Narrow Results

This paper presents the universality of the coexistence of complete synchronization with the recently new proposed inverse π-lag synchronization, in the case of mutually coupled systems of Chua's circuit family. The phenomenon of multistability and the nature of this circuit family systems are the key-points which lead to the explanation of this coexistence. For the need of this work, the most representative circuit of this circuits' family, the Chua oscillator, is used. Simulation results confirm this universality in the class of Chua's circuit family.

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.

This paper presents some analytical criteria for local activity principle in reaction–diffusion Cellular Nonlinear Network (CNN) cells [Chua, 1997, 1999] that have four local state variables with three ports. As a first application, a cellular nonlinear network model of tumor growth and immune surveillance against cancer (GISAC) is discussed, which has cells defined by the Lefever–Erneaux equations, representing the densities of alive and dead cancer cells, as well as the number of free and bound cytotoxic cells, per unit volume. Bifurcation diagrams of the GISAC CNN provide possible explanations for the mechanism of cancer diffusion, control, and elimination. Numerical simulations show that oscillatory patterns and convergent patterns (representing cancer diffusion and elimination, respectively) may emerge if selected cell parameters are located nearby or on the edge of the chaos domain. As a second application, a smoothed Chua's oscillator circuit (SCC) CNN with three ports is studied, for which the original prototype was introduced by Chua as a dual-layer two-dimensional reaction–diffusion CNN with three state variables and two ports. Bifurcation diagrams of the SCC CNN are computed, which only demonstrate active unstable domains and edges of chaos. Numerical simulations show that evolution of patterns of the state variables of the SCC CNN can exhibit divergence, periodicity, and chaos; and the second and the fourth state variables of the SCC CNNs may exhibit generalized synchronization. These 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 nearby the edge of chaos.

Modern Science is finding new methods of looking at biological, physical or social phenomena. Traditional methods of quantification are no longer sufficient and new approaches are emerging. These approaches make it apparent that the phenomena the observer is looking at are not classifiable by conventional methods. These phenomena are complex. A complex system, as Chua's oscillator, is a nonlinear configuration whose dynamical behavior is chaotic. Chua's oscillator equations allow to define the basic behavior of a dynamical system and to detect the changes in the qualitative behavior of a system when bifurcation occurs, as parameters are varied. The typical set of behavior of a dynamical system can be detailed as equilibrium points, limit cycles, strange attractors. The concepts, methods and paradigms of Dynamical Systems Theory can be applied to understand human behavior. Human behavior is emergent and behavior patterns emerge thanks to the way the parts or the processes are coordinated among themselves. In fact, the listening process in humans is complex and it develops over time as well. Sound and music can be both inside and outside humans. This tutorial concerns the translation of Chua's oscillators into music, in order to find a new way of understanding complexity by using music. By building up many computational models which allow the translation of some quantitative features of Chua's oscillator into sound and music, we have created many acoustical and musical compositions, which in turn present the characteristics of dynamical systems from a perceptual point of view. We have found interesting relationships between dynamical systems behavior and their musical translation since, in the process of listening, human subjects perceive many of the structures as possible to perceive in the behavior of Chua's oscillator. In other words, human cognitive abilities can analyze the large and complicated patterns produced by Chua's systems translated into music, achieving the cognitive economy and the coordination and synthesis of countless data at our disposal that occur in the perception of dynamic events in the real world. Music can be considered the semantics of dynamical systems, which gives us a powerful method for interpreting complexity.

Due to strong contraction in the outer regions of Chua's circuit with piecewise-linear dynamics, the system can be described by an approximate one-dimensional map. We confirm by simulation and experiment that the initial slope of this map is defined by .

The nonlinearity in Chua's oscillator is commonly implemented as a three-segment piecewise-linear resistor. The piecewise-linear nature of the element means that the implementation requires a significant amount of circuitry and the speed of operation is limited. The qualitative behavior of Chua's oscillator has also been captured using a smooth cubic nonlinearity; the implementation of the latter also requires a significant amount of circuitry and suffers from limited speed of operation.

This work describes a novel implementation of Chua's oscillator using just four transistors and a battery to produce a cubic-like nonlinearity. The circuit is simple, robust, and capable of operating at frequencies over one thousand times greater than the original Chua's oscillator.

This work presents a linguistic approach to the understanding of chaos. The idea comes from our work on translating into sounds and music the complexity of chaotic systems, as with Chua's attractors. Therefore, working with sounds, we have used a standardization criterion in order to detect only some of the features of the richness of chaos. This method involves the selection of a limited number of sounds, which, in turn, allow for the creation of other components of the system at different levels of organization, as with natural language. Thus, phonetic, morphological, syntactic and semantic levels, linked by a grammar of an "artificial chaotic language" are defined. Each linguistic unit of the artificial language presents networks of interrelated phenomena since for each of them, it is possible to detect its path length, tracing a graph of the mutual relationships of the system's components. We found that there is interaction among the different levels of the chaotic language. Furthermore, some traits of the dynamics of the evolution of language in human infants are found in the main routes to chaos. In this emergent dynamics, mutations, struggle for the fittest, natural selection, and the relative distribution of linguistic entities in genetic landscapes are observed. On the basis of this, a possible bridge of connection between the physical and the mental worlds may be developed.

