Nonlinear Dynamic Technique Research Articles

An exploratory study of chaos in human-Machine system dynamics

The human-machine system behavior and performance are dynamic, nonlinear, and possibly chaotic. Various techniques have been used to describe such dynamic and nonlinear system characteristics. However, these techniques have rarely been able to accommodate the chaotic behavior of such a nonlinear system. Therefore, this study proposes the use of nonlinear dynamic system theory as one possible technique to account for the dynamic, nonlinear, and possibly chaotic human-machine system characteristics. It briefly describes some of the available nonlinear dynamic system techniques and illustrates how their application can explain various properties of the human-machine system. A pilot's heart interbeat interval (IBI) and altitude tracking error time series data are used in the illustration. Further, the possible applications of the theory in various domains of human factors for on-line assessment, short-term prediction, and control of human-machine system behavior and performance are discussed.

Nonlinear analysis of rainfall dynamics in California's Sacramento Valley

This study investigates the dynamic nature of rainfall observed at the Sustainable Agriculture Farming Systems (SAFS) site in California's Sacramento Valley, which was established to study the benefits of winter cover cropping in Mediterranean irrigated-arid systems. Rainfall data of four different temporal scales (i.e. daily, weekly, biweekly, and monthly) are analysed to determine the dynamic nature of precipitation in time. In an arid climate with seasonal precipitation this has large implications for land and water management, both in the short term and in the long term. A nonlinear dynamic technique (correlation dimension method) that uses the phase-space reconstruction and dimension concepts is employed. Bearing in mind the possible effects of the presence of zeros (i.e. no rain) on the outcomes of this analysis, an attempt is also made to compare the dynamic nature of all-year rainfall and winter rainfall. Analysis of 15 years of data suggests that rainfall dynamics at this site are dominated by a large number of variables, regardless of the scales and seasons studied. The dimension results also suggest that: (1) rainfall dynamics at coarser resolutions are more irregular than that at finer resolutions; (2) winter rainfall has a higher variability than all-year rainfall. These results are indeed useful to gain information about the complexity of the rainfall process at this site with respect to (temporal) scales and seasons and, hence, the appropriate model (high-dimensional) type. However, in view of the potential effects of certain rainfall data characteristics (e.g. zeros, measurement errors, scale effects) on the correlation dimension analysis, the discussion also emphasizes the need for further verification, and possibly confirmation, of these results. Copyright © 2005 John Wiley & Sons, Ltd.

Nonlinear Dynamic Methods Could Predict Recurrences of Reciprocating Arrhythmia in WPW Patients After Radiofrequency Ablation of Accessory Pathways

Recent data support superiority of nonlinear dynamic measurements over traditional heart rate variability (HRV) in predicting adverse cardiac events. We tried to estimate R-R interval fluctuation in patients with the recurrence of tachyarrhythmia after RFA of accessory pathways using traditional HRV and nonlinear dynamic techniques. The study consisted of 25 WPW-patients (4 women), 31.3±12.2 y.o. Arrhythmia history was 7.6±6.8 years. Successful RFA of accessory pathways was performed in all pts. Recurrences of tachyarrhythmia were found within the first 24 hours (1 pt), 2 months (2 pts) and 9 months (2 pts) after RFA (5 pts in total). The following characteristics of chaos we used: informational dimension, fractal dimension and Lapunov's parameters (λ). λ was significantly higher 6 hours after RFA in AVRT recurrence pts comparing with post-RFA arrhythmia recurrence-free pts (4.16±0.26 vs. 3.58±0.28, p <0.05). There were no differences regarding HRV measurements and other nonlinear dynamic methods parameters noted between the groups. nonlinear dynamic methods might be considered as means to predict arrhythmia recurrence in WPW-syndrome pts who undergo RFA of accessory pathways.

