Memristive circuits and their Newtonian models $\phi ''=F(t,\phi,\phi')/m$ with memory

The prediction made by L. O. Chua 45+ years ago (see: IEEE Trans. Circuit Theory (1971) 18:507-519 and also: Proc. IEEE (2012) 100:19201927) about the existence of a passive circuit element (called memristor) that links the charge and flux variables has been confirmed by the HP lab group in its report (see: Nature (2008) 453:80-83) on a successful construction of such an element. This sparked an enormous interest in memelements, analysis of their unusual dynamical properties (i.e. pinched hysteresis loops, memory effects, etc.) and construction of their emulators. Such topics are also of interest in mechanical engineering where memdampers (or memory dampers) play the role equivalent to memristors in electronic circuits. In this paper we discuss certain properties of the oscillatory memristive circuits, including those with mixed-mode oscillations. Mathematical models of such circuits can be linked to the Newton’s law φ′′−F (t, φ, φ′)/m = 0, with φ denoting the flux or charge variables, m is a positive constant and the nonlinear non-autonomous function F (t, φ, φ′) contains memory terms. This leads further to scalar fourth-order ODEs called the jounce Newtonian equations. The jounce equations are used to construct the RC+op-amp simulation circuits in SPICE. Also, the linear parallel G-C and series R-L circuits with sinusoidal inputs are derived to match the rms values of the memristive periodic circuits.