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Professor Oleksandr M. Sharkovsky (December 7, 1936–November 21, 2022) was an honorary editorial board member of the International Journal of Bifurcation and Chaos. He left to the scientific world a famous Sharkovsky theorem established in his 1964 paper "Coexistence of Cycles of a Continuous Map of the Line into Itself," which laid the foundation of a new branch in the theory of dynamical systems — combinatorial dynamics. His contributions also include the Sharkovsky ordering, Sharkovsky space, Sharkovsky set, Sharkovsky stratification, and maximum period in the sense of Sharkovsky, among others. Professor Sharkovsky will be remembered forever. [ABSTRACT FROM AUTHOR]

In this paper, we show that parasitic elements have a significant effect on the dynamics of memristor circuits. We first show that certain -terminal elements such as memristors, memcapacitors, and meminductors can be used as nonvolatile memories, if the principle of conservation of state variables hold by open-circuiting, or short-circuiting, their terminals. We also show that a passive memristor with a strictly-increasing constitutive relation will eventually lose its stored flux when we switch off the power if there is a parasitic capacitance across the memristor. Similarly, a memcapacitor (resp., meminductor) with a positive memcapacitance (resp., meminductance) will eventually lose their stored physical states when we switch off the power, if it is connected to a parasitic resistance. We then show that the discontinuous jump that circuit engineers assumed to occur at impasse points of memristor circuits contradicts the principles of conservation of charge and flux at the time of the discontinuous jump. A parasitic element can be used to break an impasse point, resulting in the emergence of a continuous oscillation in the circuit. We also define a distance, a diameter, and a dimension, for each circuit element in order to measure the complexity order of the parasitic elements. They can be used to find higher-order parasitic elements which can break impasse points. Furthermore, we derived a memristor-based Chua's circuit from a three-element circuit containing a memristor by connecting two parasitic memcapacitances to break the impasse points. We finally show that a higher-order parasitic element can be used for breaking the impasse points on two-dimensional and three-dimensional constrained spaces. [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]

Abstract: We designed Adaptive Neuromorphic Architecture (ANA) that self-adjusts its inherent parameters (for instance, the resonant frequency) naturally following the stimuli frequency. Such an architecture is required for brain-like engineered systems because some parameters of the stimuli (for instance, the stimuli frequency) are not known in advance. Such adaptivity comes from a circuit element with memory, namely mem-inductor or mem-capacitor (memristor’s sisters), which is history-dependent in its behavior. As a hardware model of biological systems, ANA can be used to adaptively reproduce the observed biological phenomena in amoebae. [Copyright &y& Elsevier]

In this paper, we show that the dynamics of any memristor circuits can be simulated by a corresponding "dual" nonlinear RLC circuit where the memristor is substituted by a nonlinear resistor. They are in one-to-one correspondence, that is, they are duals of each other. We also propose a method for synchronizing these dual dynamic nonlinear circuits. We next define memory elements which can be characterized by charge and flux. The memory-element circuits can also be simulated by their corresponding dual nonlinear RLC circuits. We then define 2-terminal elements which are characterized by complementary pair of signals, and study them from the view point of one-to-one correspondence. We finally show an example of 2-terminal elements such that the terminal voltage and current are identical, and their time-derivatives of any order are also identical, however, their time-integrals are different. That is, these 2-terminal elements are in one-to-one correspondence except for the time-integral signals. [ABSTRACT FROM AUTHOR]

We study two-dimensional cellular automata, each cell takes three states: resting, excited and refractory. A resting cell excites if the number of excited neighbors lies in a certain interval (excitation interval). An excited cell becomes refractory independently on states of its neighbors. A refractory cell returns to a resting state only if the number of excited neighbors belong to recovery interval. The model is an excitable cellular automaton abstraction of a spatially extended semi-memristive medium where a cell's resting state symbolizes low-resistance and refractory state high-resistance. The medium is semi-memristive because only transition from high- to low-resistance is controlled by the density of local excitation. We present a phenomeno-logical classification of the automata behavior for all possible excitation intervals and recovery intervals. We describe eleven classes of cellular automata with retained refractoriness based on the criteria of space-filling ratio, morphological and generative diversity, and types of traveling localizations. [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]

