International Journal of Bifurcation and Chaos

This tutorial work identifies the characteristic properties of chaotic digital modulation schemes and shows how the unique features of chaotic basis functions can be exploited to competitive advantage in selected application domains. Two robust noncoherent chaotic digital modulation schemes (DCSK and FM-DCSK) are described in detail and the performance of FM-DCSK is examined critically for communication channels where systems using conventional coherent and narrow-band modulation schemes fail. Since the FM-DCSK signal is a wide-band signal and its demodulation is performed without synchronization, the FM-DCSK modulation scheme is especially suitable for operation in multipath channel such as Wireless Local Area Networks (WLANs).

The problem of transmitting digital information using chaotic signals over a channel with Gaussian white noise perturbation is introduced rigorously. It is shown that discrete time base-band chaotic communication systems with discrete time Gaussian white noise in the channel are sufficiently general in this context. The optimal receiver is given explicitly in terms of conditional probabilities. For the example of chaos shift keying using iterations of the tent map, the optimal classifier is constructed explicitly. Finally, it is shown how previously published methods, in particular those based on chaos synchronization, fit into this framework.

Given a time series measured (or generated) by a known or an unknown dynamical system we address a series of problems which can be considered as advanced signal processing tasks, namely: (1) section-wise approximation of the measured signal by pieces of trajectories from a chosen nonlinear dynamical system (model); (2) signal restoration when the measured signal has been corrupted e.g. by quantization; (3) signal coding and compression. These tasks can be addressed using a new approach to the shadowing problem based on nonlinear observability problem. Its goal is to reproduce initial conditions for a dynamical system under consideration (approximating waveform generator) giving rise to an orbit which is optimal in the sense of average distance from the measured (or prescribed) transient output waveform. We present the results obtained using the Chua's circuit as approximating waveform generator.

The phenomenon of chaotic synchronization is investigated from an information viewpoint. The synchronization of a "receiver" of chaos with a chaotic "source" is discussed as an ability of the receiver to recover a copy of the chaotic signal generated by the source. The main idea of the paper is to show that in terms of the information approach the condition for the chaotic synchronization is not due to the physical effect of one system on the other, but the transmission of necessary information about the chaotic process and, hence, the information carrying capacity of the "communication channel" between the source and the receiver.

We study the synchronization problem in discrete-time via an extended Kalman filter (EKF). That is, synchronization is obtained of transmitter and receiver dynamics in case the receiver is given via an EKF that is driven by a noisy drive signal from a noisy transmitter dynamics. The convergence of the filter dynamics towards the transmitter dynamics is rigorously shown using recent results in extended Kalman filtering. Two extensive simulation examples show that the filter is indeed suitable for synchronization of (noisy) chaotic transmitter dynamics. An application to private communication is also given.

We propose and illustrate a synchronization-based method to combat the channel distortions of a signal transmitted by a chaos-based communication system. In particular, we consider channel distortions like time-varying fading, multipath and time-invariant linear filtering and additive noise. The time-varying parameters are tracked at the receiver to maintain synchronization. The correct noise estimate at the receiver yields synchronization and has minimum norm. A numerical experiment illustrating the method is presented, and shows that successful recovery of the transmitted signal is possible for signal to noise ratios of order unity.

Recent work has demonstrated that symbolic representations of controlled chaotic orbits can be utilized for encoding digital information. So far, this idea has been demonstrated using systems exhibiting sustained chaotic motion on chaotic attractors. The purpose of this work is to explore digital encoding by using transient chaos naturally arising in nonlinear systems. Dynamically, transient chaos is caused by nonattracting chaotic saddles. We argue that there are two major advantages when trajectories on chaotic saddles are exploited as information source: (1) the channel capacity can in general be large, and (2) the influence of channel noise can be reduced. We present a control scheme and also a practical example of encoding a digital message.

In this paper we interpret chaos synchronization schemes within the framework of Lagrange programming networks, which form a class of continuous-time optimization methods for solving constrained nonlinear optimization problems. From this study it follows that standard synchronization schemes can be regarded as a Lagrange programming network with soft constraining, where synchronization between state vectors is defined as a constraint to the dynamical systems. New schemes are proposed then which implement synchronization by hard and soft constraints within Lagrange programming networks. A version is derived which takes into account synchronization errors within the problem formulation. Furthermore Lagrange programming networks for achieving partial and generalized synchronization are given. The methods assume the existence of potential functions for the given systems. The proposed Lagrange programming networks with hard and soft constraining show improved performance on many simulation examples for identical and nonidentical chaotic systems. The schemes are illustrated on Chua's circuit, Lorenz attractor and n-scroll circuits.

