Nonlinear Deterministic Dynamics Research Articles

Nonlinear analysis and prediction of river flow time series

In this paper we consider the problem of nonlinear modelling the discharge time series of a river in order to study the forecasting ability of a nonlinear approach. To this aim, we first check for some evidence of chaotic behaviour in the dynamics by considering a set of different procedures (phase portrait of the attractor, correlation dimension, the largest Lyapunov exponent, DVS diagram). Their joint application to our data allows us not to exclude the presence of a nonlinear deterministic dynamics of chaotic type. Secondly, we consider two kinds of nonlinear predictors: a univariate predictor, which is based only on the information of the discharges times series and a multivariate one, which also takes into account the information coming from rainfall data. By comparing these predictors with a linear predictor, we can conclude that nonlinear river flow modelling is an effective method to improve prediction in a statistically significant way. Copyright © 2000 John Wiley & Sons, Ltd.

Imparting structural instability to mortality forecasts: Testing for sensitive dependence on initial conditions with innovations*

This article explores a nontraditional approach to examining the problem of forecast uncertainty in extrapolative demographic models. It builds on prior research on stochastic time series forecast models, but diverges to examine their deterministic counterparts. The focus here is an examination of the structural integrity of the Lee‐Carter (1992) method applied to mortality forecasts. I investigate the nonlinear dynamics of the Lee‐Carter method, particularly its sensitive dependence of the forecasts on the initial conditions of the model. I examine the Lee‐Carter nonlinear demographic model, mx,t — exp (ax+ bxkt + ex,t), which is decomposed using SVD to derive a single time‐varying linear index of mortality, kt. From a 90 year time series of kt, forty nine 40 year realizations are sampled. These realizations are modeled and estimated using Box‐Jenkins techniques. The estimated parameters of these realizations and the first case of each of the samples are the initial conditions for the iterations of nonlinearized transformation of k, to exp (kt). The terminal year for each of the 49 iterated series is 2065. The deterministic nonlinear dynamics of this system of 49 iterated series is investigated by testing its Lyapunov exponents as a nonparametric diagnostic of a one dimensional dynamical system. The exponents are all negative, indicating that chaos is not prevalent in this system. The nonexistence of chaos suggests stability in the model and reaffirms the predictability of this one dimensional map. Augmenting the iterations of the initial conditions with additive stochastic innovations, {et, t ≥ 1}, creates a stochastic dynamical system of the form, k t = k t,−1 — c + ϕ flu +et. Here, et is treated as a surrogate for some unanticipated time series event (e.g. an epidemic) that impacts the deterministic map. Gaussian white noise innovations do not move the iterations far from equilibrium and only for short time intervals. So, stepping the mean of the innovations by .01 produces stable Lyapunov exponents until the mean equals .35 where some of the exponents are positive. At this point, deterministic chaos is evident, implying instability in the forecasts. The substantive implications of this instability are discussed.

Pathological tremors: Deterministic chaos or nonlinear stochastic oscillators?

Pathological tremors exhibit a nonlinear oscillation that is not strictly periodic. We investigate whether the deviation from periodicity is due to nonlinear deterministic chaotic dynamics or due to nonlinear stochastic dynamics. To do so, we apply various methods from linear and nonlinear time series analysis to tremor time series. The results of the different methods suggest that the considered types of pathological tremors represent nonlinear stochastic second order processes. Finally, we evaluate whether two earlier proposed features capturing nonlinear effects in the time series allow for a discrimination between two pathological forms of tremor for a much larger sample of time series than previously investigated. (c) 2000 American Institute of Physics.