An autoassociative memory is a device which accepts an input pattern and generates an output as the stored pattern which is most closely associated with the input. In this paper, we propose an autoassociative memory cellular neural network, which consists of one-dimensional cells with spatial derivative inputs, thresholds and memories. Computer simulations show that it exhibits good performance in face recognition: The network can retrieve the whole from a part of a face image, and can reproduce a clear version of a face image from a noisy one. For human memory, research on "visual illusions" and on "brain damaged visual perception", such as the Thatcher illusion, the hemispatial neglect syndrome, the split-brain, and the hemispheric differences in recognition of faces, has fundamental importance. We simulate them in this paper using an autoassociative memory cellular neural network. Furthermore, we generate many composite face images with spurious patterns by applying genetic algorithms to this network. We also simulate a morphing between two faces using autoassociative memory. [ABSTRACT FROM AUTHOR]

Over the past eight years, we have studied one of the simplest, yet extremely interesting, dynamical systems; namely, the one-dimensional binary Cellular Automata. The most remarkable results have been presented in a series of papers which is concluded by the present article. The final stop of our odyssey is devoted to the analysis of the second half of the 30 Bernoulli στ-shift rules, which constitute the largest among the six groups in which we classified the 256 local rules. For all these 15 rules, we present the basin-tree diagrams obtained by using each bit string with L ≤ 8 as initial state, a summary of the characteristics of their ω-limit orbits, and the space-time patterns generated from the superstring. Also, in the last section we summarize the main results we obtained by means of our "nonlinear dynamics perspective". [ABSTRACT FROM AUTHOR]

More than one third of the 88 globally-independent Cellular Automata rules exhibit robust simple Bernoulli-shift dynamics. Among them we find rule $\kern.5pt{\fboxsep2pt\fbox{170}}\kern.5pt\xspace$, which we proved to be chaotic in the previous episodes of our chronicle, and rule $\kern.5pt{\fboxsep2pt\fbox{184}}\kern.5pt\xspace$, the famous global majority rule. Therefore, we cannot overstate the importance of the Bernoulli στ-shift rules which we will present in two parts of our continuing odyssey on the Nonlinear Dynamics Perspective of Cellular Automata. This paper covers the first 15 of the 30 Bernoulli στ-shift rules. In this paper, after recalling the main concepts of Bernoulli rules — such as the role of the three Bernoulli parameters σ, τ and β — we will display the basin tree diagrams of these rules together with a convenient summary of the results extracted from them. Then, we will show that the superstring is an excellent testing signal to find the robust behavior of a given rule. Finally, we will conclude this paper with a discussion about the difference between robust and nonrobust ω-limit orbits of the Bernoulli στ-shift rules. [ABSTRACT FROM AUTHOR]

A chaotic attractor has been observed with an autonomous circuit that uses only two energy-storage elements: a linear passive inductor and a linear passive capacitor. The other element is a nonlinear active memristor. Hence, the circuit has only three circuit elements in series. We discuss this circuit topology, show several attractors and illustrate local activity via the memristor's DC vM - iM characteristic. [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]

In this paper, we design a cellular automaton and a discrete-time cellular neural network (DTCNN) using nonlinear passive memristors. They can perform a number of applications, such as logical operations, image processing operations, complex behaviors, higher brain functions, RSA algorithm, etc. By modifying the characteristics of nonlinear memristors, the memristor DTCNN can perform almost all functions of memristor cellular automaton. Furthermore, it can perform more than one function at the same time, that is, it allows multitasking. [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]

