International Journal of Bifurcation and Chaos

In recent years, a large amount of work on chaos-based cryptosystems have been published. However, many of the proposed schemes fail to explain or do not possess a number of features that are fundamentally important to all kind of cryptosystems. As a result, many proposed systems are difficult to implement in practice with a reasonable degree of security. Likewise, they are seldom accompanied by a thorough security analysis. Consequently, it is difficult for other researchers and end users to evaluate their security and performance. This work is intended to provide a common framework of basic guidelines that, if followed, could benefit every new cryptosystem. The suggested guidelines address three main issues: implementation, key management and security analysis, aiming at assisting designers of new cryptosystems to present their work in a more systematic and rigorous way to fulfill some basic cryptographic requirements. Meanwhile, several recommendations are made regarding some practical aspects of analog chaos-based secure communications, such as channel noise, limited bandwith and attenuation.

To build good models, we need to know the appropriate model size. To handle this problem, a variety of information criteria have already been proposed, each with a different background. In this paper, we consider the problem of model selection and investigate the performance of a number of proposed information criteria and whether the assumption to obtain the formulae that fitting errors are normally distributed hold or not in some conditions (different data points and noise levels). The results show that although the application of information criteria prevents over-fitting and under-fitting in most cases, there are cases where we cannot avoid even involving many data points and low noise levels in ideal situations. The results also show that the distribution of the fitting errors is not always normally distributed, although the observational noise is Gaussian, which contradicts an assumption of the information criteria.

This paper investigates the issue of robustness of complete stability of standard Cellular Neural Networks (CNNs) with respect to small perturbations of the nominally symmetric interconnections. More specifically, a class of circular one-dimensional (1-D) CNNs with nearest-neighbor interconnections only, is considered. The class has sparse interconnections and is subject to perturbations which preserve the interconnecting structure. Conditions assuring that the perturbed CNN has a unique equilibrium point at the origin, which is unstable, are provided in terms of relative magnitude of the perturbations with respect to the nominal interconnection weights. These conditions allow one to characterize regions in the perturbation parameter space where there is loss of stability for the perturbed CNN. In turn, this shows that even for sparse interconnections and structure preserving perturbations, robustness of complete stability is not guaranteed in the general case.

In the sixties, Łojasiewicz proved a fundamental inequality for vector fields defined by the gradient of an analytic function, which gives a lower bound on the norm of the gradient in a neighborhood of a (possibly) non-isolated critical point. The inequality involves a number belonging to (0, 1), which depends on the critical point, and is known as the Łojasiewicz exponent. In this paper, a class of vector fields which are defined on a hypercube of ℝⁿ, is considered. Each vector field is the gradient of a quadratic function in the interior of the hypercube, however it is discontinuous on the boundary of the hypercube. An extended Łojasiewicz inequality for this class of vector fields is proved, and it is also shown that the Łojasiewicz exponent at each point where a vector field vanishes is equal to 1/2. The considered fields include a class of vector fields which describe the dynamics of the output trajectories of a standard Cellular Neural Network (CNN) with a symmetric neuron interconnection matrix. By applying the extended Łojasiewicz inequality, it is shown that each output trajectory of a symmetric CNN has finite length, and as a consequence it converges to an equilibrium point. Furthermore, since the Łojasiewicz exponent at each equilibrium point of a symmetric CNN is equal to 1/2, it follows that each (state) trajectory, and each output trajectory, is exponentially convergent toward an equilibrium point, and this is true even in the most general case where the CNN possesses infinitely many nonisolated equilibrium points. In essence, the obtained results mean that standard symmetric CNNs enjoy the property of absolute stability of exponential convergence.

Stationary pattern formation in ensembles of coupled bistable elements is investigated both analytically and by means of numerical simulation. The models considered are similar to cellular nonlinear networks (CNNs) — a well-known class of collective dynamical systems intended mainly for image processing — but differ from them in the type of nonlinear functions contained in their equations of motion — cell output functions. The main subject of interest is the transformation of initial conditions, treated as a representation of a half-tone image, into a steady-state pattern. In the analytical part the location of attractors and their basins of attraction in the phase space are estimated for two types of CNN-like systems. In the simulation part the equations of two lattice systems — a CNN and a CNN-like system with local negative coupling — are integrated numerically with initial conditions taken in the form of a sample halftone image. The dependence of the patterns established in both systems upon parameters is studied and compared. The present paper proves, that CNN-like systems with modified cell output functions may be studied analytically, and results available for conventional CNNs may be adapted to this wider class of lattice systems. From the application point of view, it is shown, that the modification of cell output functions under certain conditions does not lead to a breakdown of the system functionality. Moreover, an example is presented, where such a modification allows to introduce new functionality into a system (namely, controlling contour lines width in an edge-detecting system).

