International Journal of Bifurcation and Chaos

Linear transformation techniques such as singular value decomposition (SVD) have been used widely to gain insight into the qualitative dynamics of data generated by dynamical systems. There have been several reports in the past that had pointed out the susceptibility of linear transformation approaches in the presence of nonlinear correlations. In this tutorial review, the local dispersion along with the surrogate testing is suggested to discriminate nonlinear correlations arising in deterministic and non-deterministic settings.

This paper presents, in a tutorial manner, nonlinear phenomena such as bifurcations and chaotic behavior in DC–DC switching converters. Our purpose is to present the different modeling approaches, the main results found in the last years and some possible practical applications. A comparison of the different models is given and their accuracy in predicting nonlinear behavior is discussed. A general Poincaré map is considered to model any multiple configuration of DC–DC switching converters and its Jacobian matrix is derived for stability analysis. More emphasis is done in the discrete-time approach as it gives more accurate prediction of bifurcations. The results are reproduced for different examples of DC–DC switching converters studied in the literature. Some methods of controlling bifurcations are applied to stabilize Unstable Periodic Orbits (UPOs) embedded in the dynamics of the system. Statistical analysis of these systems working in the chaotic regime is discussed. An extensive list of references is included.

In this paper, the possible presence of complex dynamics in nearly-symmetric standard Cellular Neural Networks (CNNs), is investigated. A one-parameter family of fourth-order CNNs is presented, which exhibits a cascade of period-doubling bifurcations leading to the birth of a complex attractor, close to some nominal symmetric CNN. Different from previous work on this topic, the bifurcations and complex dynamics are obtained for small relative errors with respect to the nominal interconnections. The fourth-order CNNs have negative (inhibitory) interconnections between distinct neurons, and are designed by a variant of a technique proposed by Smale to embed a given dynamical system within a competitive dynamical system of larger order.

In this paper, a two-neuron network with different time delays is investigated. By analyzing the associated characteristic equation, we obtain the conditions for delay-dependent and delay-independent asymptotic stability, respectively. Furthermore, we find that if the delay is used as a bifurcation parameter, Hopf bifurcation would occur. The direction and stability of the bifurcating periodic solutions are determined by using the Nyquist criterion and the graphical Hopf bifurcation theorem. Some examples are included to illustrate our results.

Consider a three-dimensional system having an invariant surface and a weak focus of order k on the invariant surface. By using bifurcation techniques and analyzing the solutions of bifurcation equations, we study the spatial bifurcation phenomena near the weak focus and obtain the conditions for the existence of periodic orbits near the weak focus. An application of our results to a SIRS epidemiological model is given.

Unstable invariant sets are important to understand mechanisms behind many dynamically important phenomenon such as chaotic transients which can be physically relevant in experiments. However, unstable invariant sets are nontrivial to find computationally. Previous techniques such as the PIM triple method [Nusse & Yorke, 1989] and simplex method variant [Moresco & Dawson, 1999], and even the step-and-stagger method [Sweet et al., 2001] have computationally inherent dimension limitations. In the current study, we explicitly investigate the landscape of an invariant set, which leads us to a simple gradient search algorithm to construct points close to the invariant set. While the calculation of the necessary derivatives can be computationally very expensive, the methods of our algorithm are not as dimension dependant as the previous techniques, as we show by examples such as the two-dimensional instability example from [Sweet et al., 2001] followed by a four-dimensional instability example, and then a nine-dimensional flow from the Yoshida equations, with a two-dimensional instability.

This paper is an attempt to give new results, by a computer-assisted study, on some global bifurcations that change the structure of the domain of feasible trajectories (bounded discrete trajectories having an ecological sense) which can be obtained by the union of all rank preimages of axes. Three two-dimensional recurrence equations (or maps) are analyzed. The two first maps are degenerated invertible maps (i.e. the inverses of them are well defined except a set of zero lebergue measure) for which the basins of attractor are obtained by the backward iteration of a stable manifold of a saddle fixed point belonging to the basin boundary, and the interior domains of feasible trajectories are given by the intersection between the basin of attractor and the first quadrant. The other is a noninvertible map which is investigated by the use of critical curves, a powerful tool for the analysis of global properties of two-dimensional maps.