This stage of our journey through the universe of one-dimensional binary Cellular Automata is devoted to period-1 rules, constituting the first of the six groups in which we systematized the 88 globally-independent CA rules. The first part of this article is mainly dedicated to reviewing the terminology and the empirical results found in the previous papers of our quest. We also introduce the concept of the ω-limit orbit with the purpose of linking our work to the classical theory of nonlinear dynamical systems. Moreover, we present the basin tree diagrams of all period-1 rules — except for rule $\kern.5pt{\fboxsep2pt\fbox{0}}\kern.5pt\xspace$, which is trivial — along with their Boolean cubes and time-1 characteristic functions. In the second part, we prove a theorem demonstrating that all rules belonging to group 1 have robust period-1 rules for any finite, and infinite, bit-string length L. This is the first time we give analytical results on the behavior of CA local rules for large values of L and, consequently, for bi-infinite bit strings. The theoretical treatment is complemented by two remarkable practical results: an explicit formula for generating isomorphic basin trees, and an algorithm for creating new periodic orbits by concatenation. We also provide several examples of both of them, showing how they help to avoid tedious simulations. [ABSTRACT FROM AUTHOR]

In this paper, we propose new difference equations which can generate the evolution rules of cellular automata. [ABSTRACT FROM AUTHOR]

Visual illusion is the fallacious perception of reality or some actually existing object. In this paper, we imitate the mechanism of Ehrenstein illusion, neon color spreading illusion, watercolor illusion, Kanizsa illusion, shifted edges illusion, and hybrid image illusion using the Open Source Computer Vision Library (OpenCV). We also imitate these illusions using Cellular Neural Networks (CNNs). These imitations suggest that some illusions are processed by high-level brain functions. We next apply the morphological gradient operation to anomalous motion illusions. The processed images are classified into two kinds of images, which correspond to the central drift illusion and the peripheral drift illusion, respectively. It demonstrates that the contrast of the colors plays an important role in the anomalous motion illusion. We also imitate the anomalous motion illusions using both OpenCV and CNN. These imitations suggest that some visual illusions may be processed by the illusory movement of animations. [ABSTRACT FROM AUTHOR]

The memristor has attracted phenomenal worldwide attention since its debut on 1 May 2008 issue of Nature in view of its many potential applications, e.g. super-dense nonvolatile computer memory and neural synapses. The Hewlett–Packard memristor is a passive nonlinear two-terminal circuit element that maintains a functional relationship between the time integrals of current and voltage, respectively, viz. charge and flux. In this paper, we derive several nonlinear oscillators from Chua's oscillators by replacing Chua's diodes with memristors. [ABSTRACT FROM AUTHOR]

The concept of isles of Eden when generalized from finite cellular automata to classical dynamical systems on ℝd calls the attention to the ZUK Theorem — Z for [Zubov, 1957], U and K for [Ura & Kimura, 1960] — a fifty years old and almost forgotten result in topological dynamics, a classification theorem for compact isolated invariant sets. We provide a simple proof and discuss various consequences including a new result on pointwise Liapunov functions. [ABSTRACT FROM AUTHOR]

Our scientific odyssey through the theory of 1-D cellular automata is enriched by the definition of quasi-ergodicity, a new empirical property discovered by analyzing the time-1 return maps of local rules. Quasi-ergodicity plays a key role in the classification of rules into six groups: in fact, it is an exclusive characteristic of complex and hyper Bernoulli-shift rules. Besides introducing quasi-ergodicity, this paper answers several questions posed in the previous chapters of our quest. To start with, we offer a rigorous explanation of the fractal behavior of the time-1 characteristic functions, finding the equations that describe this phenomenon. Then, we propose a classification of rules according to the presence of Isles of Eden, and prove that only 28 local rules out of 256 do not have any of them; this result sheds light on the importance of Isles of Eden. A section of this paper is devoted to the characterization of Bernoulli basin-tree diagrams through modular arithmetic; the formulas obtained allow us to shorten drastically the number of cases to take into consideration during numerical simulations. Last but not least, we present some theorems about additive rules, including an analytical explanation of their scale-free property. [ABSTRACT FROM AUTHOR]

A new circuit configuration, linearly conjugate to the standard Chua's circuit, is presented. Its distinctive feature is that the equations now admit an additional parameter, which controls the dissipation in the network connected to the Chua diode. In the limiting case we obtain the simplest chaotic circuit, consisting of a piecewise-linear resistor and three lossless elements. [ABSTRACT FROM AUTHOR]

