Hidden attractors in fundamental problems and engineering models

Recently a concept of self-excited and hidden attractors was suggested: an attractor is called a self-excited attractor if its basin of attraction overlaps with neighborhood of an equilibrium, otherwise it is called a hidden attractor. For example, hidden attractors are attractors in systems with no equilibria or with only one stable equilibrium (a special case of multistability and coexistence of attractors). While coexisting self-excited attractors can be found using the standard computational procedure, there is no standard way of predicting the existence or coexistence of hidden attractors in a system. In this plenary survey lecture the concept of self-excited and hidden attractors is discussed, and various corresponding examples of self-excited and hidden attractors are considered

Slow invariant manifold of heartbeat model

A new approach called Flow Curvature Method has been recently developed in a book entitled Differential Geometry Applied to Dynamical Systems. It consists in considering the trajectory curve, integral of any n-dimensional dynamical system as a curve in Euclidean n-space that enables to analytically compute the curvature of the trajectory - or the flow. Hence, it has been stated on the one hand that the location of the points where the curvature of the flow vanishes defines a manifold called flow curvature manifold and on the other hand that such a manifold associated with any n-dimensional dynamical system directly provides its slow manifold analytical equation the invariance of which has been proved according to Darboux theory. The Flow Curvature Method has been already applied to many types of autonomous dynamical systems either singularly perturbed such as Van der Pol Model, FitzHugh-Nagumo Model, Chua's Model, ...) or non-singularly perturbed such as Pikovskii-Rabinovich-Trakhtengerts Model, Rikitake Model, Lorenz Model,... More- over, it has been also applied to non-autonomous dynamical systems such as the Forced Van der Pol Model. In this article it will be used for the first time to analytically compute the slow invariant manifold analytical equation of the four-dimensional Unforced and Forced Heartbeat Model. Its slow invariant manifold equation which can be considered as a "state equation" linking all variables could then be used in heart prediction and control according to the strong correspondence between the model and the physiological cardiovascular system behavior.Comment: arXiv admin note: substantial text overlap with arXiv:1408.171

Modeling and design of memristor-based fuzzy systems

The incessant down scaling of CMOS technology has been the main driving force for the semiconductor industry over the past decades. Yet, as process variations and leakage current continue to exhibit more pronounced effect with every technology node, this down scaling paradigm is expected to saturate in the few coming years. This prospect has led the research community to seek new technologies to surpass those challenges. Amongst the promising candidates is the memristor technology recently characterized by HP Labs. The miniaturized features and the peculiar behavior exhibited by the memsitor make it very well suited in some applications. For instance, memrsitors are used as memory cells in state-of-the-art memories known as Resistive RAMs in which the non-volatility of the memristor is exploited. The programmable nature of the memristor has made it a powerful candidate in neuromorphic and fuzzy systems that, in essence, go beyond the classical Von Neumann computing paradigm. In such systems, ideas from Artificial Intelligence, that for so long have been implemented on the software level, are implemented as electronic circuitry which renders benefits such as compact area and reduced power consumption. This work focuses on memrsitor-based Fuzzy applications. First, memristor-based Min-Max circuit used in the Fuzzy Inference engine is analyzed. It is proven that memrsitor-based Min-Max circuits can be extended to an arbitrary number of inputs ‘N’ under the proper design constraints. In addition, the effect of the memristor threshold is analyzed and a closed form expression is derived. It is shown that, for a given memristor with a specific OFF resistance and threshold current, there is a trade-off between the size and the resolution of the circuit. Then, a memrsitor-based Defuzzifier circuit is proposed. A major challenge in Defuzzifiers is their area occupancy due to the use of Multiplier and Divider circuits. In this design, the memrsitor analog programmability is leveraged to reduce the multiplication operation into simple Ohm’s Law which alleviates the need for dedicated hardware for multiplier circuit and, accordingly, reduces the area occupancy

Memristor models for machine learning

In the quest for alternatives to traditional CMOS, it is being suggested that digital computing efficiency and power can be improved by matching the precision to the application. Many applications do not need the high precision that is being used today. In particular, large gains in area- and power efficiency could be achieved by dedicated analog realizations of approximate computing engines. In this work, we explore the use of memristor networks for analog approximate computation, based on a machine learning framework called reservoir computing. Most experimental investigations on the dynamics of memristors focus on their nonvolatile behavior. Hence, the volatility that is present in the developed technologies is usually unwanted and it is not included in simulation models. In contrast, in reservoir computing, volatility is not only desirable but necessary. Therefore, in this work, we propose two different ways to incorporate it into memristor simulation models. The first is an extension of Strukov's model and the second is an equivalent Wiener model approximation. We analyze and compare the dynamical properties of these models and discuss their implications for the memory and the nonlinear processing capacity of memristor networks. Our results indicate that device variability, increasingly causing problems in traditional computer design, is an asset in the context of reservoir computing. We conclude that, although both models could lead to useful memristor based reservoir computing systems, their computational performance will differ. Therefore, experimental modeling research is required for the development of accurate volatile memristor models.Comment: 4 figures, no tables. Submitted to neural computatio

Structural characterization of classical and memristive circuits with purely imaginary eigenvalues

The hyperbolicity problem in circuit theory concerns the existence of purely imaginary eigenvalues (PIEs) in the linearization of the time-domain description of the circuit dynamics. In this paper we characterize the circuit configurations which, in a strictly passive setting, yield purely imaginary eigenvalues for all values of the capacitances and inductances. Our framework is based on branch-oriented, semistate (differential-algebraic) circuit models which capture explicitly the circuit topology, and uses several notions and results from digraph theory. So-called P-structures arising in the analysis turn out to be the key element supporting our results. The analysis is shown to hold not only for classical (RLC) circuits but also for nonlinear circuits including memristors and other mem-devices

Symmetry in Chaotic Systems and Circuits

Symmetry can play an important role in the field of nonlinear systems and especially in the design of nonlinear circuits that produce chaos. Therefore, this Special Issue, titled “Symmetry in Chaotic Systems and Circuits”, presents the latest scientific advances in nonlinear chaotic systems and circuits that introduce various kinds of symmetries. Applications of chaotic systems and circuits with symmetries, or with a deliberate lack of symmetry, are also presented in this Special Issue. The volume contains 14 published papers from authors around the world. This reflects the high impact of this Special Issue