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In this paper, a new chaotic memristor system is developed based on a well-known nonlinear circuit. By replacing the nonlinear positive conductance of Shinriki's circuit with a memristor emulator, the new simple chaotic circuit can reduce the number of active elements and reduce power consumption. An analysis of equilibria and stability, dissipative properties, Lyapunov exponents, Kaplan-Yorke dimensions, and bifurcation diagrams is presented as well as numerical simulation of the new memristor chaotic system. As a final step, the electronic circuit is designed using Multisim software, which is complemented by a physical realization to demonstrate the feasibility of the chaotic system. • A new chaotic memristor system based on a well-known nonlinear circuit is proposed. • The new memristor has the advantage of reducing the number of active elements and reducing power consumption accordingly. • The proposed chaotic system is studied by analyzing its equilibrium and stability, dissipative properties, Lyapunov exponents, Kaplan-Yorke dimension, and bifurcations diagrams. • Multisim software is used to design the electronic circuit, while a physical realization is provided to demonstrate the system's feasibility. [ABSTRACT FROM AUTHOR]

In the last decade, researchers, who work in the field of nonlinear circuits, have the "dream" to use a real memristor, which is the only nonlinear fundamental circuit element, in a new or other reported nonlinear circuit in literature, in order to experimentally investigate chaos. With this intention, for the first time, a well-known nonlinear circuit, in which its nonlinear element has been replaced with a commercially available memristor (KNOWM memristor), is presented in this work. Interesting phenomena concerning chaos theory, such as period-doubling route to chaos, coexisting attractors, one-scroll and double-scroll chaotic attractors are experimentally observed. [ABSTRACT FROM AUTHOR]

We introduce in this paper an equivalence notion for submersions U → ℝ , U open in ℝ 2 , which makes it possible to identify a smooth planar curve with a unique class of submersions. This idea, which extends to the nonlinear setting the construction of a dual projective space, provides a systematic way to handle global implicit descriptions of smooth planar curves. We then apply this framework to model nonlinear electrical devices as classes of equivalent functions. In this setting, linearization naturally accommodates incremental resistances (and other analogous notions) in homogeneous terms. This approach, combined with a projectively-weighted version of the matrix-tree theorem, makes it possible to formulate and address in great generality several problems in nonlinear circuit theory. In particular, we tackle unique solvability problems in resistive circuits, and discuss a general expression for the characteristic polynomial of dynamic circuits at equilibria. Previously known results, which were derived in the literature under unnecessarily restrictive working assumptions, are simply obtained here by using dehomogenization. Our results are shown to apply also to circuits with memristors. We finally present a detailed, graph-theoretic study of certain stationary bifurcations in nonlinear circuits using the formalism here introduced. [ABSTRACT FROM AUTHOR]

The transcritical bifurcation without parameters (TBWP) describes a stability change along a line of equilibria, resulting from the loss of normal hyperbolicity at a given point of such a line. Memristive circuits systematically yield manifolds of nonisolated equilibria, and in this paper we address a systematic characterization of the TBWP in circuits with a single memristor. To achieve this we develop two mathematical results of independent interest; the first is an extension of the TBWP theorem to explicit ordinary differential equations (ODEs) in arbitrary dimension; the second result drives the characterization of this phenomenon to semiexplicit differential-algebraic equations (DAEs), which provide the appropriate framework for the analysis of circuit dynamics. In the circuit context the analysis is performed in graph-theoretic terms: in this setting, our first working scenario is restricted to passive problems (an exception is made for the bifurcating memristor), and in a second step some results are presented for the analysis of nonpassive cases. The latter context is illustrated by means of a memristive neural network model. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive taxonomy of so-called second-order memory devices, which include charge-controlled memcapacitors and flux-controlled meminductors, among other novel circuit elements. These devices, which are classified according to their differential and state orders, are necessary to get a complete extension of the family of classical nonlinear circuit elements (resistors, capacitors, and inductors) for all possible controlling variables. Using a fully nonlinear formalism, we obtain nondegeneracy conditions for a broad class of second-order mem-circuits. This class of circuits is expected to yield a rich dynamic behavior; in this regard, we explore certain bifurcation phenomena exhibited by a family of circuits including a charge-controlled memcapacitor and a flux-controlled meminductor, providing some directions for future research. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The memory-resistor or memristor is a new electrical element characterized by a nonlinear charge-flux relation. This device poses many challenging problems, in particular from the circuit modeling point of view. In this paper, we address the index analysis of certain differential-algebraic models of memristive circuits; specifically, our attention is focused on so-called branch-oriented models, which include in particular tree-based formulations of the circuit equations. Our approach combines results coming from differential-algebraic equation theory, matrix analysis and theory of digraphs. This framework should be useful in future studies of dynamical aspects of memristive circuits. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