A Nonlinear Dynamics Approach to Human Movement

Nonlinear dynamics and dynamical systems approaches and methodologies are increasingly being implemented in biomechanics and human movement research. Based on the early insights of Nicolai Bernstein (1967), a significantly different outlook on the movement control “problem” over the last few decades has emerged. From a focus on relatively simple movements has arisen a research focus with the primary goal to study movement in context, allowing the complexity of patterns to emerge. The approach taken is that the control of multiple degrees-of-freedom systems is not necessarily more difficult or complex than that of systems only comprising a few degrees of freedom. Complex patterns and dynamics might not require complex control structures. In this paper we present a tutorial overview of the mathematical underpinnings of nonlinear dynamics and some of its basic analysis tools. This should provide the reader with a basic level of understanding about the mathematical principles and concepts underlying pattern stability and change. This will be followed by an overview of dynamical systems approaches in the study of human movement. Finally, we discuss recent progress in the application of nonlinear dynamical techniques to the study of human locomotion, with particular focus on relative phase techniques for the assessment of coordination.

Inaccessibility in Online Learning of Recurrent Neural Networks

We apply nonlinear dynamical system techniques to recurrent neural networks. In particular, we numerically analyze the dynamical system characteristics of the online learning process. By introducing the notion of inaccessibility, we show that the learning process is well characterized by strong nonhyperbolicity and inaccessibility, which is a greater uncertainty than chaotic unpredictability. These results are clearly contrasted with a gradient descent dynamics, or ordinary chaos.

Characterization of chaotic dynamics in the vocalization of Cervus elaphus corsicanus (L)

Chaos, oscillations, instabilities, and intermittency represent only some nonlinear examples apparent in the natural world. These phenomena appear in any field of study, and advances in complex and nonlinear dynamic techniques bring about opportunities to better understand animal signals. In this work an analysis method is suggested based on the characterization of the vocal-fold dynamics by means of the nonlinear time-series analysis, and by the computations of the parameters typical of chaotic oscillations: Attractor reconstruction, spectrum of Lyapunov exponents, and maximum Lyapunov exponent were used to reconstruct the dynamic of the vocal folds. Identifying a sort of vocal fingerprint can be useful in biodiversity monitoring and understanding the health status of a given animal. This method was applied to the vocalization of the Cervus elaphus corsicanus, the Sardinian red deer.

Dynamics of Ship–Motion

AbstractThe ship capsizing problem is one of the major challenges in naval architecture. The IMO criterion regarding capsize stability is still the righting lever curve of static stability calculated for calm water. For the prediction of large–amplitude motions the dynamic loads have to be included. The capsizing of a ship in regular waves is resulting from a sequence of bifurcations in the ship's motion: The determination of bifurcations is possible using path‐following techniques of nonlinear dynamics. Existing tools are, however, without adaption not readily applicable for the determination of bifurcations. The main and until now unsolved problem is the necessity of including memory integrals to describe the ship hydrodynamics. First results with simple algorithms show two different scenarios leading to capsizing due to increasing wave amplitudes.

Experimental and Theoretical Studies of the Multistability in the Electroreduction of the Nickel(II)−N3-Complexes at a Streaming Mercury Electrode

Following our earlier experimental and theoretical studies of the self-organization in the Ni(II)−SCN- electroreduction at the streaming mercury electrode we describe analogous phenomena for the azide complexes of nickel(II). The complex electrochemical mechanism of this process is a source of multistability, if the appropriate serial ohmic resistance is present in the electric circuit. In dependence of the solution composition, bistability and tristability in the response of the electric current vs the applied voltage was observed. No spontaneous oscillations were reported for the studied process. The I−E characteristics of the Ni(II)−N3- electroreduction, which exhibits the region of the negative differential resistance (NDR), are the source for the bistable behavior. If the composition of the sample is adjusted so that the current of the Ni(II) electroreduction in the NDR region overlaps significantly with the extra current, originating presumably from the catalytic reduction of azide ions, the second NDR region is also formed, and these complex characteristics are a source of the tristable behavior. The theoretical description of the observed multistable phenomena was made in terms of both the theory of electrode processes at a streaming electrode and standard techniques of nonlinear dynamics. The linear stability analysis led to theoretical bifurcation diagrams similar to the experimental ones. The reported tristability seems to be one of the first experimental examples of such a behavior in electrochemical systems and one of only a few of its manifestations in chemical systems in general.

