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The study on stochastic effects occurring at the nanoscales in VLSI memristive devices is a timely topic which is gaining a growing interest. Indeed, these effects are often linked to the occurrence of nonlinear phenomena, such as spatiotemporal chaos. In this paper, we present evidence that spatiotemporal chaos in VLSI memristive systems can also be tamed by exploiting the unavoidable sources of noise affecting the dynamical behavior of the device. In particular, we aim at investigating this scenario starting from the results observed in several nonlinear oscillators, where the presence of noise facilitates their synchronization. In particular, we focus on the role of noise acting on initial conditions in memristive nonlinear oscillators. In this paper, we focus on a nonautonomous memristive oscillator characterizing its dynamics with respect to the memristor initial conditions and deriving a suitable model in which they appear as further system parameters and exploiting this dependence for the synchronization of an array of coupled nonlinear chaotic memristive oscillators. [ABSTRACT FROM AUTHOR]

We introduce a novel method revealing hidden bifurcations in the multispiral Chua attractor in the case where the parameter of bifurcation which determines the number of spiral is discrete. This method is based on the core idea of the genuine Leonov and Kuznetsov method for searching hidden attractors (i.e. applying homotopy and numerical continuation) but used in a very different way. Such hidden bifurcations are governed by a homotopy parameter whereas is maintained constant. This additional parameter which is absent from the initial problem is perfectly fitted to unfold the actual structure of the multispiral attractor. We study completely the multispiral Chua attractor, generated via sine function, and check numerically our method for odd and even values of from 1 to 12. In addition, we compare the shape of the attractors obtained for the same value of parameter while varying the parameter . [ABSTRACT FROM AUTHOR]

In this paper, we show that Hamilton's equations can be recast into the equations of dissipative memristor circuits. In these memristor circuits, the Hamiltonians can be obtained from the principles of conservation of "charge" and "flux", or the principles of conservation of "energy". Furthermore, the dynamics of memristor circuits can be recast into the dynamics of " ideal memristor" circuits. We also show that nonlinear capacitors are transformed into nonideal memristors if an exponential coordinate transformation is applied. Furthermore, we show that the zero-crossing phenomenon does not occur in some memristor circuits because the trajectories do not intersect the i = 0 axis. We next show that nonlinear circuits can be realized with fewer elements if we use memristors. For example, Van der Pol oscillator can be realized by only two elements: an inductor and a memristor. Chua's circuit can be realized by only three elements: an inductor, a capacitor, and a voltage-controlled memristor. Finally, we show an example of two-cell memristor CNNs. In this system, the neuron's activity depends partly on the supplied currents of the memristors. [ABSTRACT FROM AUTHOR]

This study presents a cellular-based mapping for a special class of dynamical systems for embedding neuron models, by exploiting an efficient memristor crossbar-based circuit for its implementation. The resultant reconfigurable memristive dynamical circuit exhibits various bifurcation phenomena, and responses that are characteristic of dynamical systems. High programmability of the circuit enables it to be applied to real-time applications, learning systems, and analytically indescribable dynamical systems. Moreover, its efficient implementation platform makes it an appropriate choice for on-chip applications and prostheses. We apply this method to the Izhikevich, and FitzHugh-Nagumo neuron models as case studies, and investigate the dynamical behaviors of these circuits. [ABSTRACT FROM AUTHOR]

We prove analytically that 2-element memristive circuits consisting of a passive linear inductor in parallel with a passive memristor, or an active memristive device, can be described explicitly by a Hamiltonian equation, whose solutions can be periodic or damped, and can be represented analytically by the constants of the motion along the circuit Hamiltonian. Generalizations to 3-element and 2N-element memristive Hamiltonian circuits are also presented where complex bifurcation phenomena including chaos, abound. [ABSTRACT FROM AUTHOR]

This 12th part of our Nonlinear Dynamics Perspective of Cellular Automata concludes a series of three articles devoted to CA local rules having robust periodic ω-limit orbits. Here, we consider only the two rules, $\kern.5pt{\fboxsep2pt\fbox{131}}\kern.5pt\xspace$ and $\kern.5pt{\fboxsep2pt\fbox{133}}\kern.5pt\xspace$, constituting the third of the six groups in which we classified the 1D binary Cellular Automata. Among the numerous theoretical results contained in this article, we emphasize the complete characterization of the ω-limit orbits, both robust and nonrobust, of these two rules and the proof that period-3 and period-6 ω-limit orbits are dense for $\kern.5pt{\fboxsep2pt\fbox{131}}\kern.5pt\xspace$ and $\kern.5pt{\fboxsep2pt\fbox{133}}\kern.5pt\xspace$, respectively. Furthermore, we will also introduce the fundamental concepts of perfect period-T orbitsets and riddled basins, and see how they emerge in rule $\kern.5pt{\fboxsep2pt\fbox{131}}\kern.5pt\xspace$. As stated in the title, we also focus on permutive rules, which have been introduced in a previous installment of our series but never thoroughly studied. Indeed, we will review some of the well-known properties of such rules, like the surjectivity, examining their implications for finite and bi-infinite Cellular Automata. Finally, we propose a new list of the 88 globally-independent local rules, which is slightly different from the one we have used so far but has the great advantage of being selected via a rigorous methodology and not an arbitrary choice. For the sake of completeness, we display in the appendix the basin tree diagrams and the portraits of the ω-limit orbits of the rules from this refined table which have not yet been reported in our previous articles. [ABSTRACT FROM AUTHOR]

