International Journal of Bifurcation and Chaos

In 2-D time-dependent fluid flows, a patch represents a localized region in space that has a significantly different average velocity compared to its surroundings. We show that one can obtain important information about the Lagrangian particle motion in such flows by studying the nature, long-term evolution, and statistical characteristics of the patchiness behavior. For example, the dispersion of passive tracers at any time is directly related to the distribution of patches in the flow. We thoroughly investigate the transport properties of the Lagrangian trajectories associated with a cellular flow previously used as a model for time-dependent Rayleigh–Bénard convection, and a kinematic model of a meandering jet (originally due to Bower [1991]). In both cases, we examine the statistical attributes of the patchiness, their relationship with the geometric features of the stable and unstable manifolds, and the effect of noise on the structure of patchiness. We uncover some interesting features associated with the origin of these patches and their influence on Lagrangian transport.

For a wide class of dynamical systems the variables involved relate to one another through a Cantor staircase function. When they are time variables, the staircases have well-known universal properties that suggest a connection with certain classical problems in Number Theory. In this paper we extend some of those universal properties to certain Cantor staircases that appear in Quantum Mechanics, where the variables involved are not time variables. We also develop some connections between the geometry of these Cantor staircases and the problem of approximating irrational numbers of rational ones, classical in Number Theory.

This paper presents an application of the local activity theory [Chua, 1998] to a specific reaction–diffusion cellular nonlinear network (CNN) with cells defined by a trimolecular model, called the Brusselator. Both the local activity domain and a subset called the "edge of chaos" are identified in the cell parameter space. Within these domains, various cell parameter points were selected to illustrate the effectiveness of the local activity theory in choosing the parameters for the emergence of complex (static and dynamic) patterns in a homogeneous lattice formed by coupled locally active cells.

The central barrier, asymmetry and random perturbation in transport and mixing by Rossby waves in a jet were investigated by simple kinematic model. Two complementary methods were used: A high-resolution Lagrangian Field Advection Model (FAM) and a finite-time Lyapunov exponent analysis. The present study revealed the following: (1) A central barrier can be formed in two Rossby waves without shear flow as well as in a jet, (2) the central barrier may occur in the region with maximum jet speed relative to the phase speed of the traveling wave, whereas the chaotic mixing most likely occurs near the critical lines; the central barrier widens as the phase speed of traveling waves relative to the jet speed increases, (3) asymmetry of wave-breaking is directly related to asymmetry of the critical line location in a jet, (4) the central barrier survives small random perturbations, (5) global bifurcation from a homoclinic orbit to a heteroclinic orbit and global chaos are two main mechanisms for the central barrier destruction. The results suggest that the small scale motions and random processes may not significantly affect the major character of Lagrangian transport and mixing by large-scale geophysical flow. Also potential vorticity mixing provides a unique kinematic and dynamic view of many features of the geophysical flow.

The spontaneous breakup of a single spiral wave of excitation into a turbulent wave pattern has been observed in both discrete element models and continuous reaction–diffusion models of spatially homogeneous 2D excitable media. These results have attracted considerable interest, since spiral breakup is thought to be an important mechanism of transition from the heart rhythm disturbance ventricular tachycardia to the fatal arrhythmia ventricular fibrillation. It is not known whether this process can occur in the absence of disease-induced spatial heterogeneity of the electrical properties of the ventricular tissue. Candidate mechanisms for spiral breakup in uniform 2D media have emerged, but the physical validity of the mechanisms and their applicability to myocardium require further scrutiny. In this letter, we examine the computer simulation results obtained in two discrete element models and show that the instability of each spiral is an artifact resulting from an unphysical dependence of wave speed on wave front curvature in the medium. We conclude that spiral breakup does not occur in these two models at the specified parameter values and that great care must be exercised in the representation of a continuous excitable medium via discrete elements.

Space and time scales are not independent in diffusion. In fact, numerical simulations show that different patterns are obtained when space and time steps (Δx and Δt) are varied independently. On the other hand, anisotropy effects due to the symmetries of the discretization lattice prevent the quantitative calibration of models. We introduce a new class of explicit difference methods for numerical integration of diffusion and reaction–diffusion equations, where the dependence on space and time scales occurs naturally. Numerical solutions approach the exact solution of the continuous diffusion equation for finite Δx and Δt, if the parameter γ_N=DΔt/(Δx)² assumes a fixed constant value, where N is an odd positive integer parametrizing the algorithm. The error between the solutions of the discrete and the continuous equations goes to zero as (Δx)^2(N+2) and the values of γ_N are dimension independent. With these new integration methods, anisotropy effects resulting from the finite differences are minimized, defining a standard for validation and calibration of numerical solutions of diffusion and reaction–diffusion equations. Comparison between numerical and analytical solutions of reaction–diffusion equations give global discretization errors of the order of 10^-6 in the sup norm. Circular patterns of traveling waves have a maximum relative random deviation from the spherical symmetry of the order of 0.2%, and the standard deviation of the fluctuations around the mean circular wave front is of the order of 10^-3.

