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This paper presents a simple oscillator using a battery and a second order memristor without the energy storage elements inductor and capacitor. The oscillating mechanism of the proposed circuit has been explained via Hopf bifurcation theorem, small signal model, local activity principle and edge of chaos theorem. This paper can be also used as a reference for explaining the intimate relationship between the super-critical Hopf bifurcation phenomenon and the edge of chaos.

This paper presents a detailed analysis of various oscillatory behaviors observed in relation to the calcium and potassium ions in the third-order Morris–Lecar model of giant barnacle muscle fiber. Since, both the calcium and potassium ions exhibit all of the characteristics of memristor fingerprints, we claim that the time-varying calcium and potassium ions in the third-order Morris–Lecar model are actually time-invariant calcium and potassium memristors in the third-order memristive Morris–Lecar model. We confirmed the existence of a small unstable limit cycle oscillation in both the second-order and the third-order Morris–Lecar model by numerically calculating the basin of attraction of the asymptotically stable equilibrium point associated with two subcritical Hopf bifurcation points. We also describe a comprehensive analysis of the generation of oscillations in third-order memristive Morris–Lecar model via small-signal circuit analysis and a subcritical Hopf bifurcation phenomenon.

The Chua Corsage Memristor is the simplest example of a passive but locally active memristor endowed with two asymptotically stable equilibrium points [Formula: see text] and [Formula: see text] when powered by an E-volt battery, where [Formula: see text]. The basin of attraction is defined by [Formula: see text], [Formula: see text] for [Formula: see text], and [Formula: see text], [Formula: see text] for [Formula: see text]. By adding an inductor of appropriate value [Formula: see text] in series with the battery, the resulting circuit undergoes a supercritical Hopf bifurcation and becomes an oscillator for [Formula: see text]. Applying a sinusoidal voltage source [Formula: see text] across the Chua corsage memristor, one finds two distinct coexisting stable periodic responses, depicted by their associated pinched hysteresis loops, of the same frequency [Formula: see text] whose basin of attraction is defined by [Formula: see text], and [Formula: see text], respectively, where [Formula: see text] depends on both amplitude A and frequency f. An in-depth and comprehensive analysis of the above global nonlinear phenomena is presented using tools from nonlinear circuit theory, such as Chua’s dynamic route method, and from nonlinear dynamics, such as phase portrait analysis and bifurcation theory.

An electronic oscillator circuit is designed by connecting an inductor in series with a locally-active PTC Memristor and a battery. The PTC Memristor is locally active on the negative resistance region of its DC [Formula: see text]–[Formula: see text] curve. A DC operating point [Formula: see text] is chosen on the locally-active region of the PTC Memristor and a small-signal equivalent circuit at [Formula: see text] is derived via Taylor series. The small-signal admittance [Formula: see text] of the composite one-port in Fig. 1 is derived using the small-signal equivalent circuit at [Formula: see text], in series with an inductor whose value is chosen such that [Formula: see text] at some [Formula: see text]. The sinusoidal oscillation computed numerically from this circuit is shown to emerge from a supercritical Hopf bifurcation.