In this work, the analytical expressions of memristances, related parameters, and time domain behavioral analysis of the fractional order memristor have been proposed. Both DC with arbitrary delay and many AC waveforms including arbitrary phase sinusoidal and cosinusoidal waveform along with arbitrary periodic waveform have been taken into account. Unlike the previous works, the formerly ignored dimensional consistency has been taken into account and the analytical modelling of the boundary effect has been performed. Moreover, both transient and asymptotic behaviors of the fractional order memristor excited by AC waveform have been distinguished and analyzed. The effect of phase of AC waveform has also been studied. The influence of the fractional order to the areas of voltage-current hysteresis loop and memristance-current lissajous curve has also been clearly discussed and the usage of fractional order memristor in the memristor based circuit has also been demonstrated.

For studying the practical memristor-based chaotic circuit with fractional-order dynamic and asserting the importance of dimensional consistency awareness, the dimensional consistency aware fractional-order generalization of a Hewlett Packard (HP) memristor-based chaotic circuit with the physical meaning of fractional time component assigned has been proposed in this work. The simplest chaotic circuit based on such practical memristor has been chosen as the candidate circuit. A novel window function dedicated to HP memristor with fractional-order dynamic i.e. fractional-order HP memristor has been adopted for modelling the boundary effect. For the dynamical analysis, the revisited version of Jumarie’s modified Riemann–Liouville fractional derivative and nonlinear transformation has been used. The generalized circuit which has been found to be the simplest fractional-order HP memristor-based chaotic circuit displays a chaotic behavior with significant differences from those of its conventional integer-order prototype and its dimensional consistency ignored counterpart; thus, the importance of dimensional consistency awareness is asserted. The realization of the generalized circuit by using the fractional-order elements is indicated. The circuit emulator has also been presented.