International Journal of Bifurcation and Chaos

We review here the nonlinear dynamical properties of the crayfish mechanoreceptor system from the hydrodynamically sensitive hairs on the tailfan through the caudal photoreceptor neurons embedded in the 6th ganglion. Emphasis is on the extraction of low dimensional behavior from the random processes (noise) that dominate this neural system. We begin with stochastic resonance in the sensory root afferents and continue with a discussion of the photoreceptor oscillator and its instabilities. Stochastic synchronization, rectification and the generation of second harmonic responses in the photoreceptors are finally discussed.

Nonlinear Dynamics has been applied to Point Process Systems profitably. Such systems manifest themselves by Series of Events in space or time that are then assimilated to Point Processes, i.e. countable collections of points in continua. Arguments are illustrated here with univariate one-dimension Point Processes representing spike trains from nerve cells. These are described fully by their "timing", i.e. the instants {… t_i-1<t_i<t_i+1 …} when events occur. Timings are described as fully by the time series of the intervals between events {…, T_i-1, T_i, T_i+1, …}; this portrayal is the most common one in Dynamics.

This communication's goal is to call attention to the peculiarities of Point Process Systems most relevant to dynamic approaches. Previously, explicit, deliberate statements to that effect have been bypassed because approaches, though justified, were extensions from Continuous Systems. The Point Process notion essential for this goal is that the ith event at t_i exclusively reflects the most recent finite epoch and the timing of the finite number of events within it. This is because real world events always have transitory consequences and cannot be repeated at intervals shorter than some dead time. Those finite quantities, though varying from event to event, have maxima (or least upper bounds), S for the most recent epochs and D for the numbers of events. Accordingly, the most involved will be spans from t_i-S to t_i and events i-D; i-D+1: S is called "integration period" and the D events "influential". Critical is the timing of the influential events plus the ith, described by the D intervals T_i-D+1, …, T_i-1 and T_i also called "influential". Therefore, it makes sense to assign to a Point Process the D-dimension state space whose coordinates are the influential intervals. The ith event is represented by point {T_i-D+1, …, T_i-1, T_i} whose location implies the timing of events i-D, i-D+1, …, i-1 and i. Successive point positions compose a discrete trajectory with countable points. Point Process trajectories have exhibited the same kinds of behaviors and attractors as continuous cases.

Point Process data are analyzed along the same steps (a–d) prescribed for continuous data but peculiarities affect the first two. Data consist, not of several variables, but of the single variable T_i. Hence, in step (a) where a representative variable is selected, T_i is the only choice. In (b) a discrete trajectory is extracted, with Point Processes simply that of the original data T_i: points stagger along it according to the system's dynamics, and need not be periodic or uniformly distributed. With Continuous Systems, discrete trajectories arise from sampling at investigator-chosen invariant time-lags; points stagger periodically and uniformly. The remaining stages are similar for Point Process and Continuous Systems: namely, (c) trajectory embedding in a space with suitable dimensionality and (d) behavior and attractor cataloging by dimensionality, nonlinearity and predictability.

The examples listed illustrate the many ways Dynamic approaches to Point Processes, as embodied by spike trains, has and will continue to contribute to the Neurosciences.

The bifurcation diagram of a truncation to six degrees of freedom of the equations for quasi-geostrophic, baroclinic flow is investigated. Period-doubling cascades and Shil'nikov bifurcations lead to chaos in this model. The low dimension of the chaotic attractor suggests the possibility to reduce the model to three degrees of freedom. In a physically comprehensible limit of the parameters this reduction is done explicitly. The bifurcation diagram of the reduced model in this limit is compared to the diagram of the six degrees of freedom model and agrees well. A numerical implementation of the graph transform is used to approximate the three-dimensional invariant manifold away from the limit case. If the six-dimensional model is reduced to a linearisation of the invariant manifold about the Hadley state, the Lorenz-84 model is found. Its parameters can then be calculated from the physical parameters of the quasi-geostrophic model. Bifurcation diagrams at physical and traditional parameter values are compared and routes to chaos in the Lorenz-84 model are described.

In this paper we present a simplified model of a three-body problem. Place three parallel lines in the plane. Place one mass on each of the lines and let their positions evolve according to Newton's inverse square law of gravitation. We prove the KAM theory applies to our model and simulations are presented. We argue that this model provides an ideal, accessible entry point into the beautiful mathematics involved in the study of the three-body problem.

