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In this paper, two issues are addressed: (1) the applicability of the delay-coordinate embedding method to transient chaotic time series analysis, and (2) the Hilbert transform methodology for chaotic signal processing.

A common practice in chaotic time series analysis has been to reconstruct the phase space by utilizing the delay-coordinate embedding technique, and then to compute dynamical invariant quantities of interest such as unstable periodic orbits, the fractal dimension of the underlying chaotic set, and its Lyapunov spectrum. As a large body of literature exists on applying the technique to time series from chaotic attractors, a relatively unexplored issue is its applicability to dynamical systems that exhibit transient chaos. Our focus will be on the analysis of transient chaotic time series. We will argue and provide numerical support that the current delay-coordinate embedding techniques for extracting unstable periodic orbits, for estimating the fractal dimension, and for computing the Lyapunov exponents can be readily adapted to transient chaotic time series.

A technique that is gaining an increasing attention is the Hilbert transform method for signal processing in nonlinear systems. The general goal of the Hilbert method is to assess the spectrum of the instantaneous frequency associated with the underlying dynamical process. To obtain physically meaningful results, it is necessary for the signal to possess a proper rotational structure in the complex plane of the analytic signal constructed by the original signal and its Hilbert transform. We will describe a recent decomposition procedure for this task and apply the technique to chaotic signals. We will also provide an example to demonstrate that the methodology can be useful for addressing some fundamental problems in chaotic dynamics.

Techniques for detecting encounters with unstable periodic orbits (UPOs) have been very successful in the analysis of noisy, experimental time series. We present here a technique for applying the topological recurrence method of UPO detection to spatially extended systems. This approach is tested on a network of diffusively coupled chaotic Rössler systems, with both symmetric and asymmetric coupling schemes. We demonstrate how to extract encounters with UPOs from such data, and present a preliminary method for analyzing the results and extracting dynamical information from the data, based on a linear correlation analysis of the spatiotemporal occurrence of encounters with these low period UPOs. This analysis can provide an insight into the coupling structure of such a spatially extended system.

We examine the dynamical roles of nonattracting chaotic sets known as chaotic saddles in an Alfvén wave system described by the driven-dissipative derivative nonlinear Schrödinger equation. These Alfvén chaotic saddles have gaps which are filled at the onset of chaos via a saddle-node bifurcation and at a chaotic transition via an interior crisis. It is shown that after an interior crisis an Alfvén chaotic attractor consists of two chaotic saddles connected by a set of coupling unstable periodic orbits.

Normally, conservative systems do not have attractors. However, in a system with escapes, the infinity acts as an attractor. Furthermore, attractors may appear as singularities at a finite distance. We consider the basins of escape in a particular Hamiltonian system with escapes and the rates of escape for various values of the parameters. Then we consider the basins of attraction of a system of two fixed black holes, with particular emphasis on the asymptotic curves of its unstable periodic orbits.

We report on the tendency of chaotic systems to be controlled onto their unstable periodic orbits in such a way that these orbits are stabilized. The resulting orbits are known as cupolets and collectively provide a rich source of qualitative information on the associated chaotic dynamical system. We show that pairs of interacting cupolets may be induced into a state of mutually sustained stabilization that requires no external intervention in order to be maintained and is thus considered bound or entangled. A number of properties of this sort of entanglement are discussed. For instance, should the interaction be disturbed, then the chaotic entanglement would be broken. Based on certain properties of chaotic systems and on examples which we present, there is further potential for chaotic entanglement to be naturally occurring. A discussion of this and of the implications of chaotic entanglement in future research investigations is also presented.

It has been reported in some dynamical systems in fluid dynamics that only a few UPOs (unstable periodic orbits) with low periods can give good approximations to mean properties of turbulent (chaotic) solutions. By employing the Kuramoto-Sivashinsky equation we compare time averaged properties of a set of UPOs embedded in a chaotic attractor and those of a set of segments of chaotic orbits, and report that the distributions of the time average of a dynamical variable along UPOs with lower and higher periods are similar to each other, and the variance of the distribution is small, in contrast with that along chaotic segments. The result is similar to those for low dimensional ordinary differential equations including Lorenz system, Rössler system and economic system.