Differential Equations of Ideal Memristors

Ideal memristor is a resistor with a memory, which adds dynamics to its behavior. The most usual char- acteristics describing this dynamics are the constitutive relation (i.e. the relation between flux and charge), or Parameter-vs-state map (PSM), mostly represented by the memristance-to-charge dependence. One of the so far unheeded tools for memristor description is its differential equation (DEM), composed exclusively of instantaneous values of voltage, current, and their derivatives. The article derives a general form of DEM that holds for any ideal memristor and shows that it is always a nonlinear equation of the first order; the PSM forms are found for memristors which are governed by DEMs of the Bernoulli and the Riccati types; a classification of memristors according to the type of their dynamics with respect to voltage and cur- rent is carried out.

[1]  L.O. Chua,et al.  Memristive devices and systems , 1976, Proceedings of the IEEE.

[2]  R. Waser,et al.  Memristors: Devices, Models, and Applications , 2012 .

[3]  Ronald Tetzlaff,et al.  Abel Dynamics of Titanium Dioxide Memristor Based on Nonlinear Ionic Drift Model , 2011 .

[4]  van der Arjan Schaft.  Representing a nonlinear state space system as a set of higher-order differential equations in the inputs and outputs , 2002 .

[5]  A. V. D. Wegen.  Representing a nonlinear state space system as a set of higher-order differential equations in the inputs and outputs , 1989 .

[6]  L. Chua.  Memristor-The missing circuit element , 1971 .

[7]  D. Stewart,et al.  The missing memristor found , 2008, Nature.

[8]  Sung-Mo Kang,et al.  Memristors: Devices, Models, and Applications [Scanning the Issue] , 2012, Proc. IEEE.

[9]  Panayiotis S. Georgiou,et al.  A mathematical framework for the analysis and modelling of memristor nanodevices , 2013 .

[10]  Stephen J. Wolf,et al.  The elusive memristor: properties of basic electrical circuits , 2008, 0807.3994.

[11]  Leon O. Chua,et al.  If it’s pinched it’s a memristor , 2014 .

[12]  Dalibor Biolek,et al.  SPICE Model of Memristor with Nonlinear Dopant Drift , 2009 .

[13]  A. Polyanin,et al.  Handbook of Exact Solutions for Ordinary Differential Equations , 1995 .

[14]  Mauricio Barahona,et al.  On memristor ideality and reciprocity , 2014, Microelectron. J..

[15]  Dalibor Biolek,et al.  Analytical Solution of Circuits Employing Voltage- and Current-Excited Memristors , 2012, IEEE Transactions on Circuits and Systems I: Regular Papers.

[16]  D. Biolek,et al.  Reliable SPICE Simulations of Memristors, Memcapacitors and Meminductors , 2013, 1307.2717.

[17]  E.M. Drakakis,et al.  Memristors and Bernoulli dynamics , 2010, 2010 12th International Workshop on Cellular Nanoscale Networks and their Applications (CNNA 2010).