In this paper we introduce the method for investigation of coupled chaotic systems using topological methods. We show that if the coupling is small then there exists independent symbolic dynamics for every coupled subsystem and in consequence the systems are not synchronized. As an example we consider coupled Hénon maps. Using computer interval arithmetic we find parameter mismatch and perturbation range for which the symbolic dynamics in the Hénon system is sustained. For coupled Hénon maps we compute the value of coupling strength for which the symbolic dynamics in every subsystem survives.

We study synchronization in arrays of chaotic circuits which are coupled via dynamic coupling elements which can themselves possess chaotic dynamics. We express the synchronization conditions in terms of properties of the underlying hypergraphs and present a result which suggests that under suitable conditions, the more connected the hypergraph is, the easier it is to synchronize the array. The special case of an array of two chaotic circuits is considered. We show that in some cases not all chaotic circuits are synchronized with each other, but clusters are formed in which circuits within a cluster are synchronized to each other.

We describe the emergence of phase clustering and collective behaviors in an ensemble of chaotic coupled map lattices, due to a mean field interaction. This kind of interaction is responsible for the appearence of a collective state, wherein the mean field evolution of each lattice undergoes a periodic behavior in space. We analyze the transition to such a state in an ensemble of one-dimensional lattices of logistic maps, showing that the resulting behavior cooperatively maximizes the energy of the mean field activity.

We investigate experimentally attractor bubbling in a system of two coupled hyperchaotic electronic circuits. The degree of synchronization over a range of coupling strengths for two different coupling schemes is measured to identify bubbling. The circuits display regimes of both attractor bubbling and high-quality synchronization. For the coupling scheme where high-quality synchronization is observed, the transition to bubbling is "soft" and its scaling with coupling strength near the transition point does not fit into the known categories of transition types. We also compare the observed behavior to several proposed criteria for estimating the regime of high-quality synchronization. It is found that none of these methods is completely satisfactory for predicting accurately the regimes of attractor bubbling and high-quality synchronization.

We consider the antiphase synchronization in symmetrically coupled self-oscillators. As model, two Chua's circuits coupled via a capacity are used. Linear analysis in the vicinity of the symmetric subspace gives the stability conditions for antiphase oscillations. Numerical oscillations demonstrate controlled antiphase synchronization at different values of the parameters of the system.

The practical stability of impulsive synchronization between two nonautonomous chaotic systems is studied in this paper, and this is equivalent to that of the origin of the synchronization error system, which is modeled by an impulsive differential equation. We develop theoretical methods of choosing the time interval between two successive synchronization impulses and strengths of synchronization impulses for restricting synchronization errors within prescribed regions around the origin. Numerical experimental results are given to demonstrate theoretical results.

The synchronization properties of a pair of Kuramoto–Sivashinsky equations are examined using only a finite number of coupling signals that are given in terms of local spatial averages. The dependence of the synchronization on the number of coupling signals and the width of the spatial average intervals is examined, including spatiotemporal intermittency of the synchronization error near the onset of synchronization.

Dynamical systems which can be decomposed into a drive–response pair sometimes display the phenomenon of generalized synchronization wherein the response state y is a function of the drive state x, y = ϕ (x). The questions of detecting when this is true in experimental data, and what are the properties of the function ϕ, require the development of statistics that are based on concepts and definitions from analysis, rather than the standard statistics literature. We show that these statistics can display when such a generalized synchronization situation exists and characterize the nature of the manifold (given by ϕ) of the response over the drive, including when that manifold is fractal.

In this article, we investigate a mechanism of explosion and extinction of strange attractors and their crisis-induced intermittency observed after the explosion in the PLL equation used as an FM-demodulator. Namely, we elucidate that, explosion of the first type P₁ (second type P₂) chaotic attractor to their marged P₁&P₂-chaotic attractor occurs due to the heteroclinic tangency between an unstable manifold of the inversely unstable fixed point, and a stable manifold of the associated directly unstable fixed point (I-D chain). We demonstrate this explosion both by computer simulation and electronic circuit experiments.