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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]

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]

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]

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]

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]

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]

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]

The practical stability of impulsive synchronization between two nonautonomous chaotic systems is studied in this paper, and this is equivalent to that of the origin of the synchronization error system, which is modeled by an impulsive differential equation. We develop theoretical methods of choosing the time interval between two successive synchronization impulses and strengths of synchronization impulses for restricting synchronization errors within prescribed regions around the origin. Numerical experimental results are given to demonstrate theoretical results. [ABSTRACT FROM AUTHOR]

A chaotic impulse radio system is an ultrawide-band communication system that uses a train of very narrow baseband impulses as a carrier. In the transmitter of a chaotic impulse radio system, a message signal is modulated by two kinds of pulse carriers. Firstly, a frequency modulation is used to modulate the message signal into a subcarrier that functions as the clock pulses of a chaotic circuit. Driven by the modulated clock pulses, the chaotic circuit outputs a chaotic impulse positioning sequence which generates the positions of the carrier impulses. The specially designed chaotic circuit in the transmitter guarantees that the time intervals between the carrier impulses are chaotic. Thus the energy of the impulse carrier is distributed evenly over the entire bandwidth. In the receiver of a chaotic impulse radio system the message signal is demodulated in two stages. At the first stage, the time interval between two consecutive impulses is recovered. At the second stage, a simple algorithm based on the knowledge of the chaotic circuit in the transmitter is used to calculate partially the locations of the inner clock pulses which in turn are used to demodulate the message signal. No synchronization at any level is needed in this chaotic impulse radio system. The security of this chaotic impulse radio system depends on the hardware parameters of the chaotic circuit and the inner clock pulse train. Simulation results are presented to illustrate the design procedure of an example of this chaotic impulse radio system. [ABSTRACT FROM AUTHOR]

In this tutorial we continue our program of clarifying chaos by examining the relationship between chaotic and stochastic processes. To do this, we construct chaotic analogs of stochastic processes, stochastic differential equations, and discuss estimation and prediction models. The conclusion of this section is that from the composition of simple nonlinear periodic dynamical systems arise chaotic dynamical systems, and from the time-series of chaotic solutions of finite-difference and differential equations are formed chaotic processes, the analogs of stochastic processes. Chaotic processes are formed from chaotic dynamical systems in at least two ways. One is by the superposition of a large class of chaotic time-series. The second is through the compression of the time-scale of a chaotic time-series. As stochastic processes that arise from uniform random variables are not constructable, and chaotic processes are constructable, we conclude that chaotic processes are primary and that stochastic processes are idealizations of chaotic processes. Also, we begin to explore the relationship between the prime numbers and the possible role they may play in the formation of chaos. [ABSTRACT FROM AUTHOR]

Generalized synchronization (GS) of two chaotic systems is a generalization of identical synchronization. Usually, the manifold of GS is much more complex than the driven system and the driving system. In this paper, we study a special case of GS in which the synchronization manifold is linear (linear GS for short). In a theorem, we present the necessary and sufficient conditions under which a linear GS can be achieved between two chaotic systems. In particular, we study the linear GS of two Chua's circuits. [ABSTRACT FROM AUTHOR]

Looks at the impulsive control of chaotic systems. Presentation of theorems on the stability of impulsive control systems; Use of the theorems to find the conditions under which chaotic systems can be asymptotically controlled; Estimation of the upper bound of the impulse interval given; Theory of impulsive synchronization of two chaotic systems.

Extends chaotic switching to general chaotic parameter modulation. G-modulation; Transmission of signals; Circuit description.

The simplest electronic circuit with a memristor was recently proposed. Chaotic attractors solution to this memristive circuit are topologically characterized and compared to Rössler-like attractors. [ABSTRACT FROM AUTHOR]

