User:Paul Murray/chaos theory example

I intend to add this to the chaos theory page when it's ready.

An example
A useful example to consider is a mass, attached to a damper and a spring, being driven with a force varying sinusoidally. The parameters of the system are the mass m of the mass, the spring constant k of the spring, the damping constant d of the damper, and the maximum force f and the frequency "r" with which that force varies.

For a given "set up" of these parameters, the current state of the system can be described fully by stating the positon p of the mass, the velocity 'v' of the mass, and the phase &theta; of the force.

The change in position is equal to the velocity: $$\frac {\partial p} {\partial t} = v$$.

The change in velocity is the sum of the damping force (the velocity times the damping constant d) and the tension on the spring (minus the displacement of the mass, times the spring constant), and the force (f sin r) divided by the mass: $$\frac {\partial v} {\partial t} = \frac { -kp -dv + f sin(tr) } m $$.

The sinusoidal motion is uninteresting - it just keeps going. The state space of this system is a three-dimensional slice, infinite in two dimensions (position and velocity of the mass), and with the third dimension being bounded at 0 and 1/r.