This paper presents the basin tree diagrams of all hyper Bernoulli στ-shift rules for string lengths L = 3, 4, ..., 8. These diagrams have revealed many global and time-asymptotic properties that we have subsequently proved to be true for all L < ∞. In particular, we have proved that local rule $\kern.5pt{\fboxsep2pt\fbox{60}}\kern.5pt\xspace$ has no Isles of Eden for all L, and that local rules $\kern.5pt{\fboxsep2pt\fbox{154}}\kern.5pt\xspace$ and $\kern.5pt{\fboxsep2pt\fbox{45}}\kern.5pt\xspace$ are inhabited by a dense set (continuum) of Isles of Eden if, and only if, L is an odd integer. A novel and powerful graph-theoretic tool, called Isles-of-Eden digraph, has been developed and can be used to test the existence of dense Isles of Eden of any local rule which satisfies certain constraints, such as rules $\kern.5pt{\fboxsep2pt\fbox{154}}\kern.5pt\xspace$, $\kern.5pt{\fboxsep2pt\fbox{45}}\kern.5pt\xspace$, $\kern.5pt{\fboxsep2pt\fbox{150}}\kern.5pt\xspace$, $\kern.5pt{\fboxsep2pt\fbox{105}}\kern.5pt\xspace$, as well as all invariant local rules, such as rules $\kern.5pt{\fboxsep2pt\fbox{170}}\kern.5pt\xspace$, $\kern.5pt{\fboxsep2pt\fbox{240}}\kern.5pt\xspace$, $\kern.5pt{\fboxsep2pt\fbox{15}}\kern.5pt\xspace$ and $\kern.5pt{\fboxsep2pt\fbox{85}}\kern.5pt\xspace$, subject to no constraints. [ABSTRACT FROM AUTHOR]

This paper continues our quest to develop a rigorous analytical theory of 1-D cellular automata via a nonlinear dynamics perspective. The 18 yet uncharacterized local rules are henceforth partitioned into ten complex Bernoulliστ-shift rules and eight hyper Bernoulliστ-shift rules, the latter including such famous rules $\kern.5pt{\fboxsep2pt\fbox{30}}\kern.5pt\xspace$ and $\kern.5pt{\fboxsep2pt\fbox{110}}\kern.5pt\xspace$. All exhibit a bizarre composite wave dynamics with arbitrarily large Bernoulli velocity σ and Bernoulli return time τ as the length L → ∞. Basin tree diagrams of all ten complex Bernoulli στ-shift rules are exhibited for lengths L = 3, 4, ..., 8. Superficial as it may seem, these basin tree diagrams suggest general qualitative properties which have since been proved to be true in general. Two such properties form the main results of this paper; namely, Explicit global state transition formulas are given for local rules $\kern.5pt{\fboxsep2pt\fbox{90}}\kern.5pt\xspace$, $\kern.5pt{\fboxsep2pt\fbox{105}}\kern.5pt\xspace$ and $\kern.5pt{\fboxsep2pt\fbox{150}}\kern.5pt\xspace$. Such formulas led to the rigorous proof of several surprising periodicity constraints for rule $\kern.5pt{\fboxsep2pt\fbox{90}}\kern.5pt\xspace$, and to the discovery of a new global, quasi-equivalence class, defined via an alternating transformation. In particular, local rules $\kern.5pt{\fboxsep2pt\fbox{105}}\kern.5pt\xspace$ and $\kern.5pt{\fboxsep2pt\fbox{150}}\kern.5pt\xspace$are globally quasi-equivalent where corresponding space-time patterns can be derived from each other by simply complementing every other row. Another important result of this paper is the discovery of a scale-free phenomenon exhibited by the local rules $\kern.5pt{\fboxsep2pt\fbox{90}}\kern.5pt\xspace$, $\kern.5pt{\fboxsep2pt\fbox{105}}\kern.5pt\xspace$ and $\kern.5pt{\fboxsep2pt\fbox{150}}\kern.5pt\xspace$. In particular, the period "T" of all attractors of rules $\kern.5pt{\fboxsep2pt\fbox{90}}\kern.5pt\xspace$, $\kern.5pt{\fboxsep2pt\fbox{105}}\kern.5pt\xspace$ and $\kern.5pt{\fboxsep2pt\fbox{150}}\kern.5pt\xspace$, as well as of all isles of Eden of rules $\kern.5pt{\fboxsep2pt\fbox{105}}\kern.5pt\xspace$ and $\kern.5pt{\fboxsep2pt\fbox{150}}\kern.5pt\xspace$, increases linearly with unit slope, in logarithmic scale, with the length L. [ABSTRACT FROM AUTHOR]

