International Journal of Bifurcation and Chaos

In this review we deal with the application of statistical test techniques for the extraction of structures in time series. Two kinds of questions are answered in this statistical framework: Are there any temporal dependences in the data? and Which kind of dynamics generate these temporal dependences? The first question is known as the problem of predictability and also considers the aspect of stationarity. The second question is deeper in the sense that it deals with the dynamical characterization of the detected temporal structures. Central to our approach is a cumulant-based measure of statistical dependences in Fourier space. The dynamical aspects are studied by means of the information flow. The theory is illustrated by artificial and real-world, stochastic and chaotic examples.

The Bonhoeffer van der Pol system, with an applied constant forcing, was invoked by Fitzhugh [1961] as a two-dimensional representation of the four-dimensional Hodgkin–Huxley system [1952], a well-known physiological model representing the electrical behavior across a nerve membrane. The system has been analyzed within a particular parameter regime relevant to the physiology (see [Fitzhugh, 1961]) and for the full parameter space with emphasis on the prediction of periodic solutions (see [Barnes & Grimshaw, 1995]). In this paper the system is considered with a time dependent sinusoidal forcing term in which form it represents a nonlinear, nonautonomous system of differential equations with five parameters. The study is motivated by physiological experiments with neurons subjected to periodic stimuli (see, e.g. [Hayashi et al., 1982]). A few aspects pertaining to the system behavior have been explored by others for particular fixed parameter combinations (and with different purposes from those here), for example [Wang, 1989; Rajasekar & Lakshmanan, 1988, 1993; Yasin et al., 1993; Braaksma & Grasman, 1993; Rabinovitch et al., 1994]). In this paper we present the numerical results of simulations for a more general parameter space and propose theoretical interpretations for a broad range of the behavior states incurred. Further, we present numerical results describing how the system dynamics change as each of the five parameters is varied, and thus predict regions where periodic, quasiperiodic or chaotic behavior can be expected.

We study theoretically the period tripling phenomenon and its role in the development of the bifurcation structure of the periodically forced Toda oscillator. The analytical studies reveal some interesting features in the context of previously known period tripling in one-parameter systems where the loss of stability of the original equilibrium used to be associated with the birth of a bifurcating saddle. In the case of Toda oscillator, a period-tripled saddlenode is born around the stable original period where the saddle approaches the original period as the dissipativity is reduced. In the conservative limit the saddle undergoes a tangential interaction with the original centre. The recurrent appearance of period tripling and doubling suggest a new self similar feature of the associated bifurcation structure. The revelation of the underlying period tripling phenomena also helps to sequentially characterize each period, cascade and substructure, and to make some qualitative predictions regarding their locations in the phase and parameter space. The period tripling may give birth to some "degenerate period n orbits" (i.e. a certain class of period n orbits with identical n). However, they are distinctly characterized on the basis of concept of period tripling and we also predict the "degeneracy" (the number of such period n orbits for a given n) analytically. The same concept is extended in the case of degenerate cascades and substructures.

We study sequences of periodic orbits and the associated phase space dynamics in a 4-D symplectic map of interest to the problem of beam stability in circular particle accelerators. The increasing period of these orbits is taken from a sequence of rational approximants to an incommensurate pair of irrational rotation numbers of an invariant torus. We find stable (elliptic–elliptic) periodic orbits of very high period and show that smooth rotational tori exist in their neighborhood, on which the motion is regular and bounded at large distances away from the origin. Perturbing these tori in parameter and/or initial condition space, we find either chains of smaller rotational tori or certain twisted tube-like tori of remarkable morphology. These tube-tori and tori chains have small scale chaotic motions in their surrounding vicinity and are formed about invariant curves of the 4-D map, which are either single loops or are composed of several disconnected loops, respectively. These smaller chaotic regions as well as the non-smoothness properties of large rotational tori under small perturbations, leading to eventual escape of orbits to infinity, are studied here by the computation of correlation dimension and Lyapunov exponents.

The dynamics and bifurcations of the Newton–Leipnik equations are presented. Numerical computations and local stability calculations suggest that the dynamics of the Newton–Leipnik equations are related to the dynamics and bifurcations of a family of odd, symmetric, bimodal maps. The numerically computed dynamics and bifurcations of the Newton–Leipnik equations are compared with the dynamics and bifurcations of a family of odd, symmetric, bimodal maps to motivate the connection between the two systems.

The dynamics and bifurcations of a family of odd, symmetric, bimodal maps, f_α are discussed. We show that for a large class of parameter values the dynamics of f_α can be described via an identification with a unimodal map u_α. In this parameter regime, a periodic orbit of period 2n + 1 of u_α corresponds to a periodic orbit of period 4n + 2 for f_α. A periodic orbit of period 2n of u_α corresponds to a pair of distinct periodic orbits also of period 2n for f_α. In a more general setting we describe the genealogy of periodic orbits in the family f_α using symbolic dynamics and kneading theory. We identify which periodic orbits of even periods are born in period-doubling bifurcations and which are born in pitchfork bifurcations and provide a method of describing the "ancestors" and "descendants" of these orbits. We also show that certain periodic orbits of odd periods are born in saddle-node bifurcations.

We study the dynamics of the traveling interface arising from a bistable piecewise linear one-way coupled map lattice. We show how the dynamics of the interfacial sites, separating the two superstable phases of the local map, is finite dimensional and equivalent to a toral map. The velocity of the traveling interface corresponds to the rotation vector of the toral map. As a consequence, a rational velocity of the traveling interface is subject to mode-locking with respect to the system parameters. We analytically compute the Arnold's tongues where particular spatiotemporal periodic orbits exist. The boundaries of the mode-locked regions correspond to border-collision bifurcations of the toral map. By varying the system parameters it is possible to increase the number of interfacial sites corresponding to a border-collision bifurcation of the interfacial attracting cycle. We finally give some generalizations towards smooth coupled map lattices whose interface dynamics is typically infinite dimensional.

