Memory effects are ubiquitous in nature and are particularly relevant at the nanoscale where the dynamical properties of electrons and ions strongly depend on the history of the system, at least within certain time scales. We review here the memory properties of various materials and systems which appear most strikingly in their non-trivial, time-dependent resistive, capacitative and inductive characteristics. We describe these characteristics within the framework of memristors, memcapacitors and meminductors, namely memory-circuit elements with properties that depend on the history and state of the system. We examine basic issues related to such systems and critically report on both theoretical and experimental progress in understanding their functionalities. We also discuss possible applications of memory effects in various areas of science and technology ranging from digital to analog electronics, biologically inspired circuits and learning. We finally discuss future research opportunities in the field.

Memory effects in complex materials and nanoscale systems

Using a systematic computer search, nine simple chaotic flows with quadratic nonlinearities were found that have the unusual feature of having a line equilibrium. Such systems belong to a newly introduced category of chaotic systems with hidden attractors that are important and potentially problematic in engineering applications.

Simple chaotic flows with a line equilibrium

An introduction to orthogonal polynomials, by Theodore S. Chihara. Pp xii, 249. £26·30. 1978. SBN 0 677 04150 0 (Gordon and Breach)

Memristor-based circuits are widely exploited to realize analog and/or digital systems for a broad scope of applications (e.g., amplifiers, filters, oscillators, logic gates, and memristor as synapses). A systematic methodology is necessary to understand complex nonlinear phenomena emerging in memristor circuits. The manuscript introduces a comprehensive analysis method of memristor circuits in the flux-charge $(\varphi,q)$ -domain. The proposed method relies on Kirchhoff Flux and Charge Laws and constitutive relations of circuit elements in terms of incremental flux and charge. The main advantages (over the approaches in the voltage-current $(v,i)$ -domain) of the formulation of circuit equations in the $(\varphi,q)$ -domain are: a) a simplified analysis of nonlinear dynamics and bifurcations by means of a smaller set of ODEs; b) a clear understanding of the influence of initial conditions. The straightforward application of the proposed method provides a full portrait of the nonlinear dynamics of the simplest memristor circuit made of one memristor connected to a capacitor. In addition, the concept of invariant manifolds permits to clarify how initial conditions give rise to bifurcations without parameters.

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Memristor Circuits: Flux—Charge Analysis Method

We study a four-dimensional system modified from a three-dimensional chaotic circuit by adding a memristor, which is a new fundamental electronic element with promising applications. Although the system has a line of infinitely many equilibria, our studies show that when the strength of the memristor increases, it can exhibit rich interesting dynamics, such as hyperchaos, long period-1 orbits, transient hyperchaos, as well as non-attractive behaviors frequently interrupting hyperchaos. To verify the existence of hyperchaos and reveal its mechanism, a horseshoe with two-directional expansion is studied rigorously in detail by the virtue of the topological horseshoe theory and the computer-assisted approach of a Poincare map. At last, the system is implemented with an electronic circuit for experimental verification. Copyright © 2013 John Wiley & Sons, Ltd.

Hyperchaos and horseshoe in a 4D memristive system with a line of equilibria and its implementation

The memristor is supposed to be the fourth fundamental electronic element in addition to the well-known resistor, inductor and capacitor. Named as a contraction for memory resistor, its theoretical...

Hopf bifurcation from lines of equilibria without parameters in memristor oscillators

We give necessary and sufficient conditions for all zeros of the palindromic polynomial R(z)=1+λ(z+z
 2+…+z
 n−1)+z
 n
 , $\lambda\in \mathbb{R}$
 , to be on the unit circle and we find $\gamma \in \mathbb{R}$
 for which S(z)=R(z)+γz
 n
 , n natural, has all its zeros in the closed unit disc.

Palindromic and perturbed polynomials: zeros location

In this paper we give necessary and sufficient conditions for all zeros of palindromic polynomial of even degree R(z) = 1 + (z + z2 + ::: + zn−1) + zn, with ∈ R, to be on the unit circle and we find  ∈ R for which S(z) = R(z)+ zn has all its zeros inside or on the unit circle.

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Some results on zeros of palindromic and perturbed polynomials of even degree

The main purpose of this paper is to investigate the distribution, simplicity and monotonicity of the roots of a class of trinomial equations that appears in certain financial mathematics problems. Furthermore, we show that it is possible to decompose some classes of trinomial equations into polynomials generated by a three term recurrence relation.

On the behavior of roots of trinomial equations

No estudo das questoes de estabilidade de metodos numericos e de fundamental importância o comportamento dos zeros do polinomio caracteristico associado ao metodo. Neste trabalho analisaremos uma conjectura que fornece aA0 estabilidade do metodo numerico e demonstraremos sua validade no caso em que o polinomio analisado e reflexivo.

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Vanessa Botta

Papers

Hopf bifurcation from lines of equilibria without parameters in memristor oscillators

Palindromic and perturbed polynomials: zeros location

Some results on zeros of palindromic and perturbed polynomials of even degree

On the behavior of roots of trinomial equations

Sobre uma Generalização do Teorema de Eneström-Kakeya: Polinômio Reflexivo