Theoretical Models of Chronotherapy: I. Periodic Perturbations in Oscillating Chemical Reactions

Dynamics resulting from periodic perturbations in oscillating chemical reaction, as theoretical models of chronotherapy techniques are studied. Replication reactions and fluctuations take place in these models giving rise to new species with autocatalitic properties. Oscillating chemical reaction models due to Hanusse (H) with three intermediaries and Bourceanu (E1, E2, E3 ) with four, five and six intermediaries, respectively, are taken for this study and are manipulated numerically using a Gear algorithm. The bifurcation diagrams are obtained applying non-linear dynamic techniques, such as, power spectrum, stroboscopic method, Lyapunov exponents and correlation dimension D 2. The dynamics of Hanusse and Bourceanu perturbed models are very similar. The main dynamic behaviours observed after perturbation are quasiperiodicity, entrainment, some Arnold tongues and situations without chemical meaning. Chaos is not observed. This study contributes to understand theoretical and practical aspects of the chronotherapy techniques.

Nonlinear elastoplastic dynamic analysis of single-layer reticulated shells subjected to earthquake excitation

A nonlinear dynamic finite element technique is developed to analyze the elastoplastic dynamic response of single-layer reticulated shells under strong earthquake excitation, in which the nonlinear three-dimensional beam elements are employed. An elastoplastic tangent stiffness matrix of three-dimensional beam element is derived by using the updated Lagrangian formulation, in which the isotropic hardening model, the Von-Mises yield criterion and the Prandtl–Reuss flow relations are applied to this study. This procedure considers both geometric and material nonlinearities. In this paper, several condensation and reduction techniques in matrices and degrees of freedom are used to simplify the analysis. An incremental-iterative technique based on the Newmark direct integration method and the modified Newton–Raphson method is employed for obtaining the solutions of the nonlinear dynamic equilibrium equations. Moreover, an accurate method is developed to compute the large rotations of space structures. As a numerical example, the elastoplastic dynamic response of a single-layer reticulated shell under strong seismic excitation is investigated. It is shown through the numerical example that the method developed in this paper is efficient for the nonlinear dynamic response analysis and plastic design of space structures.

Phase transition‐like behavior of magnetospheric substorms: Global MHD simulation results

Using nonlinear dynamical techniques, we statistically investigate whether the simulated substorms from global magnetohydrodynamic (MHD) models have a combination of global and multiscale features, revealed in substorm dynamics by Sitnov et al. [2000] and featured the phase transition‐like behavior. We simulate seven intervals of total duration of 280 hours from the data set used in the above works [Bargatze et al., 1985]. We analyze the input–output (vBs–pseudo AL index) system obtained from the global MHD model and compare the results to those inferred from the original set (vBs–observed AL index). The analysis of the coupled vBs–pseudo AL index system shows the first‐order phase transition map, which is consistent with the map obtained for the vBs–observed AL index system. Although the comparison between observations and global MHD simulations for individual events may vary, the overall global transition pattern during the substorm cycle revealed by singular spectrum analysis (SSA) is statistically consistent between simulations and observations. The coupled vBs–pseudo AL index system also shows multiscale behavior (scale‐invariant power law dependence) in SSA power spectrum. Besides, we find the critical exponent of the nonequilibrium transitions in the magnetosphere, which reflects the multiscale aspect of the substorm activity, different from power law frequency of autonomous systems. The exponent relates input and output parameters of the magnetosphere. We also discuss the limitations of the global MHD model in reproducing the multiscale behavior when compared to the real system.

