In this analysis we take the radius of the wheel to
be 1. We consider the equilibrium configuration with the wheel in
vertical position and the
second elastic (not shown in the diagram) stretched down
so as to exactly compensate the pull of the first. Since these
are identical elastics, with the same spring constant k,
they will each have been stretched
a distance L from their rest length, and each will
be pulling with a force equal to kL. An exactly
similar analysis will apply to the position where the roles of
the two elastics are reversed.
We will
apply approximations which are useful for
small angles .
When the catastrophe
machine is turned through an
angle
from its (vertical) equilibrium point,
the restoring torque is due to the stretch
of the two elastics. Elastic number one has been stretched by an amount
which is approximately the purple length
in the diagram, i.e.
,
so the force it exerts has been
increased to |

- 1. What is a mathematical catastrophe?
- 2. An algebraic version of the double well
- 3. The Cusp Catastrophe
- 4. Doctor Zeeman's Original Catastrophe Machine
- 5. The Catastrophe Machine's unperturbed potential
function is
`y=x`^{4}

© copyright 2000, American Mathematical Society.