This paper investigates the discrete-time recurrent neural networks and aims to extend the previous works with symmetric connection matrix to the asymmetric connection matrix. We provide the sufficient conditions of existence for asymptotical stability of fixed point, flip and fold bifurcations, Marotto's chaos. Besides, we state the conditions of existence for the bounded trapping region including many fixed points, and attracting set contained in bounded region and chaotic set. To demonstrate the theoretical results of the paper, several numerical examples are provided.The theorems in this paper are available more than in the previous works.

Using the method of planar dynamical systems to a higher order wave equations of KdV type, the existence of smooth and nonsmooth solitary wave, kink wave and uncountably infinite many periodic wave solutions is proved. In different regions of the parametric space, the sufficient conditions to guarantee the existence of the above solutions are given. In some spatial conditions, the exact explicit parametric representations of solitary wave solutions are determined.

In this paper, the bifurcation method of dynamical systems is employed to study the Camassa–Holm equation

In this paper, a novel method of energy analysis is developed for dynamical systems with time delays that are slightly perturbed from undamped SDOF/MDOF vibration systems. Being served frequently as the mathematical models in many applications, such systems undergo Hopf bifurcation including the classic "Hopf bifurcation" for SDOF systems and "multiple Hopf bifurcation" for MDOF systems, under certain conditions. An interesting observation of this paper is that the local dynamics near a Hopf bifurcation, including the stability of the trivial equilibrium and the bifurcating periodic solutions, of such systems, can be justified simply by the change of the total energy function. The key idea is that for the systems of concern, the total power (the total derivative of the energy function) can be estimated along an approximated solution with harmonic entries, the main part of the solution near the Hopf bifurcation. It shows that the present method works effectively for stability prediction of the trivial equilibrium and the bifurcating periodic solutions, and that it provides a high accurate estimation of the amplitudes of the bifurcating periodic solutions. Compared with the current methods such as the center manifold reduction which involves a great deal of symbolic computation, the energy analysis features a clear physical intuition and easy computation. Two illustrative examples are given to demonstrate the effectiveness of the present method.

In this paper a master-slave synchronization scheme is investigated by exploiting a dynamic output feedback control mechanism with time-delay and the influences of time-delay on synchronization are studied. Several new sufficient conditions, which are delay-independent or delay-dependent ones, are presented based on a general class of Lyapunov functions. The obtained results, expressed as matrix inequalities improve and generalize the earlier work in the literature and are readily verified via LMI control toolbox without tuning of parameters and/or matrices. The designs of the controller are implemented by solving a constrained nonlinear optimization problem. Finally, we illustrate our results on Chua's circuits and hyperchaotic attractors.

In this paper, we consider the weakened Hilbert's 16th problem for symmetric planar perturbed polynomial Hamiltonian systems. In particular, a perturbed Hamiltonian polynomial vector field of degree 9 is studied, and an example of Z₁₀-equivariant planar perturbed Hamiltonian systems is constructed. With maximal number of closed orbits, it gives rise to different configurations of limit cycles. By applying the bifurcation theory of planar dynamic systems and the method of detection functions, with the aid of numerical simulations, we show that a polynomial vector field of degree 9 with Z₁₀ symmetry can have at least 80 limit cycles, i.e. H(9) ≥ 9² - 1.

In this paper, the harmonic balance analysis of Generalized Chua's circuit exhibiting n-scroll attractors has been accomplished. The dual-input describing functions of the piecewise-linear characteristics of Chua's diode have been obtained and based on the harmonic balance principle, the existence and the locations of the n-scrolls have been verified.

We study how to employ space-time dynamics in nonlinear media to achieve distributed manipulation of objects — positioning, orienting and transporting objects by wave-fronts and patterns in excitable medium. We present the results of computational experiments of a massive parallel actuator controlled by a cellular-automaton model of an excitable medium. The model incorporates closed-loop actuation where sites of the medium can be excited not only by their closest neighbors but also by the edges of the manipulated object. We analyze motion of basic planar shapes (either initially aligned along axes or randomly oriented) induced by an actuator controlled by excitable lattice with various excitation rules. We demonstrate that space-time excitation dynamics in discrete nonlinear media bears a huge potential in terms of sensible nontrivial manipulation of planar shapes.

Inspired by the behavior of the ants in nature, we propose an optimization method, which combines the chaotic behavior of individual ants with the intelligent optimization action of an ant colony. Our method includes both effects of chaotic dynamics and swarm-based search. It is a deterministic process different from the conventional ant algorithm. The nonlinear dynamics of the proposed method are analyzed, and we show how the algorithm, called chaotic ant swarm optimization, can be applied to numerical optimization problems with encouraging results.