The primary purpose of this paper is to show that simple dissipation can bring about oscillations in certain kinds of asymptotically stable nonlinear dynamical systems; namely when the system is locally active where the dissipation is introduced. Furthermore, if these nonlinear dynamical systems are coupled with appropriate choice of diffusion coefficients, then the coupled system can exhibit spatio-temporal oscillations. The secondary purpose of this paper is to show that spatio-temporal oscillations can occur in spatially discrete reaction diffusion equations operating on the edge of chaos, provided the array size is sufficiently large. [ABSTRACT FROM AUTHOR]

Many useful and well-known image processing templates for cellular neural networks (CNN's) can be derived from neural field models, thereby providing a neural basis for the CNN paradigm. The potential ability of multitasking image processing is investigated by using these templates. Many visual illusions are simulated via CNN image processing. The ability of the CNN to mimic such high-level brain functions suggests possible applications of the CNN in cognitive engineering. Furthermore, two kinds of painting-like image processings, namely, texture generation and illustration style transformation are investigated. [ABSTRACT FROM AUTHOR]

Chaotic nonlinear networks are investigated, which are controlled by simple boids rules. They exhibit complex and emergent behaviors such as flocking behavior, separation behavior, joining behavior and obstacle avoiding behavior. [ABSTRACT FROM AUTHOR]

The emergence of complexity is investigated from the viewpoint of the energy balance property and the divergence property of reaction–diffusion cellular neural networks. [ABSTRACT FROM AUTHOR]

This paper proves, via an analytical approach, that 170 (out of 256) Boolean CA rules in a one-dimensional cellular automata (CA) are time-reversible in a generalized sense. The dynamics on each attractor of a time-reversible rule N is exactly mirrored, in both space and time, by its bilateral twin ruleN†. In particular, all 69 period-1 rules, 17 (out of 25) period-2 rules, and 84 (out of 112) Bernoulli rules are time-reversible. The remaining 86 CA rules are time-irreversible in the sense that N and N† mirror their dynamics only in space, but not in time. In this case, each attractor of N defines a unique arrow of time. A simple "time-reversal test" is given for testing whether an attractor of a CA rule is time-reversible or time-irreversible. For a time-reversible attractor of a CA rule N the past can be uniquely recovered from the future of N†, and vice versa. This remarkable property provides 170 concrete examples of CA time machines where time travel can be routinely achieved by merely hopping from one attractor to its bilateral twin attractor, and vice versa. Moreover, the time-reversal property of some local rules can be programmed to mimic the matter–antimatter "annihilation" or "pair-production" phenomenon from high-energy physics, as well as to mimic the "contraction" or "expansion" scenarios associated with the Big Bang from cosmology. Unlike the conventional laws of physics, which are based on a unique universe, most CA rules have multiple universes (i.e. attractors), each blessed with its own laws. Moreover, some CA rules are endowed with both time-reversible attractors and time-irreversible attractors. Using an analytical approach, the time-τ return map of each Bernoulli στ-shift attractor of all 112 Bernoulli rules are shown to obey an ultra-compact formula in closed form, namely,. . or its inverse map. These maps completely characterize the time-asymptotic (steady state) behavior of the nonlinear dynamics on the attractors. In-depth analysis of all but 18 global equivalence classes of CA rules have been derived, along with their basins of attraction, which characterize their transient regimes. Above all, this paper provides a rigorous nonlinear dynamics foundation for a paradigm shift from an empirical-based approach à la Wolfram to an attractor-based analytical theory of cellular automata. [ABSTRACT FROM AUTHOR]

