We recall that the solution of a classic damped linear second-order system which is forced consists of two parts; a transient solution which decays in amplitude and has the natural ringing frequency of the system, plus a steady response at the driving frequency. The double well system and the impact oscillator both have such solutions between crossings or impacts. Generalising to piecewise linear systems in the large, this is true of any system which in a certain region obeys a linear differential equation with constant coefficients. The motion can appear to be quite complex when these two parts of the solution are added together, for the two frequencies can be incommensurate and that means that the motion has elements of quasiperiodic behaviour, at least until the transient part is negligibly small.
The impacts, or crossings, or discontinuities, provide a mechanism for transferring energy from the steady state solution into the transient. If they happen at irregular intervals, due to the aforementioned complexity of the linear motions, it is not therefore surprising that the motions appear chaotic. However, the transient solution between impacts is always decaying; errors in the initial state are likewise decaying so perhaps it is not so surprising that the chaotic region solutions appear to converge.
There seems to be some advantage in linking a certain amount of analysis with both simulation and experiment. One quickly comes to the understanding that there is no such thing as a perfectly symmetric implementation, or a chaotic system in real life in which the noise is of no consequence. This particular system is accessible to all three methods of investigation by students, and yet it has complex and puzzling features even after one has thought quite deeply about it.
Synchronisation of chaotic systems is very fashionable at present. Two such chaotic double well oscillators are particularly well adapted to synchronisation experiments both in simulation and in hardware. It is clear to me that there is a tradeoff to be made between robustness in the presence of noise and imperfections, and the utility of any synchronisation scheme for secure communications applications. This particular set of circuits may be particularly well adapted to such applications, in view of the convergent nature of most of the chaotic trajectories under errors in the initial states.
To construct a synchronised system, one takes samples of the capacitor voltages on the integrators of one system at the moments at which the comparator switches, and transfers them to the capacitors on the other system. This is rather a small amount of information transfer. In a sense, we are resetting the second system with an initial state derived from the first at every impact. For non-identical systems it may happen that there is a grazing bifurcation in one but not in the other, and this will lead to rapid loss of synchronisation. This is more likely to occur if impacts or crossings are infrequent. We may therefore expect that this system is capable of displaying partial synchronisation, or what I call ``agreement", in which the time series are positively correlated (having a non-stationary correlation coefficient), even though not strictly synchronised. Whether or not there is a use for such a system remains to be seen, but it may well be a failure mode of complete synchronisation.