Consequences of deterministic and random dynamics for the course of affective disorders

Background: Uni- and bipolar affective disorders tend to be recurrent and progressive. Illness patterns can evolve from isolated episodes to more rapid, rhythmic, and “chaotic” mood patterns. Nonlinear deterministic dynamics are currently proposed to explain this progression. However, most natural systems are nonlinear and noisy, and cooperative behavior of possible clinical relevance can result. Methods: The latter issue has been studied with a mathematical model for progression of disease patterns in affective disorders. Results: Deterministic dynamics can reproduce a progression from stable, to periodic, to chaotic patterns. Noise increases the spectrum of dynamic behaviors, enhances the responsiveness to weak activations, and facilitates the occurrence of aperiodic patterns. Conclusions: Noise might amplify subclinical vulnerabilities into disease onset and could induce transitions to rapid-changing dysrhythmic mood patterns. We suggest that noise-mediated cooperative behavior, including stochastic resonance, should be considered in appropriate models for affective illness.

D.G. Frey and E.S. Deevey Review 1: Numerical tools in palaeolimnology – Progress, potentialities, and problems

In the last decade, palaeolimnology has shifted emphasis from being a predominantly qualitative, descriptive subject to being a quantitative, analytical science with the potential to address critical hypotheses concerning the impacts of environmental changes on limnic systems. This change has occurred because of (1) major developments in applied statistics, some of which have only become possible because of the extraordinary developments in computer technology, (2) increased concern about problem definition, research hypotheses, and project design, (3) the building up of high quality modern calibration data-sets, and (4) the narrowing of temporal resolution in palaeolimnological studies from centuries to decades or even single years or individual seasons. The most significant development in quantitative palaeolimnology has been the creation of many modern calibration data-sets of biotic assemblages and associated environmental data. Such calibration sets, when analysed by appropriate numerical procedures, have the potential to transform fossil biostratigraphical data into quantitative estimates of the past environment. The relevant numerical techniques are now well developed, widely tested, and perform remarkably well. The properties of these techniques are becoming better known as a result of simulation studies. The advantages and disadvantages of the preferred technique (weighted averaging partial least squares) are reviewed and the problems in model selection are discussed. The need for evaluation and validation of reconstructions is emphasised. Several statistical surprises have emerged from calibration studies. Outstanding problems remain the need for a detailed and consistent taxonomy in the calibration sets, the quality, representativeness, and inherent variability of the environmental variables of interest, and the inherent bias in the calibration models. Besides biological- environmental calibration sets, there is the potential to develop modern sediment-environment calibration sets to link sedimentary properties to catchment parameters. The adoption of a ‘dynamic calibration set’ approach may help to minimise the inherent bias in current calibration models. Modern regression techniques are available to explore the vast amount of unique ecological information about taxon-environment relationships in calibration data-sets. Hypothesis testing in palaeolimnology can be attempted directly by careful project design to take account of ‘natural experiments’ or indirectly by means of statistical testing, often involving computer- intensive permutation tests to evaluate specific null hypotheses. The validity of such tests depends on the type of permutation used in relation to the particular data-set being analysed, the sampling design, and the research questions being asked. Stratigraphical data require specific permutation tests. Several problems remain unsolved in devising permutation designs for fine-resolution stratigraphical data and for combined spatial and temporal data. Constrained linear or non-linear reduced rank regression techniques (e.g. redundancy analysis, canonical correspondence analysis and their partial counterparts) provide powerful tools for testing hypotheses in palaeolimnology. Work is needed, however, to extend their use to investigate and test for lag responses between biological assemblages and their environment. Having developed modern calibration data-sets, many palaeolimnologists are returning to the sedimentary record and are studying stratigraphical changes. In contrast to much palynological data, palaeolimnological data are often fine-resolution and as a result are often noisy, large, and diverse. Recent developments for detecting and summarising patterns in such data are reviewed, including statistical evaluation of zones, summarisation by detrended correspondence analysis, and non-parametric regression (e.g. LOESS). Techniques of time-series analysis that are free of many of the assumptions of conventional time-series analysis due to the development of permutation tests to assess statistical significance are of considerable potential in analysing fine-resolution palaeolimnological data. Such data also contain a wealth of palaeopopulation information. Robust statistical techniques are needed to help identify non-linear deterministic dynamics (chaos) from noise or random effects in fine-resolution palaeolimnological data.