This fifth installment is devoted to an in-depth study of CA Characteristic Functions, a unified global representation for all 256 one-dimensional Cellular Automata local rules. Except for eight rather special local rules whose global dynamics are described by an affine (mod 1) function of only one binary cell state variable, all characteristic functions exhibit a fractal geometry where self-similar two-dimensional substructures manifest themselves, ad infinitum, as the number of cells (I + 1) → ∞. In addition to a complete gallery of time-1 characteristic functions for all 256 local rules, an accompanying table of explicit formulas is given for generating these characteristic functions directly from binary bit-strings, as in a digital-to-analog converter. To illustrate the potential applications of these fundamental formulas, we prove rigorously that the "right-copycat" local rule ${\fbox{170}}$ is equivalent globally to the classic "left-shift" Bernoulli map. Similarly, we prove the "left-copycat" local rule ${\fbox{240}}$ is equivalent globally to the "right-shift" inverse Bernoulli map. Various geometrical and analytical properties have been identified from each characteristic function and explained rigorously. In particular, two-level stratified subpatterns found in most characteristic functions are shown to emerge if, and only if, b1 ≠ 0, where b1 is the "synaptic coefficient" associated with the cell differential equation developed in Part I. Gardens of Eden are derived from the decimal range of the characteristic function of each local rule and tabulated. Each of these binary strings has no predecessors (pre-image) and has therefore no past, but only the present and the future. Even more fascinating, many local rules are endowed with binary configurations which not only have no predecessors, but are also fixed points of the characteristic functions. To dramatize that such points have no past, and no future, they are henceforth christened "Isles of Eden". They too have been identified and tabulated. [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]

CNN templates for image processing and pattern formation are derived from neural field equations, advection equations and reaction–diffusion equations by discretizing spatial integrals and derivatives. Many useful CNN templates are derived by this approach. Furthermore, self-organization is investigated from the viewpoint of divergence of vector fields. [ABSTRACT FROM AUTHOR]

Chua's diode is widely used in nonlinear circuit analysis and modeling. In this paper, we propose canonical oscillator models having hysteretic Chua's diodes as their nonlinear elements. Many hysteresis oscillators can be classified into unified groups by using these models. We also present a method for deriving new hysteresis oscillators from known slow–fast (relaxation) oscillators. [ABSTRACT FROM AUTHOR]

By exploiting the new concepts of CA characteristic functions and their associated attractor time-τ maps, a complete characterization of the long-term time-asymptotic behaviors of all 256 one-dimensional CA rules are achieved via a single "probing" random input signal. In particular, the graphs of the time-1 maps of the 256 CA rules represent, in some sense, the generalized Green's functions for Cellular Automata. The asymptotic dynamical evolution on any CA attractor, or invariant orbit, of 206 (out of 256) CA rules can be predicted precisely, by inspection. In particular, a total of 112 CA rules are shown to obey a generalized Bernoulli στ-shift rule, which involves the shifting of any binary string on an attractor, or invariant orbit, either to the left, or to the right, by up to 3 pixels, and followed possibly by a complementation of the resulting bit string. The most intriguing result reported in this paper is the discovery that the four Turing-universal rules $\fbox{110}$, $\fbox{124}$, $\fbox{137}$, and $\fbox{193}$, and only these rules, exhibit a 1/f power spectrum. [ABSTRACT FROM AUTHOR]

In this paper, we propose a multipurpose hysteresis CNN (cellular neural network) made of first-order cells with hysteresis switches. The hysteresis CNN has applications not only in image processing, but also in pattern formation, nonlinear wave propagation and associative and dynamic memories, because each hysteresis CNN cell has two operating modes, namely, a bistable multivibrator mode and a relaxation oscillator mode. [ABSTRACT FROM AUTHOR]

We prove rigorously the four cellular automata local rules 110, 124, 137 and 193 have identical dynamic behaviors capable of universal computations. We exploit Felix Klein's remarkable Vier-ergruppe to partition the 256 local rules studied empirically by Wolfram into 89 global equivalence classes of which only 50 may exhibit complex dynamics. We define a 24-element rotation group which induces 30 local equivalence classes of nonlinear difference equations whose parameters can be mapped into each other among members of the same class. [ABSTRACT FROM AUTHOR]

