A symbolic analysis of observed time series data requires making a discrete partition of a continuous state space containing observations of the dynamics. A particular kind of partition, called "generating", preserves all dynamical information of a deterministic map in the symbolic representation, but such partitions are not obvious beyond one dimension, and existing methods to find them require significant knowledge of the dynamical evolution operator or the spectrum of unstable periodic orbits. We introduce a statistic and algorithm to refine empirical partitions for symbolic state reconstruction. This method optimizes an essential property of a generating partition: avoiding topological degeneracies. It requires only the observed time series and is sensible even in the presence of noise when no truly generating partition is possible. Because of its resemblance to a geometrical statistic frequently used for reconstructing valid time-delay embeddings, we call the algorithm "symbolic false nearest neighbors". Why might one want to represent observed time series of dynamical systems as sequences of low-precision discrete symbols? In this representation, there are often interesting techniques-often (but not exclusively) derived from information theory and its associated technologywhich may illuminate data in novel ways [1]. The initial step for all these methods requires making a partition: a coloring of the state space [2], x ∈ R d , into non-overlapping regions and associated symbols so that any x is assigned a unique symbol s in a discrete alphabet. The symbol may be represented as an integer in the set 0, 1, . . . A − 1. A partition P defines a discretization of the observed sequence x i , i = 1 . . . N into a symbolic sequence, s i , i = 1 . . . N .What partitions are "good"? Which discretizations retain the full structure of the original dynamics in the x space in the sequence of symbols? Unfortunately, the situation is unlike the remarkable time-delay embedding method for continuous dynamics: simple partitions are not generically satisfactory. The mathematics of symbolic dynamics specifies what we want: a "generating partition" (GP), where symbolic orbits uniquely identify one continuous space orbit, and thus the symbolic dynamics is fully equivalent to the continuous space dynamics.Unfortunately there is no satisfactory mathematical theory about how to find a GP as a general procedure (except for one dimensional dynamics (d = 1), where partitioning at the critical points works). Are the ad-hoc partitions often used still satisfactory? Unfortunately they are often not so. Bollt et al [3] examined the degradation in the symbolic dynamics which results from the frequently used "histogram partition", as opposed to a GP. A less optimal partition will induce improper projections or degeneracies, where a given symbolic segment may correspond to more than one topologically distinct state space orbit. This resulted in finding the wrong topological entropy. Chaotic communication with symbolic targeting works most sa...
We introduce a relaxation algorithm to estimate approximations to generating partitions for observed dynamical time series. Generating partitions preserve dynamical information of a deterministic map in the symbolic representation. Our method optimizes an essential property of a generating partition: avoiding topological degeneracies. We construct an energylike functional and use a nonequilibrium stochastic minimization algorithm to search through configuration space for the best assignment of symbols to observed data. As each observed point may be assigned a symbol, the partitions are not constrained to an arbitrary parametrization. We further show how to select particular generating partition solutions which also code low-order unstable periodic orbits in a given way, hence being able to enumerate through a number of potential generating partition solutions.
Chaotically oscillating rare-earth-doped fiber ring lasers ͑DFRLs͒ may provide an attractive way to exploit the broad bandwidth available in an optical communications system. Recent theoretical and experimental investigations have successfully shown techniques to modulate information onto the wide-band chaotic oscillations, transmit that signal along an optical fiber, and demodulate the information at the receiver. We develop a theoretical model of a DFRL and discuss an efficient numerical simulation which includes intrinsic linear and nonlinear induced birefringence, both transverse polarizations, group velocity dispersion, and a finite gain bandwidth. We analyze first a configuration with a single loop of optical fiber containing the doped fiber amplifier, and then, as suggested by Roy and VanWiggeren, we investigate a system with two rings of optical fiber-one made of passive fiber alone. The typical round-trip time for the passive optical ring connecting the erbium-doped amplifier to itself is 200 ns, so Ϸ10 5 round-trips are required to see the slow effects of the population inversion dynamics in this laser system. Over this large number of round-trips, physical effects like GVD and the Kerr nonlinearity, which may appear small at our frequencies and laser powers via conventional estimates, may accumulate and dominate the dynamics. We demonstrate from our model that chaotic oscillations of the ring laser with parameters relevant to erbium-doped fibers arises from the nonlinear Kerr effect and not from interplay between the atomic population inversion and radiation dynamics. ͓S1050-2947͑99͒08607-2͔
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