In this tutorial paper we present one of the simplest autonomous differential equations capable of generating chaotic behavior. Some of the fundamental routes to chaos and bifurcation phenomena are demonstrated with examples. A brief discussion of equilibrium points and their stability is given. For the convenience of the reader, a short computer program written in QuickBASIC is included to give the reader a possibility of quick hands-on experience with the generation of chaotic phenomena without using sophisticated numerical simulators. All the necessary parameter values and initial conditions are provided in a tabular form. Eigenvalue diagrams showing regions with particular eigenvalue patterns are given.

This paper presents a summary of the most commonly observed spatio-temporal phenomena in discrete cellular neural networks (CNNs) of dimension one and two. Among the phenomena discussed are traveling wave phenomena in chains and 2-D arrays, and spiral waves and target waves in both excitable and fluctuating media. Chua's circuit is used as the basic cell in the CNN arrays. Parameter values and initial conditions for the corresponding simulations are given so they can be reproduced with different simulators.

Abstract:As it results from many research works, the majority of real dynamical objects are fractional-order systems, although in some types of systems the order is very close to integer order. Application of fractional-order models is more adequate for the description and analysis of real dynamical systems than integer-order models, because their total entropy is greater than in integer-order models with the same number of parameters. A great deal of modern methods for investigation, monitoring and control of the dynamical processes in different areas utilize approaches based upon modeling of these processes using not only mathematical models, but also physical models. This paper is devoted to the design and analogue electronic realization of the fractional-order model of a fractional-order system, e.g., of the controlled object and/or controller, whose mathematical model is a fractional-order differential equation. The electronic realization is based on fractional-order differentiator and integrator where operational amplifiers are connected with appropriate impedance, with so called Fractional Order Element or Constant Phase Element. Presented network model approximates quite well the properties of the ideal fractional-order system compared with e.g., domino ladder networks. Along with the mathematical description, OPEN ACCESSEntropy 2013, 15 4200 circuit diagrams and design procedure, simulation and measured results are also presented.

A model of two-dimensional arrays of Chua's circuits is numerically investigated. In a certain parameter region the spatiotemporal system has both synchronized oscillation and spiral wave attractors. Feedback pinnings are suggested to migrate the system from the spiral wave state to the coherent oscillation. The influences of the pinning density, forcing strength, and different pinning distributions on the driving effect are investigated. It is shown that some properly designed control schemes may reach very high control efficiency, i.e., killing a spiral wave consisting of a huge number of cells by injecting only very few cells. The wide applications of the approach are addressed. [S0031-9007 (98)05390-3] PACS numbers: 05.45. + b, 47.54. + r, 82.20.Wt

The dynamical properties of two classical paradigms for chaotic behavior are reviewed—the Lorenz and Chua’s Equations—on a comparative basis. In terms of the mathematical structure, the Lorenz Equation is more complicated than Chua’s Equation because it requires two nonlinear functions of two variables, whereas Chua’s Equation requires only one nonlinear function of one variable. It is shown that most standard routes to cbaos and dynamical phenomena previously observed from the Lorenz Equation can be produced in Chua’s system with a cubic nonlinearity. In addition, we show other phenomena from Chua’s system which are not observed in the Lorenz system so far. Some differences in the topological geometric models are also reviewed. We present some theoretical results regarding Chua’s system which are absent for the Lorenz system. For example, it is known that Chua’s system is topologically conjugate to the class of systems with a scalar nonlinearity (except for a measure zero set) and is therefore canonical in this sense. We conclude with some reasons why Chua’s system can be considered superior or more suitable than the Lorenz system for various applications and studies.

:Chua

Empirical recurrent relations, governing the structure of the devil’s staircase in the driven Chua’s circuit are given, which reflect the self-similar structure in an algebraic form. In particular, it turns out that the same formulas hold for both winding and period numbers, but with different “initial conditions”. Some of the finer details such as period-doubling along with numerous coexistence phenomena within staircases of mode-locked states have been revealed by computing high-resolution bifurcation diagrams.

In the work [Chua, 1992], a deep intuition of its author gave rise to the choice of singularities corresponding to Chua's circuit. Therefore, it is the only one probably exhibiting three saddle points named Chua's singularities in this paper. One of the singularities is a saddle in forward time (dt > 0) of integration, whereas the other two are saddles in backward time (dt < 0) of integration. In the following, the term Chua's Chaos denotes chaos related to Chua's singularities. These singularities are the source of all special surfaces that are the subject of this contribution. We named the surface to which all other surfaces are bound as the Double-Arm Stable Manifold (DASM). The beauty and multifunctionality of this surface represents the unfathomable Intelligence in the sense of [Tolle, 2003]. The presence of the DASM in the state space is a sufficient condition for the generation of Chua's chaos or corresponding periodic windows. Since Chua's singularities are not limited by circuit morphology or the order of state equations, the research on Chua's chaos seems to be still very promising.