International Journal of Bifurcation and Chaos

This paper is part I of a series of contributions on the classification problem of chaos in three-dimensional autonomous quadratic systems. We try to classify chaos, based on the Ši'lnikov criteria, in such a large class of systems into the following four types: (1) chaos of the Ši'lnikov homoclinic orbit type; (2) chaos of the Ši'lnikov heteroclinic orbit type; (3) chaos of the hybrid type; i.e. those with both Ši'lnikov homoclinic and homoclinic orbits; (4) chaos of other types. We are especially interested in finding out all the simplest possible forms of chaotic systems for each type of chaos. Our main contributions are to develop some effective classification methods and to provide a basic classification framework under which each of the four types of chaos can be justified by some examples that are useful for describing the feasibility and procedure of the classification. In particular, we show several novel chaotic attractors, e.g. one hybrid-type chaotic attractor with three equilibria, one heteroclinic orbit and one homoclinic orbit, and one 4-scroll chaotic attractor with five equilibria and two heteroclinic orbits.

The Correlation Dimension estimation is an especially sensitive method which allows for important information to be extracted from the signal under investigation. However, due to its sensitivity, its parameters as well as details of the application process must be chosen carefully and customized to each specific problem. In this paper, we examine the application of the Correlation Dimension estimation in the case of heart rate variability signals. Specifically, we investigate issues which are not discussed in other similar studies or which are chosen differently, such as the use of the Euclidean norm and the Theiler window. We also document with detail the method of data acquisition and make a case for the importance of proper recordings the lack of which leads to controversial results.

The emergence of complexity is investigated from the viewpoint of the energy balance property and the divergence property of reaction–diffusion cellular neural networks.

A perturbation-incremental (PI) method is presented for the computation, continuation and bifurcation analysis of periodic solutions of nonlinear systems of delay differential equations (DDEs). Periodic solutions can be calculated to any desired degree of accuracy and their stabilities are determined by the Floquet theory. Branch switching at a period-doubling bifurcation is made simple by the present scheme as a parameter is simply increased from zero to a small positive value so that a solution on the new branch is obtained. Subsequent continuation of an emanating branch is also discussed. The advantage of the PI method lies in its simplicity and ease of implementation.

We present a generalized Lyapunov Schmidt (LS) reduction scheme for diffeomorphisms on a finite dimensional real vector space V which transform under real one-dimensional characters χ of an arbitrary compact group with linear action on V. Moreover we prove a normal form theorem, such that the normal form still has the desirable transformation properties with respect to χ.

In this paper we present, the design and modeling of the novel nonlinear limiter control feedback control plant [Myneni et al., 1999; Corron et al., 2000; Corron & Pethel, 2002], applied for the first time here in an aeroelastic system, and actuated as a jet reaction torquer control of a wing with potentially chaotic dynamics. This study will provide a better understanding of the nonlinear dynamics of the open/closed-loop aeroelasticity of flexible wings with either steady or unsteady aerodynamic loads. The limiter control can be applied to either the plunging or pitching characteristic of the wing or to both of them. We show that the control can effectively suppress Limit Cycle Oscillations (LCO) and chaos well beyond the nominal flutter speed. This could lead to a practical implementation of the control mechanism on actual and future generation aircraft wings via implementation of a combination of propulsive/jet type forces, micro surface effectors and fluidic devices. Analysis of this control produced favorable results in the suppression of LCO amplitude and increased flutter boundaries for plunging and pitching motion. The limiting control has asymptotically zero power, and is simply implemented, making it a feasible solution to the problem of the chaotic dynamics of the oscillating airfoil.