Lyapunov Spectral Analysis on Random Data

In this paper, an algorithm for estimating all Lyapunov exponents, or Lyapunov spectra of deterministic dynamical systems, is applied to random time series, that is, the time intervals of gamma ray emissions of cobalt. The algorithm used in this paper is that proposed by Sano and Sawada, which estimates Jacobian matrices of an assumed dynamical system from the data points on a possible attractor reconstructed. As a result, the largest Lyapunov exponent of the cobalt data is estimated to be positive, the same as the case of deterministic chaos. Although the sum of all Lyapunov exponents is positive in lower reconstructed dimensions, the values are estimated to be negative when the reconstructed dimension becomes higher. This result is also the same as deterministic chaos in dissipative dynamical systems. It is an indication that naive application of estimating algorithm for Lyapunov spectra to real data does not necessarily lead us to correct results. In order to check the appearance of positive Lyapunov exponents, firstly an analysis by the local versus global (LVG) plots is adopted. The results by LVG plots on the cobalt data are clearly different from those on typical nonlinear dynamical systems in lower dimensional reconstructed state space. Secondly, a direct approach for estimating largest Lyapunov exponents is applied. The results show that estimated Lyapunov exponents on cobalt data do not necessarily indicate existence of positive values. Lastly, a test of statistical hypothesis with surrogate data is also applied to the cobalt data under the null hypotheses that the cobalt data is produced from a linear stochastic system. The results show that statistical significances between the original and surrogate data are so small that some null hypotheses on the cobalt data cannot be rejected; the results imply that an analysis with surrogate data sets could avoid spurious interpretation that random time series, such as the cobalt data, has low-dimensional nonlinear deterministic dynamics with positive Lyapunov exponents.

Simultaneous modeling of nonlinear deterministic and stochastic dynamics

We present a method which can simultaneously model the nonlinear deterministic and stochastic dynamics underlying an observed time series. It is formulated to treat Markov processes in continuous and discrete time. The procedure, which we call Sequin (Stochastic EQUation INference) is a generalization of that which we have developed previously for continuous time systems (Borland and Haken, 1992a,b; 1993a,b; Haken 1988). The theory behind the method as well as its numerical implementation is discussed. Simulations using Sequin to analyze finite-size time series stemming from nonlinear processes with additive and multiplicative dynamical noise in continuous time are shown. Furthermore, we show that Sequin can extract the underlying dynamics of discrete time noisy chaotic processes, such as the logistic map with additive dynamical noise. The success of the method in being able to characerize a system consisting partly of a deterministic chaotic element, and partly of a stochastic element indicates its possible application to many interesting real-world problems.

Nonlinear dynamics in pulsatile secretion of parathyroid hormone in normal human subjects.

In many biological systems, information is transferred by hormonal ligands, and it is assumed that these hormonal signals encode developmental and regulatory programs in mammalian organisms. In contrast to the dogma of endocrine homeostasis, it could be shown that the biological information in hormonal networks is not only present as a constant hormone concentration in the circulation pool. Recently, it has become apparent that hormone pulses contribute to this hormonal pool, which modulates the responsiveness of receptors within the cell membrane by regulation of the receptor synthesis, movement within the membrane layer, coupling to signal transduction proteins and internalization. Phase space analysis of dynamic parathyroid hormone (PTH) secretion allowed the definition of a (in comparison to normal subjects) relatively quiet "low dynamic" secretory pattern in osteoporosis, and a "high dynamic" state in hyperparathyroidism. We now investigate whether this pulsatile secretion of PTH in healthy men exhibits characteristics of nonlinear determinism. Our findings suggest that this is conceivable, although on the basis of presently available data and techniques, no proof can be established. Nevertheless, pulsatile secretion of PTH might be a first example of nonlinear deterministic dynamics in an apparently irregular hormonal rhythm in human physiology. (c) 1995 American Institute of Physics.

