defined by sensitivity
to initial conditions: infinitesimal differences in the initial conditions of the system result
in large differences in behaviour.
Chaotic systems do not usually go out of control, but stay within bounded operating conditions.
Chaos provides a balance between flexibility and stability, adaptiveness and dependability.
It lives on the edge between order and randomness.
How do you think algorithmic complexity can be used to analysis this type of system?
What if you do not have equations describing the system and you only observe the output
of system close to its attractors?
As you saw dynamical system can have different type of attractors.
If the system evolves towards a single state and remains there, we call it fixed point.
An example is damped pendulum or a sphere at the bottom of a spherical bowl.
It will be Periodic or quasiperiodic attractor when the system evolves towards a limit cycle.
An example is undamped pendulum or a planet orbiting around the Sun.
If the system is very sensitive to initial conditions and we are not able to simply predict
its behaviour, we call it Chaotic attractor:.
An example is the Lorenz attractor and finally the system has strange attractor if it is
also very sensitive to initial conditions and we are not able to simply predict its
behaviour, but in this case, the system has the same properties like fractals.
In another words, the strange attractor represents a fractal.
An example is the Mandelbrot set.
Fractal is a geometric shape, A fractal is a non-regular geometric shape that has the
same degree of non-regularity on all scales.
Fractals can be thought of as never-ending patterns.
It has a Hausdorff dimension of its border higher than the topological dimension of the
border.
An example is a Cantor set: where the original line is divided into three parts while the
middle part is erased.
The same procedure is applied to newly created lines etc. if we are repeating this procedure
to the infinity, we obtain an infinite number of points with topological dimension 0.
The set contains n=2 copies of itself reduced to 1/3 of the original dimension (k=3).
Hausdorff dimension is log(2)/log(3)=0.6309…, which is greater than 0.
Up to now we learned dynamical systems live in phase space and develop in time, following
their dynamical law.
We saw examples of continuous dynamical system and learned how to analysis them.
In the next two lectures, we will do the same for discrete dynamical system.