Adaptive controllers are used in systems where one or more parameters are unknown. Such controllers are designed to stabilize the system using an estimate for the unknown parameters that is adapted automatically as part of the stabilization. One drawback in adaptive control design is the possibility that the closed-loop limit system is not stable. The worst situation is the existence of a destabilized limit system attracting a large open subset of initial conditions.

These situations lie behind bad behavior of the closed-loop adaptive control system. The main issue in this paper is to identify and characterize the occurrence of such bad behavior in the adaptive stabilization of first- and second-order systems with one unknown parameter. We develop normal forms for all possible cases and find the conditions that lead to bad behavior. In this context, we discuss a number of bifurcation-like phenomena.

The behavior of two-dimensional coupled map lattices is studied with respect to the global stabilization of unstable local fixed points without external control. It is numerically shown under which circumstances such inherent global stabilization can be achieved for both synchronous and asynchronous updating. Two necessary conditions for inherent global stabilization are derived analytically.

The multiple Devil's staircase reported by Qu, Wu and He in 1997 was composed of many tower-like structures. Each of which include two branches of the conventional complete Devil's staircases that connect the "single-exit" top and bottom steps. Their analytic conclusion show that all the conventional Devil's staircases are confined by two smooth curves with similar function forms W ∝ 1/ln ∊. These may be addressed as a kind of consonance. Our recent study found that it happens only in a few cases. Actually, in their system, 16 different kinds of tower branches exist in most parts of the parameter space. A lot of the steps lose the consonant property which are the so-called dissonant structures. The number of types of the corresponding dissonant branches is employed to describe the dissonance of the staircase. When the number of the discontinuous regions n, in the system develops, the dissonance of the staircase increases with 2n³ - n rule. The numerical result shows that the conclusion is valid for a general discontinuous circle map.

The aim of this paper is to present results concerning a three-dimensional model including a prey, a predator and top-predator, which we have named the Volterra–Gause model because it combines the original model of V. Volterra incorporating a logisitic limitation of the P. F. Verhulst type on growth of the prey and a limitation of the G. F. Gause type on the intensity of predation of the predator on the prey and of the top-predator on the predator. This study highlights that this model has several Hopf bifurcations and a period-doubling cascade generating a snail shell-shaped chaotic attractor.

With the aim of facilitating the choice of the simplest and most consistent model a comparison is established between this model and the so-called Rosenzweig–MacArthur and Hastings–Powell models. Many resemblances and differences are highlighted and could be used by the modellers.

The exact values of the parameters of the Hopf bifurcation are provided for each model as well as the values of the parameters making it possible to carry out the transition from a typical phase portrait characterizing one model to another (Rosenzweig–MacArthur to Hastings–Powell and vice versa).

The equations of the Volterra–Gause model cannot be derived from those of the other models, but this study shows similarities between the three models. In cases in which the top-predator has no effect on the predator and consequently on the prey, the models can be reduced to two dimensions. Under certain conditions, these models present slow–fast dynamics and their attractors are lying on a slow manifold surface, the equation of which is given.

Chua's diode is widely used in nonlinear circuit analysis and modeling. In this paper, we propose canonical oscillator models having hysteretic Chua's diodes as their nonlinear elements. Many hysteresis oscillators can be classified into unified groups by using these models. We also present a method for deriving new hysteresis oscillators from known slow–fast (relaxation) oscillators.

In this contribution, eleven different measures of the complexity of multichannel EEGs are described, and their effectiveness in discriminating between two behavioral conditions (eyes open resting versus eyes closed resting) is compared. Ten of the methods were variants of the algorithmic complexity and the covariance complexity. The eleventh measure was a multivariate complexity measure proposed by Tononi and Edelman. The most significant between-condition change was observed with Tononi–Edelman complexity which decreased in the eyes open condition. Of the algorithmic complexity measures tested, the binary Lempel–Ziv complexity and the binary Lempel–Ziv redundancy of the first principal component following mean normalization and normalization against the standard deviation gave the most significant between-group discrimination. A time-dependent generalization of the covariance complexity that can be applied to nonstationary multichannel signals is also described.