Pattern of prolactin diurnal secretion in normal humans: evidence for nonlinear dynamics.

Prolactin (PRL) circadian profiles were analyzed using methods of nonlinear dynamics, directly from the experimental data, by combining in a single time-series (432 measurements), six individual 24-hour PRL profiles (72 measurements per profile, sampling interval = 20 min), obtained from young healthy human volunteers (4 males, 2 females), under basal conditions. Significant autocorrelation exists between any given point of the time series and a limited number of its successors. Fourier analysis showed a dominant frequency of 1 cycle/24 h, without sub-24-hour harmonics. Poincaré section indicated the presence of a fractal attractor and a sketch of the attractor revealed a highly convoluted geometric structure with a conical contour. The box-counting dimension (D0), information dimension (D1) and correlation dimension (D2) of the attractor had low, fractal values, did not differ significantly from each other, and exhibited saturation at an embedding dimension of 2. The evidence taken together suggests that, under basal conditions, the daily changes in the peripheral blood levels of PRL are governed by nonlinear deterministic dynamics, with a dominant rhythm of 1 cycle/day mixed with a higher-frequency, low-amplitude signal.

Chance and chaos in population biology—Models of recurrent epidemics and food chain dynamics

In spite of their complex behaviour the dynamics of some ecological systems may be explained by deterministic laws of motion. This is clearly theoretically possible, because deductive approaches to the dynamics of ecological systems lead to nonlinear models and even very simple model ecosystems generate typically a rich spectrum of dynamics, ranging from coexisting periodic regimes to chaotic behaviour. The present study analyses the complex interplay between deterministic nonlinear dynamics and demographic stochasticity using two examples. Firstly, the search for chaos in ecology has drawn much attention to the analysis of recurrent outbreaks of childhood epidemics—in particular of measles infections—in large population centres. The second example is a compartmental model of a three-species food chain, where sudden population disappearances can be observed due to transient chaos. Our analysis lends support to the plausibility of chaos in population dynamics. However, for decreasing population size (increasing fluctuations) the distinction between chaos and stochasticity becomes more and more problematic. The interplay between deterministic dynamics and stochastic fluctuations, e.g. due to the integer structure of populations or a noisy environment, suggests an analysis combining methods from the theory of nonlinear dynamical systems as well as stochastic processes.

Fractal analysis of steady-state-flicker visual evoked potentials: feasibility.

Electrophysiological models of visual evoked potential recording have assumed that response variability is caused predominantly by random noise added to a true steady signal. Since neuronal geometry has a fractal structure, neural activity may demonstrate deterministic nonlinear dynamics, i.e., chaos. We recorded several-minute time-series traces of the visual evoked potential magnitude in response to full-field flicker from three glaucoma patients and one normal subject. When plotted in phase space, the steady-state response derived from a lock-in amplifier shows an apparent so-called strange attractor (extended nonrepeating loops) rather than the pattern expected from a signal-plus-noise model (a fuzzy dot). The fractal dimension of this attractor may be a more sensitive indicator of early optic-nerve damage than are visual evoked potential latency or amplitude measures.

The non‐linear response of the magnetosphere: 30 October 1978

Previous efforts to find evidence of deterministic nonlinear dynamics in the global geomagnetic system have treated the geomagnetic system as autonomous. However, the geomagnetic system is strongly driven by the stochastic solar wind. We consider the response of the magnetosphere, as given by the AE index, for one day when the IMF had a nearly constant southward value. Using both a series of non‐linear statistics and non‐linear prediction of the response to the input signal υBs, we find that there is some evidence for deterministic non‐linear response of the Earth's magnetosphere on that day.