In this paper, we propose an exploited single-species discrete population model with stage structure for the dynamics in a fish population for which births occur in a single pulse once per time period. Using the stroboscopic map, we obtain an exact cycle of the system, and obtain the threshold conditions for its stability. Bifurcation diagrams are constructed with the birth rate (or harvesting effort) as the bifurcation parameter, and these are observed to display complex dynamic behaviors, including chaotic bands with period windows, pitchfork and tangent bifurcation, nonunique dynamics (meaning that several attractors or attractor and chaos coexist), basins of attraction and attractor crisis. This suggests that birth pulse provides a natural period or cyclicity that makes the dynamical behaviors more complex. Moreover, we show that the timing of harvesting has a strong impact on the persistence of the fish population, on the volume of mature fish stock and on the maximum annual-sustainable yield. An interesting result is obtained that, after the birth pulses, the population can sustain much higher harvesting effort if the mature fish is removed as early in the season as possible.

The influence induced by random noise on dynamical behaviors is a classical yet challenging subject. This paper discusses the influence of Gaussian white noise on the dynamics of a self-excited triple well extended Duffing–Van der Pol oscillator already subjected to harmonic excitation. Firstly, the condition for the rise of hom/heteroclinic chaos is derived by random Melnikov's technique under its corresponding mean-square criterion and the result indicates that the threshold amplitude of harmonic excitation is lowered by the appearance of Gaussian white noise. Moreover, the threshold is decreased as the noise intensity increases. Since the Melnikov's criterion is only a necessary condition for the occurrence of chaotic motion, this prediction is tested against numerical simulations of the basins of attraction and the Lyapunov exponents. By vanishing the largest Lyapunov exponents, another criterion for the onset of chaos is obtained which is accorded with the theoretical one. Finally, how the noise effects the structure of periodic or chaotic attractor is investigated by simulating Poincare maps of the original system and rich transition states displayed by the considered extended Duffing–Van der Pol oscillator are observed.

This paper aims to provide some new results, by a computer-assisted study, on some global bifurcations that change the domains of feasible trajectories (bounded discrete trajectories having an ecological sense). It is shown that the domain boundaries of two-dimensional maps can be generally obtained by the union of all rank preimages of two axes. A discrete predator–prey ecosystem is employed to demonstrate how the global properties of persistence in a biological model can be analyzed. The main results of this paper are from the study of some global bifurcations that change the stable manifold of a saddle fixed point belonging to the domain boundary. The basin fractalization is explained by using two types of nonclassical singular sets. The first one, critical curve, separates the plane into two regions having a different number of real inverses (here zero and two). The second one is a line of nondefinition for one of the two inverses of the map, i.e. this inverse has a vanishing denominator on this line.

This paper is concerned with chaotification of discrete dynamical systems in Banach spaces via feedback control techniques. A criterion of chaos in Banach spaces is first established. This criterion extends and improves the Marotto theorem. Discussions are carried out in general and some special Banach spaces. All the controlled systems are proved to be chaotic in the sense of both Devaney and Li–Yorke. As a consequence, a controlled system described in a finite-dimensional real space studied by Wang and Chen is shown chaotic not only in the sense of Li–Yorke but also in the sense of Devaney. The original system can be driven to be chaotic by using an arbitrarily small-amplitude state feedback control in a certain space. In addition, the Chen–Lai anti-control algorithm via feedback control with mod-operation in a finite-dimensional real space is extended to a certain infinite-dimensional Banach space, and the controlled system is shown chaotic in the sense of Devaney as well as in the sense of both Li–Yorke and Wiggins. Differing from many existing results, it is not here required that the map corresponding to the original system has a fixed point in some cases. An application of the theoretical results to a class of first-order partial difference equations is given with some numerical simulations.

In this work, the evolution of a family of open-to-closed curves of saddle-node bifurcations of periodic orbits is numerically studied in Chua's equation. This family is organized by open and closed curves of Shil'nikov homoclinic connections in the presence of a nontransversal T-point. We remark the important role of cusp catastrophes of periodic orbits in the complete mechanism of creation/destruction of such closed curves of saddle-node bifurcations.