We model transmission of the Severe Acute Respiratory Syndrome (SARS) associated coronavirus (SARS-CoV) in Hong Kong with a complex small world network. Each node in the network is connected to its immediate neighbors and a random number of geographically isolated nodes. Transmission can only occur along these links. We find that this model exhibits dynamics very similar to those observed during the SARS outbreak in 2003. We derive an analytic expression for the rate of infection and confirm this expression with computational simulations. An immediate consequence of this quantity is that the severity of the SARS epidemic in Hong Kong in 2003 was due to ineffectual infection control in hospitals (i.e. nosocomial transmission). If all infectious individuals were isolated as rapidly as they were identified the severity of the outbreak would have been minimal.

We study probability distributions of permutations and binary words, which arise in symbolic analysis of time series and their differences. Under the assumptions that the series is stationary and independent we show that these probability distributions are universal and we derive a recursive algorithm for computing the distribution of binary words. This provides a general framework for performing chi square tests of goodness of fit of empirical distributions versus universal ones. We apply these methods to analyze heartbeat time series; in particular, we measure the extent to which atrial fibrillation can be modeled as an independent sequence.

The detrended fluctuation analysis (DFA) [Peng et al., 1994] and its extensions (MF-DFA) [Kantelhardt et al., 2002] have been used extensively to determine possible long-range correlations in self-affine signals. While the DFA has been claimed to be a superior technique, recent reports have indicated its susceptibility to trends in the data. In this report, a smoothing filter is proposed to minimize the effect of sinusoidal trends and distortion in the log–log plots obtained by DFA and MF-DFA techniques.

A system of ODE's is used to attempt an approximation of the dynamics of two delayed coupled FitzHugh–Nagumo excitable units, described by delay-differential equations. It is shown that the codimension 2 generalized Hopf bifurcation acts as the organizing center for the dynamics of ODE's for small time-lags. Furthermore, this is used to explain important qualitative properties, like the phenomenon of oscillator death, of the exact dynamics for small time-delays.

This paper presents a study on the problem of Hopf bifurcation control of time delayed systems with weak nonlinearity via delayed feedback control. It focusses on two control objectives: one is to annihilate the periodic solution, namely to perform a linear delayed feedback control so that the trivial equilibrium is asymptotically stable, and the other is to obtain an asymptotically stable periodic solution with given amplitude via linear or nonlinear delayed feedback control. On the basis of the averaging method and the center manifold reduction for delayed differential equations, an effective method is developed for this problem. It has been shown that a linear delayed feedback can always stabilize the unstable trivial equilibrium of the system, and a linear or nonlinear delayed feedback control can always achieve an asymptotically stable periodic solution with desired amplitude. The illustrative example shows that the theoretical prediction is in very good agreement with the simulation results, and that the method is valid with high accuracy not only for delayed systems with weak nonlinearity and via weak feedback control, but also for those when the nonlinearity and feedback control are not small.

This paper studies the nonlinear equation for ventilation flows in a single opening enclosure. Periodic solutions, the global attractive region, periods and amplitudes of the periodic solutions, and quasi-periodic solutions in the nonlinear equation are investigated.

We consider the problem of transitions between states of a bistable mechanical system induced by a harmonically, and a randomly oscillating barrier. We study the problem both experimentally and numerically. The system consists of a mass attached to a rotating disk, nonlinearly coupled to a spring subject to an external force. For periodically driven external forces, the transition rate across the barrier increases linearly with the frequency of the driving force until the system reaches the boundary of a chaotic region. Beyond this boundary, the rate drops sharply until it is fully suppressed at the end of this region. When the driving force is stochastic we find that the transition rate also has a maximum, and that its position depends on the rate of switching of the random force.

This letter uses a singular perturbation method to introduce a double-scroll version of the Rossler system. Its generation and basic dynamic behavior are briefly described.

In this paper we revisit the well-known Chua's circuit and give a discussion on entropy of this circuit. We present a formula for the topological entropy of a Chua's circuit in terms of the Poincaré map derived from the ordinary differential equations of this Chua's circuit by computer simulation arguments.

This paper introduces a new application of the Field Programmable Analog Array. Ease of design and reprogrammability are the advantages offered by this class of analog circuits, making them an ideal environment for the implementation of chaotic circuits and their experimental characterization.

Chua's circuit, a well-known paradigm of nonlinear circuits, has been used as an example of application. Experimental results show the suitability of the approach, highlighting the features of the new implementation: a fully on the fly programmable Chua's circuit has been obtained.