Conditions for realization of multiple pinch-off points in generalized fractional-order memory elements

Memelements (memristor, meminductor, and memcapacitor) have gained wide interests due to their useful hysteresis and nonlinear properties. In this work, fractional-order current-controlled memelements are modeled using an n$$ n $$-order polynomial. By decomposing the n$$ n $$-order polynomial into some 2n+1$$ 2n+1 $$ sets of current-controlled functions, we were able to theoretically analyze and establish the general conditions necessary for all memelements' voltage-current dynamics to create up to n+1$$ n+1 $$ number of lobes and hence n$$ n $$ pinch-off points in their v-i$$ v-i $$ hysteresis. The exact coordinate locations of the pinch-off points on the v$$ v $$-axis and i$$ i $$-axis are calculated in all cases. The observations made in this work are novel and contributes to how all multiple-lobed memory elements are analyzed. A general emulator circuit is then designed to verify the theoretical deductions, the emulator output agrees with the numerical results.