In order to operate in higher dimensional form of autonomous and nonautonomous Chua's circuits keeping their original chaotic behaviors, we have experimentally modified VOA (Voltage Mode Operational Amplifier)-based autonomous Chua's circuit and nonautonomous MLC [Murali–Lakshmanan–Chua] circuit by using a simple experimental method. After introducing this experimental method, we will present PSpice simulation and experimental results of modified high dimensional autonomous and nonautonomous Chua's circuits.

In this paper, we study the existence of a new class of chaotic attractors, namely the rank-one attractors, in the MLC (Murali–Lakshmanan–Chua) circuit [Murali et al., 1994] by numerical simulations based on a theory of rank-one maps developed in [Wang & Young, 2005]. With the guidance of the theory in [Wang & Young, 2005], weakly stable limit cycles, found through Hopf bifurcations and other numerical means, are subjected to periodic pulses with long relaxation periods to produce rank-one attractors. The periodic pulses are applied directly as an input. Periodic pulses have been used before in various schemes of chaos. However, for this scheme of creating rank-one attractors to work, the applied periodic pulses must have short pulse widths and long relaxation periods. This is one of the key components in creating this new class of chaotic attractors.

This Letter studies adaptive chaos synchronization between two coupled nonlinear electrical circuits using a linear state feedback. A simple and yet easily applicable algorithm is derived for adaptive chaos synchronization based on the Lyapunov stability theory. An adaptive linear state feedback is derived such that the coupled systems are globally synchronized. Numerical simulations show the effectiveness of the proposed synchronization scheme. A Pspice simulation of the adaptive synchronization process is performed to confirm the numerical results.

This paper proposes an adaptive observer-based synchronization of continuous-time chaotic systems. Our method concerns any chaotic system that comprises uncertainties. Using the Lyapunov stability theory, we show that the synchronization of two coupled chaotic systems can be achieved by means of a linear coupling. A remarkable feature of the proposed approach is that it is robust with respect to noise and parameter mismatch. Numerical simulation of the Chua's circuit verifies the efficiency and effectiveness of the proposed adaptive synchronization scheme.

We report on the experimental observation of dynamic behaviors in diode pumped passively Q-switched Nd:GdVO₄ and Nd:YVO₄ lasers. In addition to the period-doubling route to chaos, various periodic windows have also been observed. Based on an extended rate equation model the chaotic nature of the lasers is numerically simulated. It was found that under the experimental condition the model could qualitatively reproduce the experimental observations.

In this paper, the motion law of some classical spatial physical motion systems is described, and a uniform motion system is obtained from them. The uniform motion system is analyzed and compared with the currently popular model, i.e. coupled map lattice (CML) model. CML is just one special type of the uniform motion system. In the uniform motion system, an iterative periodic orbit is constructed and the spatial chaotic behavior in the sense of Li–Yorke is studied. The proposed uniform motion system provides a reliable theoretic foundation in the application of the classical motion systems used in the field of engineering technology.

Recently, we have developed a phase field model to describe Marangoni convection with evaporation in a compressible fluid of van der Waals type away from criticality [Eur. Phys. J. B44 (2005)]. Using this model, we report now 2D fully nonlinear simulations where we emphasize the influence of evaporation on convective patterns.

Momentum distributions and temporal power spectra of nonzero temperature Bose–Einstein condensates are calculated using a Gross–Pitaevskii model. The distributions are obtained for micro-canonical ensembles (conservative Gross–Pitaevskii equation) and for grand-canonical ensembles (Gross–Pitaevskii equation with fluctuations and dissipation terms). Use is made of equivalence between statistics of the solutions of conservative Gross–Pitaevskii and dissipative complex Ginzburg–Landau equations. In all cases the occupation numbers of modes follow a 〈N_k〉 ∝ k^-2 dependence, which corresponds in the long wavelength limit (k → 0) to Bose–Einstein distributions. The temporal power spectra are of 1/f^α form, where: α = 2 - D/2 with D the dimension of space.

