International Journal of Bifurcation and Chaos

Brain-like information processing has become a challenge to modern computer science and chip technology. The CNN (Cellular Neural Network) Universal Chip is the first fully programmable industrial-sized brain-like stored-program dynamic array computer which dates back to an invention of Leon O. Chua and Lin Yang in Berkeley in 1988. Since then, many papers have been written on the mathematical foundations and technical applications of CNN chips. They are already used to model artificial, physical, chemical, as well as living biological systems. CNN is now a new computing paradigm of interdisciplinary interest. In this state of development a textbook is needed in order to recruit new generations of students and researchers from different fields of research. Thus, Chua's and Roska's textbook is a timely and historic publication. On the background of their teaching experience, they have aimed at undergraduate students from engineering, physics, chemistry, as well as biology departments. But, actually, it offers more. It is a brilliant introduction to the foundations and applications of CNN which is distinguished with conceptual and mathematical precision as well as with detailed explanations of applications in visual computing.

We show that two coupled pendulums that are coupled and can synchronize, are mathematically equivalent to one "horizontal" parametrically driven pendulum. We have fabricated a horizontal pendulum and present data from this horizontal pendulum which we believe to be the first physical realization of such a mechanical "synchronization machine." A description of intermittent synchronization that can occur when two coupled pendulums are in a chaotic state is given in terms of the data from the horizontal pendulum. We discuss the relationship between the modes of the horizontal pendulum and the corresponding synchronization of the two coupled pendulums. Finally, we show that when a horizontal pendulum is driven by any random source, not necessarily chaotic, intermittent synchronization can occur.

A computationally efficient method is proposed for computing the simplest normal forms of vector fields. A simple, explicit recursive formula is obtained for general differential equations. The most important feature of the approach is to obtain the "simplest" formula which reduces the computation demand to minimum. At each order of the normal form computation, the formula generates a set of algebraic equations for computing the normal form and nonlinear transformation. Moreover, the new recursive method is not required for solving large matrix equations, instead it solves linear algebraic equations one by one. Thus the new method is computationally efficient. In addition, unlike the conventional normal form theory which uses separate nonlinear transformations at each order, this approach uses a consistent nonlinear transformation through all order computations. This enables one to obtain a convenient, one step transformation between the original system and the simplest normal form. The new method can treat general differential equations which are not necessarily assumed in a conventional normal form. The method is applied to consider Hopf and Bogdanov–Takens singularities, with examples to show the computation efficiency. Maple programs have been developed to provide an "automatic" procedure for applications.

The original Hilbert's 16th problem can be split into four parts consisting of Problems A–D. In this paper, the progress of study on Hilbert's 16th problem is presented, and the relationship between Hilbert's 16th problem and bifurcations of planar vector fields is discussed. The material is presented in eight sections.

Section 1: Introduction: what is Hilbert's 16th problem?

Section 2: The first part of Hilbert's 16th problem.

Section 3: The second part of Hilbert's 16th problem: introduction.

Section 4: Focal values, saddle values and finite cyclicity in a fine focus, closed orbit and homoclinic loop.

Section 5: Finiteness problem.

Section 6: The weakened Hilbert's 16th problem.

Section 7: Global and local bifurcations of Z_q–equivariant vector fields.

Section 8: The rate of growth of Hilbert number H(n) with n.

The classical dynamics of a hydrogen atom in a generalized van der Waals potential is investigated. In order to carry out the analytical and numerical investigations for a range of parametric values, we removed the singularity of the problem using Levi–Civita regularization and converted the problem into that of two coupled sextic anharmonic oscillators. We give a complete description of the real phase space structure of the converted system and give also an explicit periodic solution for singular common-level sets of the first integrals. All generic bifurcations of Liouville tori were determined theoretically. Numerical investigations are carried out for all generic bifurcations and we observe chaos-order-chaos transition when one of the system parameters is varied.

This study demonstrates the devil's staircase structure of topological entropy functions for one-dimensional symmetric unimodal maps with a gap inside. The results are obtained by using kneading theory and are helpful in investigating the communication of chaos.