We give an overview of all codim 1 bifurcations in generic planar discontinuous piecewise smooth autonomous systems, here called Filippov systems. Bifurcations are defined using the classical approach of topological equivalence. This allows the development of a simple geometric criterion for classifying sliding bifurcations, i.e. bifurcations in which some sliding on the discontinuity boundary is critically involved. The full catalog of local and global bifurcations is given, together with explicit topological normal forms for the local ones. Moreover, for each bifurcation, a defining system is proposed that can be used to numerically compute the corresponding bifurcation curve with standard continuation techniques. A problem of exploitation of a predator–prey community is analyzed with the proposed methods.

This paper presents some analytical criteria for local activity principle in reaction–diffusion Cellular Nonlinear Network (CNN) cells [Chua, 1997, 1999] that have four local state variables with three ports. As a first application, a cellular nonlinear network model of tumor growth and immune surveillance against cancer (GISAC) is discussed, which has cells defined by the Lefever–Erneaux equations, representing the densities of alive and dead cancer cells, as well as the number of free and bound cytotoxic cells, per unit volume. Bifurcation diagrams of the GISAC CNN provide possible explanations for the mechanism of cancer diffusion, control, and elimination. Numerical simulations show that oscillatory patterns and convergent patterns (representing cancer diffusion and elimination, respectively) may emerge if selected cell parameters are located nearby or on the edge of the chaos domain. As a second application, a smoothed Chua's oscillator circuit (SCC) CNN with three ports is studied, for which the original prototype was introduced by Chua as a dual-layer two-dimensional reaction–diffusion CNN with three state variables and two ports. Bifurcation diagrams of the SCC CNN are computed, which only demonstrate active unstable domains and edges of chaos. Numerical simulations show that evolution of patterns of the state variables of the SCC CNN can exhibit divergence, periodicity, and chaos; and the second and the fourth state variables of the SCC CNNs may exhibit generalized synchronization. These results demonstrate once again Chua's assertion that a wide spectrum of complex behaviors may exist if the corresponding CNN cell parameters are chosen in or nearby the edge of chaos.

In this report, a clustering approach is presented to detect dynamical nonlinearity in a stationary time series. The one-dimensional time series sampled from a dynamical system is mapped on to the m-dimensional phase space by the method of delays. The vectors in the phase space are partitioned by k-means clustering technique. The local trajectory matrix for the vectors in each of the clusters is determined. The eigenvalues of the local trajectory matrices represent the variation along the principal directions and are obtained by singular value decomposition (SVD). The product of the eigenvalues represents the generalized variance and is a measure of the local dispersion in the phase space. The sum of the local dispersions (g^T) is used as the discriminant statistic to classify data sets obtained from deterministic and stochastic settings. The surrogates were generated by the Iterated Amplitude Adjusted Fourier Transform (IAAFT) technique.

This paper is the second part of an earlier work devoted to the properties specific to maps of the plane characterized by the presence of a vanishing denominator, which gives rise to the generation of new types of singularities, called set of nondefinition, focal points and prefocal curves. A prefocal curve is a set of points which are mapped (or "focalized") into a single point, called focal point, by the inverse map when it is invertible, or by at least one of the inverses when it is noninvertible. In the case of noninvertible maps, the previous text dealt with the simplest geometrical situation, which is nongeneric. To be more precise this situation occurs when several focal points are associated with a given prefocal curve. The present paper defines the generic case for which only one focal point is associated with a given prefocal curve. This is due to the fact that only one inverse of the map has the property of focalization, but with properties different from those of invertible maps. Then the noninvertible maps of the previous Part I appear as resulting from a bifurcation leading to the merging of two prefocal curves, without merging of two focal points.

We introduce shrubs in this paper in order to present a first approach to studying the structure of the Mandelbrot set. Primary, secondary… and N-ary shrubs are analyzed. We have experimentally obtained formulae to calculate the periods of the hyperbolic component representatives of the structural branches, and the preperiods and periods of both the nodes — where structural branches emanate — and the tips — where the shrub enters in crisis. A generalization allows us to give in each case one formula to calculate representative periods and preperiods and periods of nodes and tips.

A large number of simultaneous frequency and amplitude data from an electronic chaotic circuit (Chua's circuit) have been obtained. These acquisitions are validated by plotting the bifurcation diagrams of the experimental data versus the bifurcation parameter. We introduce a topological parallel between the Colpitts oscillator and Chua's circuit, and look for similar behavior of the frequency fluctuations using the Allan deviation.

