Signals in chaos: A method for the cancellation of deterministic noise from discrete signals

In this paper we describe a method for separating a combination of a chaotic time series and a discrete-valued signal. The method uses a feedback technique stabilised using knowledge of the chaotic system and the discrete nature of the signal. It is not based on a linearization of the dynamics, and therefore, unlike previously proposed separation methods, does not require that the signal be small. Nor does it make any assumptions about the relative timescales of the signal and the chaotic time series.     

Scaling and interleaving of subsystem Lyapunov exponents for spatio-temporal systems

The computation of the entire Lyapunov spectrum for extended dynamical systems is a very time consuming task. If the system is in a chaotic spatio-temporal regime it is possible to approximately reconstruct the Lyapunov spectrum from the spectrum of a subsystem by a suitable rescaling in a very cost effective way. We compute the Lyapunov spectrum for the subsystem by truncating the original Jacobian without modifying the original dynamics and thus taking into account only a portion of the information of the entire system. In doing so we notice that the Lyapunov spectra for consecutive subsystem sizes are interleaved and we discuss the possible ways in which this may arise. We also present a new rescaling method, which gives a significantly better fit to the original Lyapunov spectrum. We evaluate the performance of our rescaling method by comparing it to the conventional rescaling (dividing by the relative subsystem volume) for one- and two-dimensional lattices in spatio-temporal chaotic regimes. Finally, we use the new rescaling to approximate quantities derived from the Lyapunov spectrum (largest Lyapunov exponent, Lyapunov dimension, and Kolmogorov-Sinai entropy), finding better convergence as the subsystem size is increased than with conventional rescaling. (c) 1999 American Institute of Physics.     

Statistical properties in the chaotic regime of the Maxwell-Bloch equations

The statistical properties of the electric field solution of the Maxwell-Bloch equations describing a single mode, homogeneously broadened laser in the mean field limit are investigated in the strange attractor regime. The electric field distribution sis calculated and it is found that the low order intensity moments are somewhat higher than those for thermal-chaotic light whilst the higher order moments are substantially lower. The field and intensity correlation functions are also calculated and found to exhibit radically different behaviour. The results are interpreted in terms of iterative map which is dederived using multiple time-scale perturbation theory. It is shown that a simple random phasor model is compatible with the numerical data.     

The dissipation of restricted rotation

The decay of a classical hindered rotor over the whole domain of motion from almost free overall rotation to harmonic vibration in a well is examined. The decay is described by a modification of a projection operator formalism involving the separation of dynamical and lattice time scales, and depends on a canonical transformation into a hindered rotating frame that rotates and oscillates in accord with the unperturbed equation of motion of two coaxial dipoles.     

On the Fine Structure of Stationary Measures in Systems Which Contract-on-Average

Suppose {f ₁,...,f _m } is a set of Lipschitz maps of ^d . We form the iterated function system (IFS) by independently choosing the maps so that the map f _i is chosen with probability p _i ( ^m _i=1 p _i =1). We assume that the IFS contracts on average. We give an upper bound for the upper Hausdorff dimension of the invariant measure induced on ^d and as a corollary show that the measure will be singular if the modulus of the entropy _i p _i log p _i is less than d times the modulus of the Lyapunov exponent of the system. Using a version of Shannon's Theorem for random walks on semigroups we improve this estimate and show that it is actually attainable for certain cases of affine mappings of .     

Discrete derived categories I: homomorphisms,autoequivalences and t-structures

Discrete derived categories were studied initially by Vossieck (J Algebra 243:168–176, 2001) and later by Bobiński et al. (Cent Eur J Math 2:19–49, 2004). In this article, we describe the homomorphism hammocks and autoequivalences on these categories. We classify silting objects and bounded t-structures.     

Delay Embeddings for Forced Systems. II. Stochastic Forcing

Takens Embedding Theorem forms the basis of virtually all approaches to the analysis of time series generated by nonlinear deterministic dynamical systems. It typically allows us to reconstruct an unknown dynamical system which gave rise to a given observed scalar time series simply by constructing a new state space out of successive values of the time series. This provides the theoretical foundation for many popular techniques, including those for the measurement of fractal dimensions and Liapunov exponents, for the prediction of future behaviour, for noise reduction and signal separation, and most recently for control and targeting. Current versions of Takens Theorem assume that the underlying system is autonomous (and noise-free). Unfortunately this is not the case for many real systems. In a previous paper, one of us showed how to extend Takens Theorem to deterministically forced systems. Here, we use similar techniques to prove a number of delay embedding theorems for arbitrarily and stochastically forced systems. As a special case, we obtain embedding results for Iterated Functions Systems, and we also briefly consider noisy observations.     

Quantitative chromatographic analysis using rectified radio-frequency methods: Lithium,sodium and potassium

The blake Zone Detector has been shown to be a very useful apparatus for the location of ion zones separated by paper Chromatography. In the case of the alkali metals, these zones can be estimated by measuring the impedance of their aqueous extract. An alternative method, that of measuring the area under the detector curve, does not provide a practical means of quantitative measurement, A new developing solvent is described for the paper Chromatographic separation of lithium, sodium and potassium chlorides.