Two methods, based on nonlinear dynamics, are described for broad- and narrow-band communication systems. The first approach employs hyperchaotic discrete time series carrier sequences generated with mixed linear–nonlinear coupled differential equations [Irving & Dewson, 1997]. In this first approach it is assumed that the solution of the coupled ordinary differential equations can be represented as a multichannel Volterra functional expansion. Identification, synchronization and the potential unmasking of hyperchaotic communications systems, are achieved by generating a tractable hierarchy of ascending order time series moment equations by operating on a suitably truncated Volterra functional expansion. The estimated moment hierarchy facilitates the calculation of the coefficients of the coupled differential equations that generate the multichannel carrier that is to be modulated by the message. Due to its ability to easily and accurately estimate the coefficients of the governing differential equations the method is applied to the multichannel carrier sequences of data generated by a set of hyperchaotic equations. The inherent problem of resynchronization in chaotic communication systems cannot be avoided in general because of the evolutionary, time dependent, nature of the time series carriers used. This time dependence problem that results in repeated resynchronization can be avoided by considering an alternative and innovative approach that employs time series carrier sequences generated by an input–output multichannel Volterra kernel expansion [Dewson & Irving, 1996]. In this second method, mixed order multichannel Volterra kernel functions are used to generate the carrier sequence that is then modulated by the message. The Volterra kernel function values used to generate the carrier can be identified with the moment hierarchy method [Dewson & Irving, 1996] enabling the message to be deconvolved from the transmitted signal. The modulated nonlinear multichannel signal is transmitted to the receiver. The effect of transmission is to distort and corrupt the signal. In this work it is assumed that the distortions to the transmitted signal remain small, with the case where they become significant being treated elsewhere. First, the case where the signal is received free from both noise and distortion, for example the decryption of an encrypted e-mail, is first considered. Then signal reception in the presence of additive noise is considered. Each of the two proposed methods has been investigated to determine their accuracy and robustness in the presence of additive noise.

Hide [1997] has introduced a number of different nonlinear models to describe the behavior of n-coupled self-exciting Faraday disk homopolar dynamos. The hierarchy of dynamos based upon the Hide et al. [1996] study has already received much attention in the literature (see [Moroz, 2001] for a review). In this paper we focus upon the remaining dynamo, namely Case 3 of [Hide, 1997] for the particular limit in which the Malkus–Robbins dynamo [Malkus, 1972; Robbins, 1997] obtains, but now modified by the presence of a linear series motor. We compare and contrast the linear and the nonlinear behaviors of the two types of dynamo.

This paper investigates the complex dynamics, synchronization and control of chaos in a system of strongly connected Wilson–Cowan neural oscillators. Some typical synchronized periodic solutions are analyzed by using the Poincaré mapping method, for which bifurcation diagrams are obtained. It is shown that topological change of the synchronization mode is mainly caused and carried out by the Neimark–Sacker bifurcation. Finally, a simple feedback control method is presented for stabilizing an in-phase synchronizing periodic solution embedded in the chaotic attractor of a higher-dimensional model of such coupled neural oscillators.

Feedback coupling through an interaction term proportional to the difference in the value of some behavioral characteristics of two systems is a very common structural setting that leads to synchronization of the behavior of both systems. The degree of synchronization attained depends on the strength of the interaction term and on the mutual interdependency of the structures of both systems. In this paper, we show that two chaotic systems linked through a feedback coupling interaction term of gain parameter k reach a synchronized regime characterized by a vector of variable errors which tends towards zero with parameter k while the interaction term tends towards a finite nonzero permanent regime. This means that maintaining a certain degree of synchronization has a cost. In the limit, complete synchronization occurs at a finite limit cost. We show that feedback coupling in itself brings about conditions permitting that systems with a degree of structural parameter flexibility evolve close towards each other structures in order to facilitate the maintenance of the synchronized regime. In this paper, we deduce parameter adaptive laws for any family of homochaotic systems provided they are previously forced to work, via feedback coupling, within an appropriate degree of synchronization. The laws are global in the space of parameters and lead eventually to identical synchronization at no interaction cost. We illustrate this point with homochaotic systems from the Lorenz, Rössler and Chua families.