We generalize a method of noise estimation for chaotic time series due to [Schreiber, 1993] in cases where the noise level is relatively large. The noise estimation is based on the correlation integral, which, for small amounts of noise, is not affected by the attractor's curvature effects. When the noise is large, however, one has to increase the range of the correlation integral and this brings about significant inaccuracies in its evaluation due to both curvature effects and noise. In this Letter, we present a modification of Schreiber's noise level estimation method, which uses a robust error estimator based on L_-∞ (rather than the usual L₂) norm in the computations. Since L_-∞ was proved less sensitive to curvature effects, it gives a more accurate estimation of the noise standard deviation compared with Schreiber's results. Here, we illustrate our approach on the Hénon map corrupted by Gaussian white noise with zero mean, as well as on real data obtained from the Nasdaq Composite time series of daily returns.

Based on a quantum dynamical invariant of motion, I, we study the classical limit of a semiclassical Hamiltonian composed by a full quantum harmonic oscillator plus a classical particle plus a "semiclassical" coupling quartic term. The motion-invariant is closely related to the uncertainty principle. The classical limit (CL) is determined by the relationship between I and the total energy of the system, defining an adimensional invariant E_r. We find that the CL coincides with the results of a purely classical treatment. Both invariants allow to follow the transit between quantum nonchaotic to the classical chaotic regime. Particularly, with E_r we define the threshold above which chaos appears, and the interval during which both regimes co-exist.

In a recent paper [Ossendrijver, 2000] numerical simulations of a 2D mean-field model where shown to produce grand minima, typical of the long-term behavior of solar magnetic activity. The model consisted of a dynamo that features an α effect based on the buoyancy instability of magnetic flux tubes, which gives rise to the switching back and forth from grand minima to "regular" solar behavior. In this Letter, we report evidence from a time-series analysis of the model for the presence of crisis-induced intermittency due to attractor-widening. We support this finding by showing that the average duration of the minima, <τ>, follows the theoretically predicted scaling , where C_δα is the bifurcation parameter of interest, together with other strong statistical evidence. As far as we are aware, this is the first time concrete and detailed evidence has been produced for the occurrence of this type of crisis-induced intermittency — due to attractor widening — for such dynamo models.

In this paper a novel framework for synchronizing hyperchaotic systems with one or several nonlinearities is presented. The proposed technique exploits observer design and time-division multiplexing of the transmitted signal. A remarkable feature of the approach is that only a scalar signal is required for achieving hyperchaos synchronization. Simulation results are carried out for eighth order circuits with one nonlinearity as well as sixth order circuits with two nonlinearities.

This paper studies chaos synchronization between two coupled chaotic systems using linear state feedback control. A new, simple and yet easily applicable criterion is derived for chaos synchronization based on the Lyapunov stability theory and the Linear Quadratic Optimal Control theory. This criterion is given in terms of some simple algebraic inequalities established by the Gerschgorin theorem, and it is easily applicable to a large class of chaotic systems. As an example, the familiar Chua's circuit is simulated for demonstration.

Recently, there has been a good deal of interest in the use of chaotic signals for direct sequence code division multiple access (DS-CDMA) communication systems. The capacity of DS-CDMA systems is interference-limited, and can therefore be increased by techniques that suppress interference. This letter is devoted to the evaluation of the impact of blind multiuser detection techniques on chaos based DS-CDMA systems. Blind receivers can suppress multiple access interference and do not require knowledge of the code sequences and propagation channels of the interference. We demonstrate that, for chaotic sequence-based communications, blind multiuser receivers significantly improve the BER with respect to single-user receivers, and that their use is practically essential with a high number of users.

This Letter shows some counter-intuitive simulation results that for some filter parameters in the extended boundaries of the stability triangle, the state vector will converge to a periodic orbit after some iterations, no matter what the initial conditions. Also, a new pattern, which looks like a rotated letter "X", is found. The center of the rotated letter is located at the origin and the slopes of the "straight lines" of the rotated letter are equal to the values of the pole locations.

The cyclic A/D converting process is equivalent to iterations of a chaotic may using symbolic dynamics. The principle can be extended to construct "general cyclic ADCs" based on chaotic iterations. In this paper, the two necessary conditions of iterated maps are presented, and the new attributes of these converters are also discussed. By establishing a noise model, the analytic expressions of the random and system errors in general cyclic ADCs are derived and verified by simulations. With the same level of additive white noise, the optimal iterative maps for general cyclic ADCs are deduced.