Harmonic oscillations in a second order system modelled by a two-integrator loop with damping are treated in every undergraduate mathematics course. Two initial conditions are needed; we call these (for convenience) displacement and velocity. If such a system is forced, time must also be specified and we have the third order coupled system well known to the non-linear dynamics community consisting of three coupled first order differential equations. The system and its mathematics are illustrated in Figure 1.
Figure 1: The doublewell system and its maths
In our double well system,
the discontinuity is at
the origin of displacement. To determine the
conditions for bifurcation we examine the time
(modulo 2
) at impact and the velocity at impact.
Loosely speaking, in a double well oscillator
which is driven at a frequency above resonance
and has positive damping, the oscillations on one
side grow from the offset centre under the influence
of the drive until the displacement reaches zero.
The velocity is continuous across the discontinuity
but effectively the drive (which was adding
energy in the LH section) extracts energy in the
RH section because the restoring force discontinuously
changes sign. The trajectory spirals towards the
centre on the RH side, grows again and crosses
the discontinuity in the opposite direction (R
L).
Large period motions are easily obtained in the system. This leads to the speculation that two-centre systems of ions in solids might be used to divide down laser light by large integer division ratios.
Whether this results in chaotic motion is determined by the Liapunov exponent of the time series of the eigenvalues of the varied (velocity, time) matrix evaluated at each crossing. There are also so-called ``grazing bifurcations" [5,7,3] in this system which are not predicted by the eigenvalue calculation.