After the successful solid state implementation of the memristor, memristor-based circuits have received a lot of attention. In this paper, a memristor with cubic nonlinear characteristics is employed in the modified canonical Chua's circuit. A systematic study of hyperchaotic behavior in this circuit is performed with the help of nonlinear tools such as Lyapunov exponents, phase portraits and bifurcation diagrams. In particular, an imitative memristor circuit is examined to reveal the construction of hyperchaotic attractors. [ABSTRACT FROM AUTHOR]

The memory-resistor or memristor is a new electrical device governed by a nonlinear flux-charge relation. Its existence was predicted by Leon Chua in 1971, and the report in 2008 of a physical device with such a constitutive relation has driven a lot of attention to this circuit element. The memristor and related devices are expected to play a very relevant role in electronics in the near future, specially at the nanometer scale. The special form of the voltage-current characteristic, which reads as either v = M(q)i or i = W(ψ)v, implies that any equilibrium point is embedded into a center manifold of equilibria whose dimension is defined by the total number of memristors in the circuit. We characterize the normal hyperbolicity of these manifolds of equilibria in graph-theoretic terms. Moreover, when the assumptions supporting the normal hyperbolicity of such manifolds fail, the differential-algebraic nature of circuit models is shown to lead to certain bifurcations without parameters not exhibited by explicit ODEs. The results are illustrated by several circuit examples, some of which arise in the design of superconducting quantum bits based on the Josephson junction. [ABSTRACT FROM AUTHOR]

Abstract: The recent design of a nanoscale device with a memristive characteristic has had a great impact in nonlinear circuit theory. Such a device, whose existence was predicted by Leon Chua in 1971, is governed by a charge-dependent voltage–current relation of the form . In this paper, we show that allowing for a fully nonlinear characteristic in memristive devices provides a general framework for modeling and analyzing a very broad family of electrical and electronic circuits; Chua’s memristors are particular instances in which is linear in . We examine several dynamical features of circuits with fully nonlinear memristors, accommodating not only charge-controlled but also flux-controlled ones, with a characteristic of the form . Our results apply in particular to Chua’s memristive circuits; certain properties of these can be seen as a consequence of the special form of the elastance and reluctance matrices displayed by Chua’s memristors. [Copyright &y& Elsevier]

The recent discovery of a physical device behaving as a memristor has driven a lot of attention to memristive systems, which are likely to play a relevant role in electronics in the near future, especially at the nanometer scale. The derivation of explicit ODE models for these systems is important because it opens a way for the study of the dynamics of general memristive circuits, including e.g. stability aspects, oscillations, bifurcations or chaotic phenomena. We tackle this problem as a reduction of implicit ODE (differential-algebraic) models, and show how tree-based approaches can be adapted in order to accommodate memristors. Specifically, we prove that the derivation of a tree-based explicit ODE model is feasible for strictly passive memristive systems under broad coupling effects and without a priori current/voltage control assumptions on tree/cotree elements. Our framework applies in particular to topologically degenerate circuits and accommodates a wide class of controlled sources. We also discuss a quasilinear reduction of nonpassive problems, which do not admit an explicit ODE description in the presence of singularities; some related bifurcations are addressed in this context. [ABSTRACT FROM AUTHOR]