Controlling chaos in a fluid flow past a movable cylinder

The model of a two-dimensional fluid flow past a cylinder is a relatively simple problem with a strong impact in many applied fields, such as aerodynamics or chemical sciences, although most of the involved physical mechanisms are not yet well known. This paper analyzes the fluid flow past a cylinder in a laminar regime with Reynolds number, Re, around 200, where two vortices appear behind the cylinder, by using an appropriate time-dependent stream function and applying non-linear dynamics techniques. The goal of the paper is to analyze under which circumstances the chaoticity in the wake of the cylinder might be modified, or even suppressed. And this has been achieved with the help of some indicators of the complexity of the trajectories for the cases of a rotating cylinder and an oscillating cylinder.

A new framework for investigating both normal and abnormal eye movements

Any comprehensive framework for understanding eye movements has to include both normal and abnormal eye movement behaviour. One approach which is applicable to the entire range of oculomotor behaviour is provided by the techniques of nonlinear dynamics. The stability of models of the oculomotor system can be analysed in terms of the characteristics of their fixed points and periodic orbits, and the method of delays can be used to recover such parameters from measurements of eye position. Within this framework, quantitative comparisons can be made between the predictions of different models, and both normal and clinical eye movement recordings.

Strain-rate sensitivity limit diagrams and plastic instabilities in a 6xxx series aluminum alloy Part I: Analysis of temporal stress–strain serrations

A method for the construction of “processing windows” to avoid negative strain-rate sensitivity and associated serrated flow in some aluminum alloys is described. The method is based on modern approaches and techniques of non-linear dynamics – linear stability analysis, amplitude Ginzburg–Landau equations, and bifurcation diagrams. The general instability area is identified as possessing a negative strain-rate sensitivity coefficient (SRS). The mathematical technique developed in the present paper was applied to a specific aluminum alloy, Al–0.4%Mg–0.2%Si, and yielded good results in terms of predicting the negative strain-rate sensitivity regions in the “strain rate–temperature” parameter space. It was clearly demonstrated that even though the instability area is located in the region of intermediate strain rates, a qualitative difference exists between the areas of (relatively) fast and (relatively) slow strain rates: while in the first case the dynamic behavior of the system is supercritical, in the second case it is subcritical. The second case is highly undesirable because it causes a sudden onset of stable stress serrations that are difficult to suppress (at a given temperature), while in the first case the development of instability is gradual and, consequently, more easily controllable.

Dynamics of children with torsional anomalies of the lower limb joints

In this study we used the techniques of nonlinear dynamics to analyze the stability of normal and pathological gait in children. We based the analysis on the assumption that a human at steady-state locomotion can be represented as a nonlinear periodic system. Kinematic data for the lower limb joints were used to construct phase plane portraits and first return maps for the hip, the knee and the ankle joints. Anomalies in the joint rotations of pathological individuals were graphically depicted by comparing the phase plane portraits and first return maps. Using the Floquet theory, an index of dynamic stability was used to compare normal and pathological gait.

Stochastic modeling and prediction of experimental seizures in Sprague-Dawley rats.

Most seizure prediction methods are based on nonlinear dynamic techniques, which are highly computationally expensive, thus limiting their clinical usefulness. The authors propose a different approach for prediction that uses a stochastic Markov chain model. Seizure (Ts) and interictal (Ti) durations were measured from 11 rats treated with 3-mercaptopropionic acid. The duration of a seizure Ts was used to predict the time (Ti2) to the next one. Ts and Ti were distributed bimodally into short (S) and long (L), generating four probable transitions: S --> S, S --> L, L --> S, and L --> L. The joint probability density f (Ts, Ti2) was modeled, and was used to predict Ti2 given Ts. An identical model predicted Ts given the duration Ti1 of the preceding interictal interval. The median prediction error was 3.0 +/- 3.5 seconds for Ts (given Ti1) and 6.5 +/- 2.0 seconds for Ti2 (given Ts). In comparison, ranges for observed values were 2.3 seconds < Ts < 120 seconds and 6.6 seconds < Ti < 782 seconds. These results suggest that stochastic models are potentially useful tools for the prediction of seizures. Further investigation of the probable temporal interdependence between the ictal and interictal states may provide valuable insight into the dynamics of the epileptic brain.

