On the action parameter and one-period loops of oscillatory memristive circuits

Abstract

We consider memelements (memristors and memductors) with special periodic responses (mixed-mode oscillations) and 2D one-period loops yielding constant parameters describing the memelements as single units or components of oscillating circuits. One of the parameters is the action parameter having the dimensions of energy $\times$ time and the SI unit Joule $\times$ second. The remaining loops and parameters correspond to energy of magnetic and electric fields, power and rms current and voltage values. Special mixed one-period loops are also analyzed with pairs of signals associated with two different components of the circuits. The areas enclosed by various loops result in special equations which can be derived from the underlying ODE model of the circuits. The action of a memelement is equivalent to the time integral of the Lagrangian $L(w,w')$, where w is the internal state variable of a memelement. The analysis of memristive circuits and their parameters is considered in the framework of mixed-mode oscillations. Also, the unit of action for memelements is proposed to be called Chua ($=$ Joule $\,\times$ second) to honor L.O. Chua for his work on memristors and memristive circuits.

Appendix

Equation (3) describes the memristive circuits with MMOs shown in Fig. 7. For bifurcation diagrams, more time-domain responses and pinched hysteresis loops see [7].

Fig. 7

Two dual memristive circuits described by (3) with $x=\eta \overline{x}$. a The M+CLRC circuit with memductance g(w) for $w=\phi$, b The M+LGCL circuit with memristance g(w) for $w=q$

Finally, suppose that we use [x(0), y(0), z(0), w(0)] as the nonzero initial conditions of (3). It is possible to transform system (3) into a singularly perturbed scalar ODE in variable w(t) and use it in a SPICE [22] realization of the circuits in Fig. 7a, b. The scalar equation is (the modified version of the second equation in (3) with $a_s$ is used as discussed in Sect. 2):

$$\begin{aligned}&\epsilon w''''+\{s_c(a+3bw^2)-\epsilon s_c\alpha K\}w'''\nonumber \\&\quad +\,\{ s_c^2\alpha (1-K(a+3bw^2))-\epsilon s_c^2\alpha \beta \nonumber \\&\quad +\,18s_cbww'\}w''-\left\{ s_c^3\alpha \beta (a+3bw^2)\right. \\&\quad \left. -\,6s_cbw'^2+6s_c^2\alpha b Kw w'\right\} w'=0.\nonumber \end{aligned}$$

(10)

In the process of deriving (10) we also obtain the following initial conditions in addition to w(0)

$$\begin{aligned}&w'(0)=s_c\gamma x(0)\nonumber \\&w''(0)=-s_c^2[y(0)+\gamma (a+3bw^2(0))x(0)]/\epsilon \nonumber \\&w'''(0)=\left\{ \left[ -s_c^3\alpha \gamma - s_c(a+3bw^2(0))\right. \right. \nonumber \\&\left. \quad -\,6s_c^3\gamma ^2bw(0)x(0)\right] x(0)\nonumber \\&\quad +\,(s_c^3\alpha K+s_c^3(a+3bw^2(0))y(0)/\epsilon \nonumber \\&\left. \quad +\,s_c^3\alpha z(0)\mp s_c^3\alpha a_s\right\} \big /\epsilon . \end{aligned}$$

(11)

A similar approach was successfully used in [17, 21] for a third-order jerk equation yielding MMOs in a circuit with a nonlinear element of third-degree polynomial current–voltage characteristic. Moreover, it was shown that some of the singularly perturbed jerk equations were Newtonian, since they were obtained after differentiation of the Newton’s second law of the type $w''=F(t,w,w')/m$ with m being constant. In such a situation, the $w''$ has the meaning of acceleration, $F(t,w,w')$ is a nonlinear force (with a memory term) and $w'''$, being derivative of $w''$ has the meaning of jerk. If the same analysis can be extended to (10), then $w''''$ will have the meaning of jounce, the second derivative of acceleration $w''$. Such an approach will link the memristive MMOs circuits with their equivalent mechanical realizations.