This paper focuses on the numerical study of chaotic dynamics via the Adomian decomposition method. The approach, which provides series solutions of the system equations, is first applied to Chua's circuit and Chua's oscillator, both with cubic nonlinearity. Successively, the method is utilized for obtaining hyperchaotic multiscroll attractors in a ring of three Chua's circuits, where the smooth nonlinearities are Hermite interpolating polynomials. The reported examples show that the approach presents two main features, i.e. the system nonlinearity is preserved and the chaotic solution is provided in a closed form.

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.

This work shows the experimental implementation of a chaotic communication system based on two Chua's oscillators which are synchronized by Hamiltonian forms and observer approach. The chaotic communication scheme is realized by using the commercially available positive-type second generation current conveyor (CCII+), which is included into the AD844 device. As a result, experimental measurements are provided to demonstrate the suitability of the CCII+ to implement chaotic communication systems.

The memristor is supposed to be the fourth fundamental electronic element in addition to the well-known resistor, inductor and capacitor. Named as a contraction for memory resistor, its theoretical existence was postulated in 1971 by L. O. Chua, based on symmetrical and logical properties observed in some electronic circuits. On the other hand its physical realization was announced only recently in a paper published on May 2008 issue of Nature by a research team from Hewlett–Packard Company. In this work, we present the bifurcation analysis of two memristor oscillators mathematical models, given by three-dimensional five-parameter piecewise-linear and cubic systems of ordinary differential equations. We show that depending on the parameter values, the systems may present the coexistence of both infinitely many stable periodic orbits and stable equilibrium points. The periodic orbits arise from the change in local stability of equilibrium points on a line of equilibria, for a fixed set of parameter values. This phenomenon is a kind of Hopf bifurcation without parameters. We have numerical evidences that such stable periodic orbits form an invariant surface, which is an attractor of the systems solutions. The results obtained imply that even for a fixed set of parameters the two systems studied may or may not present oscillations, depending on the initial condition considered in the phase space. Moreover, when they exist, the amplitude of the oscillations also depends on the initial conditions.

In this paper, we develop a parameter matching technique using adaptive synchronization with a chaotic attractor. Two Chua's oscillators, identical in every parameter except for one, are set-up in a master/slave configuration where the slave's mismatched parameter is adaptable. Using Lyapunov functions and incorporating the presence of chaotic signals, adaptive control laws are designed to ensure exact parameter matching. One of the derived adaptive controllers is experimentally validated by using an adaptive inductor-gyrator composed of current feedback op-amps (CFOAs). The experimental results are compared to high-fidelity SPICE simulations, and performance of adaptive controllers are compared over a wide range of parameter values using MATLAB.

This paper investigates the global lagged finite-time synchronization of the master-slave Lur’e systems subject to time delay of signal transmission. By designing a variable-substitution and feedback controller, a master-slave finite-time synchronization scheme for the Lur’e systems with time delay is built up. Two delay-independent global lagged finite-time synchronization criteria are proved in the forms of linear matrix inequalities (LMIs), and the corresponding settling time of synchronization is analytically estimated. The obtained LMI criteria are applied to Chua’s oscillators, obtaining some easily implemented algebraic criteria under various single-variable-substitution and feedback controller, which are then optimized to improve their conservative property. Finally, several numerical examples are illustrated to verify the effectiveness of the optimized criteria.

We study a van der Pol-like memristor oscillator, obtained by substituting a Chua’s diode with an active controlled memristor in a van der Pol oscillator with Chua’s diode. The mathematical model for the studied circuit is given by a three-dimensional piecewise linear system of ordinary differential equations, depending on five parameters. We show that this system has a line of equilibria given by the $z$-axis and the phase space ${ℝ}^3$ is foliated by invariant planes transverse to this line, which implies that the dynamics is essentially two-dimensional. We also show that in each of these invariant planes may occur limit cycles and relaxation oscillations (that is, nonsinusoidal repetitive (periodic) solutions), depending on the parameter values. Hence, the oscillator studied here, constructed with a memristor, is also a relaxation oscillator, as the original van der Pol oscillator, although with a main difference: in the case of the memristor oscillator, an infinity of oscillations are produced, one in each invariant plane, depending on the initial condition considered. We also give conditions for the nonexistence of oscillations, depending on the position of the invariant planes in the phase space.