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The prediction made by L. O. Chua 45+ years ago (see: IEEE Trans. CircuitTheory (1971) 18:507-519 and also: Proc. IEEE (2012) 100:1920-1927) about theexistence of a passive circuit element (called memristor) that links the chargeand 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. Thissparked an enormous interest in mem-elements, analysis of their unusualdynamical properties (i.e. pinched hysteresis loops, memory effects, etc.) andconstruction of their emulators. Such topics are also of interest in mechanicalengineering where memdampers (or memory dampers) play the role equivalent tomemristors in electronic circuits. In this paper we discuss certain propertiesof the oscillatory memristive circuits, including those with mixed-modeoscillations. Mathematical models of such circuits can be linked to theNewton's law $\phi''\!-\!F(t,\phi,\phi')/m=0$, with $\phi$ denoting the flux orcharge variables, $m$ is a positive constant and the nonlinear non-autonomousfunction $F(t,\phi,\phi')$ contains memory terms. This leads further to scalarfourth-order ODEs called the jounce Newtonian equations. The jounce equationsare used to construct the $RC$+op-amp simulation circuits in SPICE. Also, thelinear parallel $G$-$C$ and series $R$-$L$ circuits with sinusoidal inputs arederived to match the $rms$ values of the memristive periodic circuits.

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