## La memfractance : un cadre mathématique pour une loi d’Ohm généralisée aux éléments de circuits électriques à mémoire (memristors)

University of Nice-Sophia Antipolis, France, Laboratoire J.A. Dieudonné, UMR 7351

In this joint work with M.-S. Abdelouahab and L. O. Chua, we define a common mathematical framework which includes the modelling of the behavior of the new electronic circuit elements with memory recently discovered. In this general frame we extend Ohm’s law to any case.

Circuit elements that store information without the need of a power source would represent a paradigm change in electronics, allowing for low-power computation and storage.

One such circuit element is the memory-resistor (memristor for short) which was postulated by Leon O. Chua, from Berkeley University (U.S.A.) in 1971 by analyzing mathematical relations between pairs of fundamental circuit variables. Considering that resistor links voltage to intensity (Ohm’s law), capacitor associates voltage to charge and inductor intensity to magnetic flux (defined mathematically as the time integral of the voltage, which needs not have a magnetic flux interpretation), a fourth element is missing: the one linking the charge and the flux. While resistors, capacitors, inductors are macroscopic elements, there is no obvious macroscopic device corresponding to memristors. In the years following Chua’s work, little research was directed towards the memristor concept until the birth of nanotechnology some years ago. In 2008 a group at the Helwett-Packard (HP) lab managed to construct a physical component acting as a memristor (in 2008) using thin films of TiO2. Co-developers HP and Hynix plan to bring memristor technology to market in summer 2014. The technology is eventually expected to replace flash, DRAM and even hard drives.

Turning back to his theoretical research on this device, 40 years after his genuine publication, L. O. Chua extended the notion of memristive systems to capacitive and inductive elements, namely, capacitors and inductors whose properties depend on state and history of the system.

Besides the discovery of these new electronic elements, by means of the mathematical concept of fractional derivative and the Laplace transform, some authors around ’90 eventually extended the generalized Ohm’s law (which links voltage to intensity in circuits encompassing resistors, capacitors and inductors) to any impedance, introducing the paradigm of fractance.

The most important conceptual difficulty raised by the generalization of fractance to memory elements, is the fact that it is necessary to use an interpolation of two parameters instead of one. Using such a two-parameter interpolation, we introduce Memfractance, a common mathematical framework which encompasses electronic circuit elements with or without memory (resistors, capacitors, inductors, memresistors, memcapacitors and meminductors). A new element, the memfractor, interpolates all these devices. Ultimately, in this new paradigm we generalize Ohm’s law allowing to model the behavior of every electronic device which can be built using nanotechnology.

Moreover, it has been recently shown and experimentally demonstrated that many nano scale existing memristive devices can behave chaotically. We will present some new results on chaotic dynamics of memfractor.