  Networks with memristive devices are a potential basis for the next
generation of computing devices. They are also an important model system for
basic science, from modeling nanoscale conductivity to providing insight into
the information-processing of neurons. The resistance in a memristive device
depends on the history of the applied bias and thus displays a type of memory.
The interplay of this memory with the dynamic properties of the network can
give rise to new behavior, offering many fascinating theoretical challenges.
But methods to analyze general memristive circuits are not well described in
the literature. In this paper we develop a general circuit analysis for
networks that combine memristive devices alongside resistors, capacitors and
inductors and under various types of control. We derive equations of motion for
the memory parameters of these circuits and describe the conditions for which a
network should display properties characteristic of a resonator system. For the
case of a purely memresistive network, we derive Lyapunov functions, which can
be used to study the stability of the network dynamics. Surprisingly, analysis
of the Lyapunov functions show that these circuits do not always have a stable
equilibrium in the case of nonlinear resistance and window functions. The
Lyapunov function allows us to study circuit invariances, wherein different
circuits give rise to similar equations of motion, which manifest through a
gauge freedom and node permutations. Finally, we identify the relation between
the graph Laplacian and the operators governing the dynamics of memristor
networks operators, and we use these tools to study the correlations between
distant memristive devices through the effective resistance.