This paper addresses time-delayed feedback control (DFC) of time-delay chaotic systems. To extend the DFC approach to time-delay chaotic system, alter having been successfully used in chaotic systems without time-delays, the standard feedback control (SFC) method is firstly employed to show the main control technique in this paper based on one error control system. Then sufficient conditions for stabilization and tracking problems via DFC are derived from the results based on SFC. Also, the systematic and analytic controller design method can be obtained to stabilize the system to an unstable fixed point and to tracking an unstable periodic orbit, respectively. Some numerical examples are provided to demonstrate the effectiveness of the presented method.

This paper studies a master-slave type of chaos synchronization problem for a general form of Lur'e systems by a time-delay feedback control technique, improving some results of Yalçin et al. [2001]. Some fairly simple algebraic conditions are derived for easier verification, facilitating the design and applications of such chaos synchronization systems.

This paper proposes a fuzzy model-based chaotic encryption approach using synchronization. The cryptosystem uses T–S fuzzy models to exactly represent discrete-time chaotic systems into separate linear systems. Then the synchronization problem is solved using linear matrix inequalities. The advantages of this approach are: the general and systematic T–S fuzzy model design methodology suitable for well-known Luré type discrete-time chaotic systems; flexibility in selection of chaotic signals for cryptosystem secure key generator; and multiuser capabilities. Especially taking a chaotic superincreasing sequence as an encryption key enhances the chaotic communication structure to a higher-level of security compared to traditional masking methods. In addition, numerical simulations and DSP-based experiments are carried out to verify the validity of theoretical results.

In this Letter, a new demodulation approach to communication using chaotic signals is presented. Information signal is modulated by applying it as an input to a chaotic system. At the receiver, a synchronous subsystem is constructed for estimates on the states of the chaotic system. The key innovation in our communication system is a new robust filter that recovers the information signal from the scalar transmitted signal.

An electronic receiver based on stochastic resonance is presented to rescue subthreshold modulated digital data. In real experiment, it is shown that a complete data restoration is achieved for both uniform and Gaussian white noise.

The recently proposed Highly Optimized Tolerance (H.O.T.) model [Carlson & Doyle, 1999, 2000], which aims to describe the statistics of robust complex systems in uncertain environments, is compared with data from the Western United States (W.S.C.C.) power distribution system. We use for comparison a 15-year record of all power outages occurring on the grid, measured in the size of megawatts lost and the number of customers without service. In applying the model to the power grid data, we find that the problem of determining how the resources in the system scale with event size is nontrivial given the assumptions of the model and the information about how the power grid actually operates. Further, we observe that the model agrees closely with the W.S.C.C. data for the megawatts but not the customers, and consequently propose that the assumption in the model of optimal resource distribution is not valid in general when more than one measure of event size is used. A modified H.O.T. model which allows for resource misallocation is introduced and we find that this model can be made to fit both data sets reasonably well.

Using the method of qualitative analysis we show that five perturbed cubic Hamiltonian systems have the same distribution of limit cycles and have 11 limit cycles for some parameters. The accurate location of each limit cycle is given by numerical exploration. In other words, we demonstrate the existence of 11 limit cycles and their distribution in five perturbed systems in two ways, the results obtained from both ways are the same.

A general method for the creation of strange nonchaotic attractor is proposed. As an example, the strange nonchaotic attractor in a high-dimensional globally coupled neural system is studied numerically. For such an attractor, the time interval of continuously positive finite-time Lyapunov exponent must be smaller than the period of the driving stimulus, although it has a long-time positive tail. The intermittency between laminar and burst behavior is a characteristic dynamic of the strange nonchaotic attractors. Simulation results show that the chaotic phase occurs only within a small region around the origin in the parameter space. More than half of the large nonchaotic region is the strange nonchaotic phase. The chaotic phase is typically surrounded by strange nonchaotic attractors. This result also suggests that some biological signals that have a strange structure may be nonchaotic rather than chaotic.

This Letter introduces a relatively simple three-dimensional continuous autonomous chaotic system, which can display complex 2- and 4-scroll attractors in simulations. Its generation and basic dynamical behaviors are briefly described.