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.