The 11th part of our tour through one-dimensional binary Cellular Automata concerns period-2 rules, which form the second group in our classification of the 88 globally-independent CA rules according to the properties of their periodic orbits. In this article, we display the basin tree diagrams of all period-2 rules along with their time-2 characteristic functions, and then we prove that all rules belonging to group 2 have robust period-2 ω-limit orbits for any finite, and infinite, bit string length. This rigorous result, which pairs with the one about period-1 rules given in the tenth installment of our chronicle, confirms what we stated about period-2 rules on the basis of empirical evidence. In the second part of this tutorial, we introduce the notion of quasi global-equivalence and prove that there are only 82 quasi globally-independent CA rules. For the first time, we show that the space-time patterns of globally-independent local rules can depend on each other, and we present an example of quasi-global transformation. We also define the super string 턞, and its unique decimal representation x턞, dubbed the super decimal, which provides a completely transparent yet rigorous proof that rule $\kern.5pt{\fboxsep2pt\fbox{170}}\kern.5pt\xspace$ is chaotic when L → ∞. Moreover, we present the basin tree generation formulas, which uncover the analytical relationships between basin trees of globally-equivalent rules. Last but not least, for pedagogical and epistemological reasons, we conclude this paper with the selection of rule$\kern.5pt{\fboxsep2pt\fbox{137}}\kern.5pt\xspace$, instead of rule $\kern.5pt{\fboxsep2pt\fbox{110}}\kern.5pt\xspace$, as the prototypic universal Turing machine for our future discourse. [ABSTRACT FROM AUTHOR]

Many scientists have struggled to uncover the elusive origin of "complexity", and its many equivalent jargons, such as emergence, self-organization, synergetics, collective behaviors, nonequilibrium phenomena, etc. They have provided some qualitative, but not quantitative, characterizations of numerous fascinating examples from many disciplines. For example, Schrödinger had identified "the exchange of energy" from open systems as a necessary condition for complexity. Prigogine has argued for the need to introduce a new principle of nature which he dubbed "the instability of the homogeneous". Turing had proposed "symmetry breaking" as an origin of morphogenesis. Smale had asked what "axiomatic" properties must a reaction–diffusion system possess to make the Turing interacting system oscillate. The purpose of this paper is to show that all the jargons and issues cited above are mere manifestations of a new fundamental principle called local activity, which is mathematically precise and testable. The local activity theorem provides the quantitative characterization of Prigogine's "instability of the homogeneous" and Smale's quest for an axiomatic principle on Turing instability. Among other things, a mathematical proof is given which shows none of the complexity-related jargons cited above is possible without local activity. Explicit mathematical criteria are given to identify a relatively small subset of the locally-active parameter region, called the edge of chaos, where most complex phenomena emerge. [ABSTRACT FROM AUTHOR]

Introduces a real-time programmable, spatial-temporal bifurcation test bed. Presentation of the experimental analysis of an antisymmetric template class; Production of spatial-temporal patterns that exhibit complex dynamics; Dependence of the characteristic of the propagating patterns on the self-feedback and on the sign of the coupling below the self-feedback template element.

This paper presents analytical criteria for local activity in reaction-diffusion Cellular Nonlinear Network (CNN) cells [Chua, 1997, 1999] with four local state variables. As a first application, we apply the criteria to a Hodgkin-Huxley CNN, which has cells defined by the equations of the cardiac Purkinje fiber model of morphogenesis that was first introduced in [Noble, 1962] to describe the long-lasting action and pace-maker potentials of the Purkinje fiber of the heart. The bifurcation diagrams of the Hodgkin-Huxley CNN's supply a possible explanation for why a heart with a normal heart-rate may stop beating suddenly: The cell parameter of a normal heart is located in a locally active unstable domain and just nearby an edge of chaos. The membrane potential along a fiber is simulated in a Hodgkin-Huxley CNN by a computer. As a second application, we present a smoothed Chua's circuit (SCC) CNN. The bifurcation diagrams of the SCC CNN's show that there does not exist a locally passive domain, and the edges of chaos corresponding to different fixed-cell parameters are significantly different. Our computer simulations show that oscillatory patterns, chaotic patterns, or divergent patterns may emerge if the selected cell parameters are located in locally active domains but nearby the edge of chaos. This research demonstrates once again the effectiveness of the local activity theory in choosing the parameters for the emergence of complex (static and dynamic) patterns in a homogeneous lattice formed by coupled locally active cells. [ABSTRACT FROM AUTHOR]

In the present work explicit formulas for analyzing the birth of limit cycles arising in the Chua's circuit through a Hopf bifurcation is provided. A local amplitude equation is derived using a frequency domain approach and harmonic balance approximations. Furthermore, the first Lyapunov index used to detect degenerate Hopf bifurcations is derived in terms of the parameters of the nonlinear circuit. A perspective for analyzing other bifurcations using this frequency domain approach is discussed. [ABSTRACT FROM AUTHOR]

The dynamic behavior of the standard Cellular Neural Network (CNN) was studied via explicit solutions of the CNN vector state equations for certain piecewise linear domains Ω. The concepts of Ω-globally attractive domain, Ω-globally attractive equilibrium point, and Ω-globally expelling domain for the CNN state equations are presented. The terminology "Ω-global" indicates that the property is global relative to a domain Ω. It is shown that every "saturated domain" D (i.e. for every vector X ∈ D, each corresponding component of X has the same sign as all others and its absolute value is greater than or equal to 1), is either Ω-globally attractive or Ω-globally expelling. Attractive equilibrium points in another special domain were also studied. Sufficient conditions for convergence to such equilibria are given, along with concrete new computer simulation examples which demonstrate this theory. Finally, several definitions and simulation results are given to explicate the complex dynamical relationships which were observed between initial states in the linear domain Ω(0) (i.e. every vector X ∈ Ω(0) has the property that the absolute value of each component is less that l) and Ω-globally attractive equilibrium points in the saturated domain. [ABSTRACT FROM AUTHOR]