Concerning the problem of shape and pattern description, a compact formula (Gielis' formula) has recently been proposed that generates a vast diversity of natural shapes. However, this formula is a modified version of the equation for the circle, so that it is expressed in terms of trigonometric functions which are inherent to linear phenomena but rarely appear in the description of nonlinear phenomena. In this work, two examples of simple mathematical formulas which are natural nonlinear modifications (one being a generalization) of Gielis' formula are discussed. These formulas involve a comparable number of parameters and provide non-Platonic representations of a vast diversity of natural shapes and patterns by incorporating diverse aspects of asymmetry and seeming disorder which are absent in the original Gielis' formula. It is also shown how diverse sequences resembling some natural-world pattern evolutions are also generated by such nonlinear formulas.

The dynamics of the convergence to the critical attractor for the logistic map is investigated. At the border of chaos, when the Lyapunov exponent is zero, the use of the nonextensive statistical mechanics formalism allows to define a weak sensitivity or insensitivity to initial conditions. Using this formalism we analyze how a set of initial conditions spread all over the phase space converges to the critical attractor in the case of infinite bifurcation and tangent bifurcation points. We show that the phenomena is governed in both cases by a power-law regime but the critical exponents depend on the type of bifurcation and may also depend on the numerical experiment set-up. Differences and similarities between the two cases are also discussed.

A version of Sharkovskii's cycle coexistence theorem is formulated for a composition of connectivity G_δ-relations with closed values. Thus, a multivalued version in [Andres & Pastor, 2005] holding with at most two exceptions for M-maps, jointly with a single-valued version in [Szuca, 2003], for functions with a connectivity G_δ-graph, are generalized. In particular, our statement is applicable to differential inclusions as well as to some discontinuous functions.

This paper considers the globally exponential synchronization (GES) of the family of Rössler chaotic systems. One pair of the six transmitter-receiver systems is specifically studied, and algebraic criterion for the GES is obtained via proper nonlinear feedback controls. Based on the study of the systems' structures, appropriate Lyapunov functions are constructed for error systems. The method presented in this paper provides a convenient tool in the practical use of chaos control and synchronization. Numerical simulations are provided to demonstrate the theoretical results.

We consider the effect of random variation in the material parameters in a model for machine tool vibrations, specifically regenerative chatter. We show that fluctuations in these parameters appear as both multiplicative and additive noise in the model. We focus on the effect of additive noise in amplifying small vibrations which appear in subcritical regimes. Coherence resonance is demonstrated through computations, and is proposed as a route for transitions to larger vibrations. The dynamics also exhibit scaling laws observed in the analysis of general stochastic delay differential models.

An impulsive T–S fuzzy model is presented in this paper. The stability of impulsive controlled T–S fuzzy system has been analyzed theoretically. The proposed impulsive control scheme seems to have a simple control structure and may need less control energy than the normal continuous ones for the stabilization of T–S fuzzy system. Some typical chaotic systems, such as Chua's circuit, Lorenz system and Chen's chaotic system, are considered as illustrations to demonstrate the effectiveness of the proposed control scheme.

This paper presents a simulation of a biological olfactory neural system with a KIII set, which is a high-dimensional chaotic neural network. The KIII set differs from conventional artificial neural networks by use of chaotic attractors for memory locations that are accessed by, chaotic trajectories. It was designed to simulate the patterns of action potentials and EEG waveforms observed in electrophysiological experiments, and has proved its utility as a model for biological intelligence in pattern classification. An application to recognition of handwritten numerals is presented here, in which the classification performance of the KIII network under different noise levels was investigated.

In this paper a bifurcation analysis of a piecewise-affine discrete-time dynamical system is carried out. Such a system derives from a well-known map which has good features from its circuit implementation point of view and good statistical properties in the generation of pseudo-random sequences. The considered map is a generalization of it and the bifurcation parameters take into account some common circuit implementation nonidealities or mismatches. It will be shown that several different dynamic situations may arise, which will be completely characterized as a function of three parameters. In particular, it will be shown that chaotic intervals may coexist, may be cyclical, and may undergo several global bifurcations. All the global bifurcation curves and surfaces will be obtained either analytically or numerically by studying the critical points of the map (i.e. extremum points and discontinuity points) and their iterates. In view of a robust design of the map, this bifurcation analysis should come before a statistical analysis, to find a set of parameters ensuring both robust chaotic dynamics and robust statistical properties.

In this paper we demonstrate hyperchaotic dynamics in a very simple Cellular Neural Network (CNN) which is a one-dimensional regular array of four cells. The Lyapunov spectrum is calculated in a range of parameters, and the bifurcation plot is presented as well.