The Earth's magnetosphere under continued forcing: Substorm activity during the passage of an interplanetary magnetic cloud

In this third paper on magnetic clouds and their interaction with the terrestrial magnetosphere we focus on substorm activity attending the 30‐hour‐long passage of the magnetic cloud seen at 1 AU on January 14/15, 1988. Our main aim is to describe high resolution field and plasma data from six spacecraft at geostationary orbit and beyond (up to 9 RE), supplemented by ground magnetograms, in order to estimate the number of substorm onsets that occurred during this passage. This analysis, attempted here for the first time over such an extended data interval, enables us to obtain a comprehensive view of the magnetospheric behavior. We then relate the interplanetary field and plasma input to the magnetospheric and ionospheric output. The paper is a case study of the response of the nightside magnetosphere to particularly well‐behaved interplanetary conditions: the variations with time of the field and plasma parameters in the magnetic cloud are simple; and the north‐south (Bz) and east‐west (By) components of the magnetic field range slowly over ∼20 nT on both sides of zero. From a synopsis of the substorm observations we find that during the earlier 11‐h interval when Bz was continuously positive, the magnetosphere was quiescent, whereas in the later 18‐h interval, when Bz was uninterruptedly negative, a large magnetic storm was set off. In this latter interval the substorm onsets recur on average every 50 min. Their average recurrence frequency remains relatively undiminished even when the magnetic cloud Bz and other measures of the interplanetary energy input have decreased considerably. We find these results to concur with the current models of magnetospheric substorms based on deterministic nonlinear dynamics. Using ancillary information from HF radar observations of ionospheric convection, we investigate the magnetospheric conditions preceding the first substorm onset in the long substorm sequence. We find that it occurred when the cloud's magnetic field had a persistent northward component but was predominantly westward pointing.

Is there chaos in plankton dynamics?

A controversial issue in ecosystem modeling is whether the irregular fluctuations that one observes in nature are due solely to random environmental factors or whether, at least partially, a deterministic mechanism is responsible for the unpredictable behavior. This second alternative is called deterministic chaos and the issue in this paper is to decide if actual plankton time series can vindicate the hypothesis of chaotic dynamics. The near-neighbor forecasting method is a recent technique for detecting determinism in a time series and we apply it to measurements of phytoplankton and zooplankton biomass obtained at a single station in the Middle Atlantic Bight. Although the results do not conclude the presence of chaos, they do give some support to the idea that deterministic non-linear trophic dynamics may account for at least some of the variability that is seen in the data, particularly in terms of inferring zooplankton oscillations from those of phytoplankton.

A nonlinear dynamical analogue model of geomagnetic activity

The solar wind‐magnetosphere interaction is discussed within the framework of deterministic nonlinear dynamics. Linear prediction filter studies have shown that the magnetospheric response to energy transfer from the solar wind contains both directly driven and unloading components. These studies have also shown that the response is significantly nonlinear and, thus, the filter technique and other correlative techniques cannot give a complete description of that response. Phase space reconstruction studies have shown that the evolution of the nonlinear solar wind‐magnetosphere system is dominated by only a few degrees of freedom; the system approaches a low‐dimensional attractor on which its behavior can be described using a relatively simple nonlinear dynamical model. An earlier dripping faucet analogue model of the low‐dimensional solar wind‐magnetosphere system is briefly reviewed, and then a plasma physical counterpart to that model is constructed. A Faraday loop in the magnetotail is considered, and the relationship of electric potentials on the loop to changes in the magnetic flux threading the loop is developed. This approach leads to a model of geomagnetic activity which is similar to the earlier mechanical model but described in terms of the geometry and plasma contents of the magnetotail. The model is best characterized as an elementary time‐dependent global convection model. The convection evolves within a magnetotail shape that varies in a prescribed manner in response to the dynamical evolution of the convection. The result is a nonlinear model capable of exhibiting a transition from regular to chaotic loading and unloading. The behavior of the model under steady loading and also some elementary forms of time‐dependent loading is discussed. The model appears to properly account for all macrophysical aspects of magnetotail geomagnetic activity, it incorporates both the directly driven and the unloading components of geomagnetic activity, and it includes, in a fundamental way, the inherent nonlinearity of the solar wind‐magnetosphere interaction.