A model of a simple electric power supply network involving two generators connected by a transmission network to a bus is studied by numerical simulation. In this model, the bus is supposed to maintain a voltage of fixed amplitude, but with a small periodic fluctuation in the phase angle. In such a case, traditional analysis using direct methods is not applicable. The frequency of the periodic fluctuation is varied over a range of values near a nonlinear resonance of the two-generator network. When the bus fluctuation frequency is away from resonance, the system has several attractors; one is a small-amplitude periodic oscillation corresponding to synchronized, quasi-normal operation (slightly swinging), while others are large amplitude periodic oscillations which, if realized, would correspond to one or both generators operating in a desynchronized steady state.

When the bus fluctuation frequency approaches resonance, a new periodic attractor with large amplitude oscillations appears. Although it does correspond to a synchronized steady state, this attractor has a disastrously large amplitude of oscillation, and represents an unacceptable condition for the network. Basin portraits show that this resonant attractor erodes large, complicated regions of the basin of the safe operating condition. Under conditions of small periodic fluctuation in bus voltage, this basin erosion would not be detected by traditional analysis using direct methods. Further understanding of such complicated basin structures will be essential to correctly predict the stability of electric power supply systems.

This paper is the first part of a study of the Hénon and Heiles problem in three dimensions. Due to the axial symmetry of the Hamiltonian, the third component of the angular momentum is an integral and the system is considered as a Hamiltonian with two degrees of freedom. As functions of that integral, we show the existence of three circular trajectories around the axis Oz and a domain for which we have bounded motions. In the part of that domain near the origin, the corresponding dynamical system is treated as a perturbed harmonic oscillator in 1–1–1 resonance. We present some numerical studies searching for periodic orbits, showing the corresponding Poincaré surfaces of section. In addition, we obtain some natural families of periodic orbits associated with the relative equilibria of the fourth order normalized system.

The second part of a research on the Hénon and Heiles system in three dimensions is presented. We focus on motions around the origin, where the system may be treated as a case of three perturbed isotropic harmonic oscillators. It is the only Hamiltonian of the family of axially symmetric cubic potentials in 1–1–1 resonance, which needs at least order four in the normalization to reach the features of the system. The fourth order normalized system is analyzed, considering orbits at any inclination. We study and classify the relative equilibria of the system finding pitchfork, centre-saddle and cusp bifurcations in the reduced orbit space.

The technique of surrogate data has been used as a method to test for membership of particular classes of linear systems. We suggest an obvious extension of this to classes of nonlinear parametric models and demonstrate our methods with respiratory data from sleeping human infants. Although our data are clearly distinct from the different classes of linear systems we are unable to distinguish between our data and surrogates generated by nonlinear models. Hence we conclude that human respiration is likely to be a nonlinear system with more than two degrees of freedom with a limit cycle that is driven by high dimensional dynamics or noise.

A complete analysis of various behaviors in discretization of a sliding mode control system is given. These behaviors are classified in three types: inherent periodic behaviors, convergent periodic behaviors and unstable behaviors. In particular, there are two classes of periodic behavior patterns: One class is a set of discrete points; the other a set of dense ellipses. These classes are determined by the sampling period which acts as an additional dimension of freedom to the discretized sliding mode control system. The relation between the sampling period and periodic behaviors is explored and the bifurcation point that distinguishes the stable periodic behaviors and irregular behaviors is given. The analysis is accompanied by various simulations.

In this paper, methods are shown how to adapt invertible two-dimensional chaotic maps on a torus or on a square to create new symmetric block encryption schemes. A chaotic map is first generalized by introducing parameters and then discretized to a finite square lattice of points which represent pixels or some other data items. Although the discretized map is a permutation and thus cannot be chaotic, it shares certain properties with its continuous counterpart as long as the number of iterations remains small. The discretized map is further extended to three dimensions and composed with a simple diffusion mechanism. As a result, a symmetric block product encryption scheme is obtained. To encrypt an N×N image, the ciphering map is iteratively applied to the image. The construction of the cipher and its security is explained with the two-dimensional Baker map. It is shown that the permutations induced by the Baker map behave as typical random permutations. Computer simulations indicate that the cipher has good diffusion properties with respect to the plain-text and the key. A nontraditional pseudo-random number generator based on the encryption scheme is described and studied. Examples of some other two-dimensional chaotic maps are given and their suitability for secure encryption is discussed. The paper closes with a brief discussion of a possible relationship between discretized chaos and cryptosystems.