We propose a new prediction method for nonlinear time series based on the paradigm of deterministic chaos. Introducing a stochastically equivalent dynamical system to an original map, a prediction method is derived by minimizing a random term that defines intervals in which a good prediction performance is obtained. The use of the present method is illustrated for some chaotic systems with particular emphasis on issues of choices of variable time steps that are necessary when discretizing the stochastic differential equation. Applying to some systems, it is found that the present method works better than traditional chaotic methods.

This paper presents a new class of chaotic and hyperchaotic low dimensional cellular neural networks modeled by ordinary differential equations with some simple connection matrices. The chaoticity of these neural networks is indicated by positive Lyapunov exponents calculated by a computer.

This paper addresses qualitative properties of equilibrium points in a class of delayed neural networks. We derive a sufficient condition for the local exponential stability of equilibrium points, and give an estimate on the domains of attraction of locally exponentially stable equilibrium points. Our condition and estimate are formulated in terms of the network parameters, the neurons' activation functions and the associated equilibrium point; hence, they are easily checkable. Another advantage of our results is that they neither depend on monotonicity of the activation functions nor on symmetry of the interconnection matrix. Our work has practical importance in evaluating the performance of the related associative memory. To our knowledge, this is the first time to present an estimate on the domains of attraction of equilibrium points for delayed neural networks.

The human activity (e.g. fish farming) may result in the supply of undesired species (with poor heredity) to the wild, which may deteriorate the quality of the wild species of the same kind or even cause their extinction. We present a generic three-variable model focused on this problem. The model includes wild and cultivated preys and predators. The coexistence of these species is predicted to be possible provided that the supply of the cultivated preys is below the critical value. If this condition is not fulfilled, the wild preys asymptotically disappear. In the case of temporal mean-field kinetics, this transition representing mathematically a saddle-node bifurcation depends on the rate constants of the steps describing the interplay of the species. In the spatio-temporal case with the supply of undesired species at the boundary of the habitat, the situation depends also on the rate of migration of species and the habitat size.

This study reports the results of the characterization of an air-water two-phase experimental apparatus and the preliminary analyses of the experimental time series. The test section of the apparatus consists of a vertical pipe equipped with an impedance void fraction sensor. The carrying frequency of the impedance sensor has been chosen in order to operate it as a resistive sensor. The calibration of the sensor has been performed through comparison of the instantaneous two-phase mixture conductivity signal and the local actual dimension of the bubble as estimated from high resolution photograph. The calibration curve allows, therefore, reliable estimation of the void fraction time series.

A preliminary analysis of the time series has been performed both in time and frequency domains, evaluating also the time series autocorrelation. These analyses have pointed out the inadequacy of linear tools for the characterization of two-phase flow dynamics, which are nonetheless characterized by strong recurrence and autocorrelation, which need to be further exploited by mean of nonlinear analysis in phase space. Phase space representation of different typical flow patterns, corresponding to a succession of bifurcation, shows the high potential of nonlinear analytical tools, to be adopted in order to exploit the system dynamics.

The so-called bottleneck or ghost can appear after a saddle-node bifurcation, leaving a region in phase space by which the flow is attracted although no fixed points are present. Such ghosts, displayed by some dynamical systems, actually cause a delay of the flow. In this paper, we analyze a saddle-node ghost found in a biological model for the two-member hypercycle dynamics. The model predicts a scaling law of the dynamic delay caused by the ghost near the threshold: τ ~ ϕ^-1/2, consistent with previous results in physical systems. Possible biological meanings for such a dynamical phenomenon are outlined.

We study the dynamical behavior of the collective field of chaotic systems on small world lattices. Coupled neuronal systems as well as coupled logistic maps are investigated. We observe that significant changes in dynamical properties occur only at a reasonably high strength of nonlocal coupling. Further, spectral features, such as signal-to-noise ratio (SNR), change monotonically with respect to the fraction of random rewiring, i.e. there is no optimal value of the rewiring fraction for which spectral properties are most pronounced. We also observe that for small rewiring, results are similar to those obtained by adding small noise.