The study of chemical reactions with oscillating kinetics has drawn increasing interest over the last few decades because it also contributes towards a deeper understanding of the complex phenomena of temporal and spatial organizations in biological systems. The Cellular Nonlinear Network (CNN) local activity principle introduced by Chua [1997, 2005] has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice formed by coupled cells. Recently, Yang and Epstein proposed a reaction–diffusion Oregonator model with five variables for mimicking the Belousov–Zhabotinskii reaction. The Yang–Epstein model can generate oscillatory Turing patterns, including the twinkling eye, localized spiral and concentric wave structures. In this paper, we first propose a modified Yang–Epstein's Oregonator model by introducing a controller, and then map the revised Oregonator reaction–diffusion system into a reaction–diffusion Oregonator CNN. The Oregonator CNN has two cell equilibrium points Q1 = (0, 0, 0, 0, 0) and Q2, representing the "original" equilibrium point and an additional equilibrium point, respectively. The bifurcation diagrams of the Oregonator CNN are calculated using the analytical criteria for local activity. The bifurcation diagrams of the Oregonator CNN at Q1 have only locally active and unstable regions; and the ones at Q2 have locally passive regions, locally active and unstable regions, and edge of chaos regions. The calculated results show that the parameter groups of the Oregonator CNN which generate complex patterns are located on the edge of chaos regions, or on locally active unstable regions near the edge of chaos boundary. Numerical simulations show also that the Oregonator CNNs can generate similar dynamics patterns if the parameter groups are selected the same as those of the Yang–Epstein model. In particular, the parameters of the Yang–Epstein model which exhibit twinkling-eye patterns, and pinwheel patterns are located on the edges of chaos regions near the boundaries of locally active unstable regions with respect to Q2. The parameters of the Yang–Epstein models which exhibit labyrinthine stripelike patterns are located on the locally active unstable regions near the boundaries of the edge of chaos regions with respect to Q2. However the parameter group of the Yang–Epstein model with isolated spiral patterns is in the locally passive region near the boundary with edge of chaos with respect to Q2, whose trajectories tend to the equilibrium point Q2. Choosing a kind of triggering initial conditions given in [Chua, 1997], and the parameters of the Oregonator equations with the twinkling-eye patterns, pinwheel patterns, labyrinthine stripelike patterns, and isolated spiral patterns, three kinds of new spiral waves generated by the Oregonator CNNs were observed by numerical simulations. They seem to be essentially different patterns to those generated by the Oregonator CNNs with initial conditions consisting of equilibrium points plus small random perturbations. Our results demonstrate once again Chua's assertion that a wide spectrum of complex behaviors may exist if the corresponding CNN cell parameters are chosen in or near the edge of chaos region. [ABSTRACT FROM AUTHOR]

This paper presents a time-delayed impulsive feedback approach to the problem of stabilization of periodic orbits in chaotic hybrid systems. The rigorous stability analysis of the proposed method is given. Using the time-delayed impulsive feedback method, we analyze the problem of detecting various periodic orbits in a special class of hybrid system, a switched arrival system, which is a prototype model of many manufacturing systems and computer systems where a large amount of work is processed in a unit time. We also consider the problem of stabilization of periodic orbits of chaotic piecewise affine systems, especially Chua's circuit, which is another important special class of hybrid systems. [ABSTRACT FROM AUTHOR]

Experimental results show that chaotic and hyperchaotic systems can be synchronized by impulses sampled from one or two state variables. In this paper, we study the conditions under which chaotic and hyperchaotic systems can be synchronized by impulses sampled from a part of their state variables. By calculating the Lyapunov exponents of variational synchronization error systems, we show that this kind of impulsive synchronization can be applied to almost all hyperchaotic systems. We also study the selective synchronization of chaotic systems. In a selective synchronization scheme, the synchronizing signal is chosen in the time periods when the Lyapunov exponents of variational synchronization error systems are negative. Since only driving signals during the time periods when synchronization error can be reduced are applied to reduce the synchronization error, and no signal is applied during the time periods when synchronization error can be increased, selective synchronization scheme can be used to achieve synchronization even in the case when continuous synchronization schemes fail to work. [ABSTRACT FROM AUTHOR]

In this paper we show how to relate a form of high-dimensional complexity to chaotic and other types of dynamical systems. The derivation shows how "near-chaotic" complexity can arise without the presence of homoclinic tangles or positive Lyapunov exponents. The relationship we derive follows from the observation that the elements of invariant finite integer lattices of high-dimensional dynamical systems can, themselves, be viewed as single integers rather than coordinates of a point in n-space. From this observation it is possible to construct high-dimensional dynamical systems which have properties of shifts but for which there is no conventional topological conjugacy to a shift. The particular manner in which the shift appears in high-dimensional dynamical systems suggests that some forms of complexity arise from the presence of chaotic dynamics which are obscured by the large dimensionality of the system domain. [ABSTRACT FROM AUTHOR]