The global phase portrait of structurally stable two-cell cellular neural networks is studied. The configuration of equilibrium points, the number of limit cycles and their locations are investigated systematically. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Star cellular neural network (Star CNN) for associative and dynamic memories. A Star CNN consists of local oscillators and a central system. All oscillators are connected to a central system in the shape of a Star, and communicate with each other through a central system. A Star CNN can store and retrieve given patterns in the form of synchronized chaotic states with appropriate phase relations between the oscillators (associative memories). Furthermore, the output pattern can occasionally travel around the stored patterns, their reverse patterns, and new relevant patterns which are called spurious patterns (dynamic memories). [ABSTRACT FROM AUTHOR]

A systematic design methodology for finding CNN parameters with prescribed functions is proposed. A given function (task) is translated into several local operations, and they are realized as stable states of the CNN system. Many CNN parameters (CNN genes) with the same functions can be easily derived by using this design methodology. A genetic algorithm based CNN gene design methodology is also proposed. Two new genetic "activation and inactivation" operations are introduced to generate CNN genes effectively. Many useful CNN genes can be obtained systematically from known genes by using these genetic operations. Furthermore, the signal propagation property for activated and inactivated CNN genes is studied. [ABSTRACT FROM AUTHOR]

Wolfram's celebrated three-input Cellular Automata is further developed and extended from the perspective of neural networks. A single explicit formula involving two nested absolute-value functions and eight adjustable parameters called synaptic weights, is presented. Such a neuron is proved to be universal by specifying the synaptic weights of all 256 local rules. Applying the nonlinear dynamics concepts developed from Part I of this multipart series of papers, we present the rational for partitioning the entire set of 256 local rules into 16 distinct gene families, each composed of 16 gene siblings. Such a partitioning allows us to explain, if not predict, the pattern features generated from each local rule. Finally, these 16 gene families of Cellular Automata rules are encoded onto a new compact and insightful representation called the "double-helix torus." [ABSTRACT FROM AUTHOR]

In 1992, a new spatial temporal computing idea had been proposed, the CNN Universal Machine. It turned out that a new paradigm in computing on image flows, a Universal Machine on Flows, has ignited the intellect of hundreds of researchers. Today, visual microprocessors based on this idea can perform about TeraOPS computing power and 10 000 frames per second. In this paper, after a brief description of the history of the invention, architectural advances, physical implementation, algorithmic developments, as well as the biology relevance, theoretical aspects, mission critical applications, and new directions are reviewed. [ABSTRACT FROM AUTHOR]

In this paper, canonical isolated CNN cell models are proposed by using implicit differential equations. A number of equivalent but distinct CNN cell models are derived from these canonical models. Almost every known CNN cell model can be classified into one or more groups via constrained conditions. This approach is also applied to discrete-time CNN cell models. Pattern formation mechanisms are investigated from the viewpoint of equivalent templates and genetic algorithms. A strange wave propagation phenomenon in nonuniform CNN cells is also presented in this paper. Finally, chaotic associative memories are proposed. [ABSTRACT FROM AUTHOR]

This tutorial provides a nonlinear dynamics perspective to Wolfram's monumental work on A New Kind of Science. By mapping a Boolean local Rule, or truth table, onto the point attractors of a specially tailored nonlinear dynamical system, we show how some of Wolfram's empirical observations can be justified on firm ground. The advantage of this new approach for studying Cellular Automata phenomena is that it is based on concepts from nonlinear dynamics and attractors where many fuzzy concepts introduced by Wolfram via brute force observations can be defined and justified via mathematical analysis. The main result of Part I is the introduction of a fundamental concept called linear separability and a complexity index κ for each local Rule which characterizes the intrinsic geometrical structure of an induced "Boolean cube" in three-dimensional Euclidean space. In particular, Wolfram's seductive idea of a "threshold of complexity" is identified with the class of local Rules having a complexity index equal to 2. [ABSTRACT FROM AUTHOR]

In this paper, we demonstrate that some hyperchaotic circuits can be synchronized by using only one state variable. We applied three kinds of synchronization schemes, a continuous synchronization, an impulsive synchronization, and a selective synchronization to these hyperchaotic circuits. Their performance is examined from the viewpoint of synchronization stability and convergence time. [ABSTRACT FROM AUTHOR]

