  In this paper, we show that the dynamics of a wide variety of nonlinear
systems such as engineering, physical, chemical, biological, and ecological
systems, can be simulated or modeled by the dynamics of memristor circuits. It
has the advantage that we can apply nonlinear circuit theory to analyze the
dynamics of memristor circuits. Applying an external source to these memristor
circuits, they exhibit complex behavior, such as chaos and non-periodic
oscillation. If the memristor circuits have an integral invariant, they can
exhibit quasi-periodic or non-periodic behavior by the sinusoidal forcing.
Their behavior greatly depends on the initial conditions, the parameters, and
the maximum step size of the numerical integration. Furthermore, an overflow is
likely to occur due to the numerical instability in long-time simulations. In
order to generate a non-periodic oscillation, we have to choose the initial
conditions, the parameters, and the maximum step size, carefully. We also show
that we can reconstruct chaotic attractors by using the terminal voltage and
current of the memristor. Furthermore, in many memristor circuits, the active
memristor switches between passive and active modes of operation, depending on
its terminal voltage. We can measure its complexity order by defining the
binary coding for the operation modes. By using this coding, we show that in
the forced memristor Toda lattice equations, the memristor's operation modes
exhibit the higher complexity. Furthermore, in the memristor Chua circuit, the
memristor has the special operation modes.