In this work we deal with slow-fast autonomous dynamical systems. We initially define them as being modeled by systems of differential equations having a small parameter multiplying one of their velocity components. In order to analyze their solutions, some being chaotic, we have proposed a mathematical analytic method based on an iterative approach [Rossetto et al., 1998]. Under some conditions, this method allows us to give an analytic equation of the slow manifold. This equation is obtained by considering that the slow manifold is locally defined by a plane orthogonal to the tangent system's left fast eigenvector. In this paper, we give another method to compute the slow manifold equation by using the tangent system's slow eigenvectors. This method allows us to give a geometrical characterization of the attractor and a global qualitative description of its dynamics. The method used to compute the equation of the slow manifold has been extended to systems having a real and negative eigenvalue in a large domain of the phase space, as it is the case with the Lorenz system. Indeed, we give the Lorenz slow manifold equation and this allows us to make a qualitative study comparing this model and Chua's model. Finally, we apply our results to derive the slow manifold equations of a nonlinear optical slow-fast system, namely, the optical parametric oscillator model. [ABSTRACT FROM AUTHOR]

This paper presents analytic criteria for local activity in one-port Cellular Nonlinear Network (CNN) cells [Chua, 1997, 1999], and gives the applications to the Oregonator CNN defined by the kinetic chemical reaction model of morphogenesis first introduced in [Field & Noyes, 1974]. Locally active domains, locally passive domains, and the edge of chaos are identified in the cell parameter space. Computer simulations of the dynamics of several Oregonator CNN's with specific selected cell parameters in the above-mentioned domains show genesis and the emergence of complexity. Furthermore, a novel phenomena is observed that "extremely high energy" is concentrated only on a few cells in the dynamic patterns of some Oregonator CNN's whose cell parameters are located in active domains; furthermore, relaxation oscillations and "transient oscillations" can exist if the cell parameters of the Oregonator CNN are located nearby or on the edge of chaos. This research illustrates once again the effectiveness of the local activity theory in choosing the system parameters for the emergence of complex patterns (static and dynamic) in a homogeneous lattice formed by coupled cells. [ABSTRACT FROM AUTHOR]

In this paper, experimental results on impulsive synchronization of two kinds of chaotic circuits; namely, Chua's oscillator and a hyperchaotic circuit, are presented. To impulsively synchronize two Chua's oscillators, synchronization impulses sampled from one state variable of the driving circuit are transmitted to the driven circuit. To impulsively synchronize two hyperchaotic circuits, synchronizing impulses sampled from two signals of the driving circuit are sent to the driven circuit. Our experimental results show that the accuracy of impulsive synchronization depends on both the period and the width of the impulse. The ratio between the impulse width and impulse period for "almost-identical" synchronization increases as the impulse period increases. The robustness of impulsive synchronization to additive noise is also experimentally studied. For sufficiently short impulse periods, no significant differences are observed between impulsive and continuous synchronizations. The performance of chaotic spread spectrum communication systems based on impulsive synchronization is also studied experimentally. [ABSTRACT FROM AUTHOR]

Recently it was shown that in the dynamical model of Chua circuit both the classical self-excited and hidden chaotic attractors can be found. In this paper the dynamics of the Chua circuit is revisited. The scenario of the chaotic dynamics development and the birth of self-excited and hidden attractors is studied. A pitchfork bifurcation is shown in which a pair of symmetric attractors coexist and merge into one symmetric attractor through an attractor-merging bifurcation and a splitting of a single attractor into two attractors. The scenario relating the subcritical Hopf bifurcation near equilibrium points and the birth of hidden attractors is discussed. [ABSTRACT FROM AUTHOR]

This paper presents a possible realization of a (CD)²MA multiuser communication system. The state variables of two synchronized chaotic Chua's circuits are thereby used to generate identical continuous valued spreader sequences in the transmitter and the receiver. To create a higher degree of randomness in the signatures, the state variables are processed by a k-shift cypher. The performance to be expected for such a system as opposed to designs using binary spreaders can be estimated theoretically. For verification, extensive simulations were run that allow for a direct comparison of the chaotic spreader with a traditional binary Gold sequence or a sequence of continuous random numbers, respectively, in terms of the achievable bit error ratio for a given data rate and number of simultaneous users per channel. A brief description of the system and channel models implemented in the simulation program is provided. The simulation results are presented and discussed. [ABSTRACT FROM AUTHOR]