The paper analyzes a model of immune system developed by different authors (Perelson, De Boer, Weisbuch and others). The model describes interactions among B-lymphocytes. It does not consider antibodies as interaction intermediaries, although it uses a typical activation curve. The relevant parameters are: an influx term, a threshold value, a proliferation rate, and a decay parameter. The study of the n-dimensional extension of the model and a bifurcation analysis of the stationary states with respect to the influx parameter show that the influx value for which biologically acceptable solutions exist decreases as n increases. When the influx term is neglected the stationary states are obtained analytically and their stability is discussed. Moreover, it is discussed how the stable solutions can be considered as "selective states", that is, with only one high idiotypic concentration, when we suppose a complete connectivity.

The phenomenon of spatiotemporal chaos as it arises in systems of reaction–diffusion equations has been the subject of much study over recent years. Initially, chaotic responses in these systems were found to occur when the ratio of the diffusion coefficients was chosen sufficiently large (greater than one). Subsequently, such behavior has also been found when this ratio is taken to be unity with the conclusion that this represents a new mechanism for inducing chaos. In this paper we discuss numerical simulations of a system of reaction–diffusion equations which has been derived to study the large-scale growth patterns of fungal colonies. We show that, for this reaction–diffusion system at least, these apparently different mechanisms represent two ways of entering the same "chaotic region" in parameter space. We also discuss other wave-induced behavior which characterize the routes to chaos.

Rotating vortices are easily pinned to any defects in excitable media. Moderate advective fields cannot remove the pinned vortices. We have found and tested experimentally a mechanism for unpinning a vortex. To unpin a vortex, we created additional waves close to its core. A vortex can be unpinned when a new created wave is pinned to the same obstacle and has a topological defect. This mechanism works perfectly both in experiment and in numerical simulations. A hypothesis is put forward that similar mechanisms may underlay pacing for tachycardia control used in cardiac clinics.

Deterministic chaos has been studied extensively in various fields. Some of the ideas emerging out of these studies have been put to novel applications. However, it is unknown whether natural ecological systems support chaotic dynamics. There is no concrete evidence which suggests that ecosystem evolution is chaotic in certain situations. This is very intriguing because ecosystems do possess all the necessary qualifications to be able to support such a dynamical behavior.

The present paper attempts to answer the above question with the help of a few systems modeling different but very common ecological situations. A new methodology for the analysis of a class of model ecological systems is presented. Simulation experiments suggest that natural terrestrial systems are not suitable candidates where one should look for chaos. Additionally, our study also points out that the failure of attempts to observe chaos in natural populations might have resulted because biological interactions are not conducive for such a behavior to be supported. The cause of these failures may not be the poor data quality or demerits in the analysis techniques.

Two sinusoidal oscillator circuits, that employ an operational amplifier (op amp) as a current negative impedance converter (CNIC), are modified for chaos using a nonlinear resistor of anti-symmetrical current-voltage characteristics formed by a junction field effect transistor (JFET) operating in the triode region. The internal op amp dominant pole is found to contribute significantly to the chaotic nature of one circuit while the other circuit develops different chaotic attractors when cubic and fifth power odd symmetrical nonlinearities are used. Mathematical models of the two generators are presented. Experimental laboratory results, circuit simulations and numerical simulations of the mathematical models well agree and are included.

Oscillatory and chaotic regimes arising from Andronov-Hopf bifurcation are obtained in a model of fiber-optic interferometric strain gauge with delayed nonlinear feedback as a result of analytical and numerical investigation. Complicated behavior of solutions in the case of infinite number of equilibrium points is reported.

In this work, we extend the periodic impulsive method to high-dimensional conservative system, and use the coupled standard map as an example to illustrate how to control the high-dimensional Hamiltonian chaos. When the periodic pulse is applied to one variable of the system, different periodic windows, quasiperiodic orbit and some special structure can be stabilized, even the approximate KAM curve. And the largest Lyapunov exponent spectra corresponding to different initial points are given.

This Letter proposes a new synchronization theorem for a subclass of chaotic Lur'e systems. We take a specific state variable of the drive system as the driving signal. We prove that globally synchronization can be attained via the simple linear error feedback. The approach is illustrated using Chua's chaotic oscillator and a hyperchaotic oscillator.

In this paper we consider impulsive control of master-slave synchronization schemes that consist of identical Lur'e systems. Impulsive control laws are investigated which make use of linear or nonlinear dynamic measurement feedback. A sufficient condition for global asymptotic stability is presented which is characterized by a set of matrix inequalities. Synchronization is proven for the error between the output signals. The method is illustrated on Chua's circuit and a hyperchaotic system with coupled Chua's circuits.