  We give analytical solutions to the titanium dioxide memristor with arbitary
order of window functions, which assumes a nonlinear ionic drift model. As the
achieved solution, the characteristic curve of state is demonstrated to be a
useful tool for determining the operation point, waveform and saturation level.
By using this characterizing tool, it is revealed that the same input signal
can output completely different u-i orbital dynamics under different initial
conditions, which is the uniqueness of memristors. The approach can be regarded
as an analogy to using the characteristic curve for the BJT or MOS
transisitors. Based on this model, we further propose a class of analytically
solvable class of memristive systems that conform to Abel Differential
Equations. The equations of state (EOS) of the titanium dioxide memristor based
on both linear and nonlinear ionic drift models are typical integrable
examples, which can be categorized into this Abel memristor class. This large
family of Abel memristive systems offers a frame for obtaining and analyzing
the solutions in the closed form, which facilitate their characterization at a
more deterministic level.