Discrete-time CNN systems are studied in this paper by the application of Chua's local activity principle. These systems are locally active everywhere except for one isolated parameter value. As a result, nonhomogeneous spatiotemporal patterns may be induced by any initial setting of the CNN system when the strength of the system diffusion coupling exceeds a critical threshold. The critical coupling coefficient can be derived from the loaded cell impedance of the CNN system. Three well-known 1D map CNN's (namely, the logistic map CNN, the magnetic vortex pinning map CNN, and the spiral wave reproducing map CNN) are introduced to illustrate the applications of the local activity principle. In addition, we use the cell impedance to demonstrate the period-doubling scenario in the logistic and the magnetic vortex pinning maps. [ABSTRACT FROM AUTHOR]

In this paper we analyze the replication properties of additive or parity Cellular Automata (CA). The objective of the paper is twofold. Firstly, to review and extend the existing results in the cases of one- and two-dimensional CA. Secondly, to report a general result that states the replication properties of an n-dimensional CA, whose evolution law depends on an arbitrary m number of neighbor cells. [ABSTRACT FROM AUTHOR]

Verifies the tunneling junction that can be modeled by a piecewise linear voltage-charge relation using physics-based Monte Carlo simulations which from the circuit-theoretic perspective is a nonlinear capacitor. Physical schematics of a single-electron tunneling junction; Simulation of a current-biased single-electron tunneling junction; Simulation of a voltage-biased single-electron tunneling junction.

In this paper we study the local activity, local passivity and edge of chaos of continuous-time reaction-diffusion cellular nonlinear networks (CNN) with one-port first-order, one-port secondorder, two-port second-order, two-port third-order and three-port third-order cells. We prove that the local passive regions determined by cell impedance Z[sub Q](s) and cell admittance Y[sub Q](s) for first- and second-order cells are equivalent to each other. We also present an efficient procedure to determine the edge-of-chaos parameter region by combining the local active regions derived from Y[sub Q](s) and the pole locations of Z[sub Q](s). In order to characterize the fundamental limitations of local passivity on the emergence of complexity we study the local active property from a parameter space spanned by both the cell parameters and the external excitations called the cell's port currents (in view of its interpretations from classical circuit theory). Analytical results of locally passive, restricted locally passive, edge-of-chaos and locally active parameter regions for CNN cells modeled by cubic nonlinearities are presented. We illustrate our results by analyzing CNN cells modeled by Chua's circuits with a cubic nonlinearity. We find that the morphology of the edge-of-chaos and the local active parameter regions have a close connection to the pattern formation behaviors of CNNs. Simulation results are presented to verify our theoretical results. [ABSTRACT FROM AUTHOR]

As a unique paradigm for chaos, the various versions of Chua's circuits and equations consists of a three-dimensional autonomous system with a three-segment piecewise-linear function which gives rise to three equilibrium points. This paper considers the possibility of simplifying the system configurations of piecewise-linear chaotic systems based on the structures of Chua's systems. We study a new class of piecewise-linear three-dimensional autonomous system with a three-segment piecewise-linear function. However, unlike Chua's systems, the systems we study in this paper have only single equilibrium points. To find chaotic attractors from this class of systems, we use a systematic random-search process to search the parameter space. The searching process consists of three stages. For the first stage, we simply count the number of points on a Poincaré section and find candidates for chaotic attractors. At the second stage, Lyapunov exponents are calculated for selecting chaotic attractors from the candidates. Finally, bifurcation diagrams constructed around the located chaotic attractors are used to find different types of chaotic attractors. Many qualitatively different chaotic attractors of this class of systems had been found and presented in this paper. Another method to simplify the configurations of a piecewise-linear chaotic system is to reduce the number of segments of the piecewise-linear function. We have developed some chaotic systems with a two-segment piecewise-linear function and which gives rise to two equilibrium points. Many color illustrations of chaotic attractors and bifurcation diagrams are presented. [ABSTRACT FROM AUTHOR]