Complexity and abundant dynamics may arise in locally-active systems only,in which locally-active elements are essential to amplify infinitesimal fluctuation signals and maintain oscillating.It has been recently found that some mem-ristors may act as locally-active elements under suitable biasing.A number of important engineering applications would benefit from locally-active memristors.The aim of this paper is to show that locally-active memristor-based circuits can generate periodic and chaotic oscillations.To this end,we propose a non-volatile locally-active memristor,which has two asymptotically stable equilibrium points(or two non-volatile memristances)and globally-passive but locally-active char-acteristic.At an operating point in the locally-active region,a small-signal equivalent circuit is derived for describing the characteristics of the memristor near the operating point.By using the small-signal equivalent circuit,we show that the memristor possesses an edge of chaos in a voltage range,and that the memristor,when connected in series with an inductor,can oscillate about a locally-active operating point in the edge of chaos.And the oscillating frequency and the external inductance are determined by the small-signal admittance Y(iω).Furthermore,if the parasitic capacitor in parallel with the memristor is considered in the periodic oscillating circuit,the circuit generates chaotic oscillations.