SummaryThis paper presents a comprehensive taxonomy of so‐called second‐order memory devices, which include charge‐controlled memcapacitors and flux‐controlled meminductors, among other novel circuit elements. These devices, which are classified according to their differential and state orders, are necessary to get a complete extension of the family of classical nonlinear circuit elements (resistors, capacitors, and inductors) for all possible controlling variables. Using a fully nonlinear formalism, we obtain nondegeneracy conditions for a broad class of second‐order mem‐circuits. This class of circuits is expected to yield a rich dynamic behavior; in this regard, we explore certain bifurcation phenomena exhibited by a family of circuits including a charge‐controlled memcapacitor and a flux‐controlled meminductor, providing some directions for future research. Copyright © 2014 John Wiley & Sons, Ltd.