Embedding Phase Space Research Articles

Smoothness Implies Determinism: A Method to Detect It in Time Series

Continuity on an embedded phase space is enough to imply determinism in time series. Also, it is possible to define infinitely many arbitrary vector fields over an attractor. We exploit this arbitrariness to generate a detector of smoothness and therefore of determinism in time series. We report results of numerical studies of both flow and map examples and of a chaotic experimental system.

Improvement of K2-entropy calculations by means of dimension scaled distances

The K 2-entropy is evaluated for experimental time series as well as for numerically simulated ones. The conventional calculation procedure leads to results that can be extrapolated to obtain a K 2-value for the time series, but in the case of experimental systems the outcome is hardly significant for a classification of the states. The reason for this is a distortion of distances due to the phase space embedding procedure by time-delay coordinates. Therefore, a K 2-calculation based on dimension scaled distances (DSD) is suggested. A comparison to the often suggested use of the maximum norm is discussed. By the application of DSD the convergence problems are avoided and clear results were obtained in all investigated situations.

Application of reconstructed phase-space techniques for correlation-dimension calculations in a spatially extended dynamical system

The correlation dimension is calculated for data from a partial-differential-equation model of a magnetic-domain wall---the Bloch wall in a magnetic thin film. The data were extracted from different locations along the height of the wall and two types of embedding were used: a time-delay method from data taken at a single location in the wall (reconstructed phase space) and data taken from ten locations in the wall simultaneously (physical phase space). Three chaotic attractors of the Bloch wall were studied. The time-delay method for some attractors gives a dependence of the correlation dimension on the location at which the data were extracted. This dependence has a specific symmetry with respect to the film in which the Bloch wall resides and a surface (or boundary) effect is seen---the correlation dimension is different at the surfaces of the film. Several scaling regions are found in some cases. The physical phase-space embedding yields a correlation dimension which is an average of the spatial dependence obtained in the time-delay method, but usually this embedding gives a somewhat lower value of the correlation dimension.

Are sequences of volcanic eruptions deterministically chaotic?

In order to address the question posed in the title, two tests have been applied to the sequence of repose periods between eruptions in two volcanoes (Piton de la Fournaise in La Réunion Island and Mauna Loa and Kilauea in Hawaii). In the first test a phase space is constructed from the sequence of repose periods, and the dimension D of the attractor is computed as a function of the dimension d of the embedding phase space. D(d) is found to saturate to a constant value around 2 (La Réunion) and around 4 (Hawaii) for large values of d. This signals the existence of a few degrees of freedom (of order 2 and 4), respectively, of the dynamical system underlying the time evolution of volcanic eruptions. Following this first observation we have then attempted to explain the deterministic dynamic by constructing a so‐called “return map” of the repose periods. This is done by extracting a subsequence from the full sequence of repose periods between eruptions. Different schemes for obtaining these subsequences, corresponding to various Poincaré sections, have been examined. One particular method is inspired by the specific Poincaré section, consisting of selecting the maxima of one of the variables, which is used to analyze the Rössler and Lorenz attractors in the nonlinear physics literature. The return maps are found to be in reasonable agreement with what one should expect from a deterministic dynamical system spoiled, however, by noise. The analysis is not perfect because of additional fluctuations which are, however, not strong enough to blur completely the relevant information. These return maps are compared to those obtained from a set of repose periods artificially synthesized with a pseudorandom number generator possessing the same distribution. However, the information provided by the return maps is not sufficient to determine the dynamical equations underlying the eruptions. Therefore a general procedure to test further the determinism of the data set, which generalizes the construction of the return map to arbitrary dimension, is presented. A three‐dimensional representation of the attractor (for the Piton de la Fournaise volcano) is given and compared with the same representation for a set of repose periods artificially synthesized with a pseudorandom number generator. We find an attractor which presents a global structure reminiscent of the Rössler attractor band in the Duffing system. In contrast, the reconstructed “attractor” using a pseudorandom generator does not exhibit a coherent structure. Adding together these different pieces of evidence, we conclude that volcanic eruptions might well be controlled by deterministic dynamics of low dimensions. While chaos places a fundamental limit on long‐term prediction, it suggests the very interesting possibility of making efficient nonlinear short‐term forecasting of new eruptions from the previous series of eruptions. We have not attempted to develop a nonlinear forecasting procedure from previous eruptions because, in order to be efficient, one needs first to identify as precisely as possible the geometrical structure of the reconstructed attractor. At present, the results do not have the necessary quality to carry out this project. We think that in the future, important improvements could be achieved by monitoring and recording a complete set of data related to the evolution of volcanoes.

An approach to quanitzation of constrained systems

The dynamics of a classical system, specified by a HamiltonianH0(q,p) and a number of constraint functionsϕk(q,p) in Euclidean phase space (q,p), is described by means of a new HamiltonianH(q,p), which is an invariant of the (closed) Poisson-bracket Lie algebra (H0,ϕk). Fixed values ofϕk (not necessarily zero) are given by initial conditions, and they are conserved along the trajectories determined by the Hamilton equations. The quantization is performed by the standard Heisenberg commutation relations in the embedding phase space, while the constraint functions are put in correspondence with constraint operators\(\hat \phi _k\) which generate the Lie algebra of quantum commutators. A subset of commuting constraint operators may be chosen to have certain values in the initial state; and as soon as the Hamiltonian operatorĤ is an invariant of the Lie algebra, these conditions are maintained permanently. Simple examples are presented. Systems with both Bose and Fermi degrees of freedom (and constraints) can be treated universally.

Estimating the Dimensions of Weather and Climate Attractors

The dimensions of attractors are estimated from phase space trajectories of observed weather and climate variables (local surface pressure and relative sunshine duration, zonal wave amplitude; a δ18O-record). They provide primary information for descriptions of properties of the attractors of dynamical systems and give a lower limit to the number of the essential variables necessary to model the dynamics. These estimates are based on distance distributions of pairs of points on the single variable trajectory evolving in phase spaces which embed the attractor. One observes a low fractal dimensionality between three and four for the weather attractor, if interannual variability and seasonal changes are eliminated. The physical interpretation is based on the three dominating scales of cyclones, cyclone families and index-cycle; the irregularity of the flow and strong dependence on initial conditions amount for the fractal value. The climate variable also reveals a low dimensionality (between four and five) of the climate attractor. This is supported by an independent estimate based on eigenfunction expansion of the embedded phase space trajectory. These types of analyses suggest how to extend the standard data evaluation and model verification techniques to an analysis of the phase space behavior of observed and simulated dynamical systems.