  We examine in theory and by numerical simulation, the dynamic process of
switching from a zero voltage to a finite voltage state in a Josephson junction
circuit. The theoretical model describes small capacitance Josephson junctions
which are overdamped at high frequencies, and can be applied to detection of
the quantum state of a qubit circuit. We show that the speed and fidelity of
the readout are strongly influenced by the topology of the phase space
attractors. The readout will be close to optimal when choosing the circuit
parameters so as to avoid having an unstable limiting cycle which separates the
two basins of attraction.
