Emergence of current branches in a series array of negative differential resistance circuit elements

We study a series array of nonlinear electrical circuit elements that possess negative differential resistance and find that \emph{heterogeneity} in the element properties leads to the presence of multiple branches in current-voltage curves and a non-uniform distribution of voltages across the elements. An inhomogeneity parameter $r_{max}$ is introduced to characterize the extent to which the individual element voltages deviate from one another, and it is found to be strongly dependent on the rate of change of applied voltage. Analytical expressions are derived for the dependence of $r_{max}$ on voltage ramping rate in the limit of fast ramping and are confirmed by direct numerical simulation.