Period doubling route to chaos has been shown to occur in the voltage controlled DC/DC buck converter, both experimentally and numerically. A chaotic attractor was found at the end of the sequence, suddenly followed by an increase of its size. In this paper new secondary bifurcations and high periodic phenomena, coexisting with the main sequence are detected and analyzed over the same range of parameters. A(synchronous)-switching and stroboscopic maps, unstable orbits, bifurcation diagrams, invariant manifolds and basins of attraction are outlined. These tools are put together to reveal the dynamical richness of this nonsmooth system.

A new method for the computation of periodic solutions of differential-algebraic equations (DAEs) in the neighborhood of Hopf bifurcation points of DAEs is proposed. The method is based on an asymptotic energy analysis of the dynamic behavior near the Hopf bifurcation point. The algorithm is described in more detail in combination with the method of harmonic balance which is used to compute the periodic solutions. Furthermore the usability of the method also in the case of nonclassical Hopf bifurcation points is explained. Finally numerical results using an implementation of the method in an electrical circuit simulator are given.

In this paper, the structure, principle and framework of chaotic digital code-division multiple access ((CD)² MA) communication systems are presented. Unlike the existing CDMA systems, (CD)²MA systems use continuous pseudo-random time series to spread the spectrum of message signal and the spread signal is then directly sent through a channel to the receiver. In this sense, the carrier used in (CD)²MA is a continuous pseudo-random signal instead of a single tone as used in CDMA. We give the statistical properties of the noise-like carriers. In a (CD)²MA system, every mobile station has the same structure and parameters, only different initial conditions are assigned to different mobile stations. Instead of synchronizing two binary pseudo-random sequences as in CDMA systems, we use an impulsive control scheme to synchronize two chaotic systems in (CD)²MA. The simulation results show that the channel capacity of (CD)²MA is twice as large than that of CDMA.

A detailed description of the stationary patterns observed in one-, two- and three-dimensional bistable reaction–difusion media is presented. Because of the intrinsic nature of the system, no spatial constraints are found and completely new structures arise. The mechanism stabilizing these patterns is similar to the Turing structures case for a monostable media. The most important factors determining the final stationary structures observed are also stressed out.

We consider a pendulum subjected to feedforward and feedback control, in which chaotic motions occur when the feedback gain is small. We apply two control techniques recently proposed by Ott and his coworkers so that the pendulum can exhibit the desired motion. The techniques use some dynamical properties relating to chaos. Numerical examples are given and the effectiveness of these control techniques is demonstrated.

In our previous paper, it was confirmed that the unstable periodic orbit embedded in the chaotic attractor in magneto-elastic beam system can be stabilized by delayed feedback control experimentally. It seems an advantage that the control method does not require any exact model of the system. However, the application of the control raises the problem that we cannot predict the stabilized unstable periodic orbit until it converges. In this paper, an "onset window" is introduced to determine the onset timing for targeting the desired orbit embedded in the chaotic attractor experimentally. Moreover, the dependence of the stabilization on the delay and the gain parameters is also discussed based on the experimental results.

This paper presents a global approach for controlling the Lorenz system. The basic idea is to partially cancel the nonlinear cross-coupling terms such that the stability of the resulting system can be guaranteed by sequentially proving the stability of each individual state. The method combines ideas from feedback linearization, classical control theory, and Lyapunov's second method. Robust behavior with respect to model uncertainties in the feedback loop is proven. The performance of partial linearization compared to input-state linearization is illustrated on tracking of several trajectories including a periodic orbit and a steady state.

Optimal control of chaotic systems is discussed in this paper. Since often only open-loop control can be obtained from the minimum principle, the output of the controlled system may be dramatically affected by noise. By using chaos synchronization, we successfully keep the output of the controlled chaotic system in the designed optimal trajectory even in noisy background. The numerical experiments of the Rössler system and the Hénon system are presented to demonstrate its effectiveness.

A new technique for synchronization of chaotic circuits is proposed. This technique, based on varying a circuit's overall topology rather than varying a set of continuous parameters, offers a possible resolution to the tradeoff between security and synchronizability inherent in existing chaotic systems. The encryption key is represented by a mapping from a set of nodes to a set of switches in the circuit. This method significantly improves reliability and can be easily interfaced to digital control circuits.

The present paper describes the effects of the system noise on the chaotic behavior of the Roessler type signal from the point of view of controlling chaos. We have investigated the system equations by a variable transformation method and have then carried out computer simulations to verify our theoretical consideration for a DC bias as system noise. These approaches reveal that the DC bias brings about the direct effect on the system parameters in the equations, while the random noise hardly affects the chaotic system.

While day-to-day atmospheric fluctuations are not predictable after two weeks or so because of the chaotic nature of the atmospheric dynamics, there is evidence that certain aspects of the atmospheric circulation are predictable on time scales much longer than two weeks in the presence of a time dependent climate forcing. This paper illustrates distinctions between weather predictability and climate predictability by using a truncated atmospheric model in an explicitly chaotic parameter regime. An external forcing with time scale longer than the weather tends to modulate the statistical characteristics of the chaotic atmospheric variations to enhance the predictability of time-averaged climate states. Moreover, there exist climate variables which are more predictable than others.

In this note we indicate the manner in which synchronization criteria may be developed for master-slave connected Lur'e systems. For flexibility we incorporate linear, static state feedback. The criteria presented are based on the generation of Lur'e–Postnikov type Lyapunov functions for the error system.