Drifter dispersion in the Adriatic Sea: Lagrangian data and chaotic model

Abstract. We analyze characteristics of drifter trajectories from the Adriatic Sea with recently introduced nonlinear dynamics techniques. We discuss how in quasi-enclosed basins, relative dispersion as a function of time, a standard analysis tool in this context, may give a distorted picture of the dynamics. We further show that useful information may be obtained by using two related non-asymptotic indicators, the Finite-Scale Lyapunov Exponent (FSLE) and the Lagrangian Structure Function (LSF), which both describe intrinsic physical properties at a given scale. We introduce a simple chaotic model for drifter motion in this system, and show by comparison with the model that Lagrangian dispersion is mainly driven by advection at sub-basin scales until saturation sets in.Key words. Oceanography: General (marginal and semi-closed seas) – Oceanography: Physical (turbulence, diffusion, and mixing processes; upper ocean processes)

Evidence of low-dimensional dynamics in a traffic noise time series

The noise level measurements were made at 10 s interval near a busy road which admits one-way traffic only. The time series thus obtained was analysed using non-linear dynamical techniques. The results of the analysis suggest that the underlying dynamical process could be deterministic. It appears that the data support periodic and quasiperiodic attractors. The presence of colour fluctuations in the time series could be attributed to shuttling of the dynamics between these two attractors.

Fuzzy methods in tremor assessment, prediction, and rehabilitation

Tremor is a disabling condition for a large segment of population, mainly elderly. To the present date, there are no adequate tools to diagnose and help rehabilitation of subjects with tremor, but recently there is a tremendous surge of interest in the research in the field. We report on the use of fuzzy methods in applications for rehabilitation, namely in tremor diagnosing and control. We synthesize our results regarding the characterization of the tremor by means of nonlinear dynamics techniques and fuzzy logic, and the prediction of tremor movements in view of rehabilitation purposes. Based on linear and nonlinear analysis of tremor, and using fuzzy aggregation, the fusing of tremor parameters in global functional disabling factors is proposed. Nonlinear dynamic parameters, namely correlation dimension and Lyapunov exponent is used in order to improve the assessment of tremor. The benefits of the fuzzy fused tremor parameters rely on more complete and accurate assessment of the functional impairment and on improved feedback for rehabilitation, based on the fused parameters of the tremor. Further steps in rehabilitation may require external muscular control. In turn, the control of tremor by electrical stimulation requires movement prediction. Several neural and neuro-fuzzy predictors are compared and a neuro-fuzzy predictor is presented, allowing us five-step ahead prediction, with an RMS error of the order of 10%. The benefits of the neuro-fuzzy predictor are good prediction capability, versatility, and apparently a high robustness to individual variations of the tremor. The reported research, which extended over several years and included development of sensors, equipment, and software, has been aimed to development of products. The results may also open new ways in tremor rehabilitation.

Analysis and Design Optimization of Flexible Pavement

A project-level optimization approach was developed to minimize total pavement cost within an analysis period. Using this approach, the designer is able to select the optimum initial pavement thickness, overlay thickness, and overlay timing. The model in this approach is capable of predicting both pavement performance and condition in terms of roughness, fatigue cracking, and rutting. The developed model combines the American Association of State Highway and Transportation Officials (AASHTO) design procedure and the mechanistic multilayer elastic solution. The Optimization for Pavement Analysis (OPA) computer program was developed using the prescribed approach. The OPA program incorporates the AASHTO equations, the multilayer elastic system ELSYM5 model, and the nonlinear dynamic programming optimization technique. The program is PC-based and can run in either a Windows 3.1 or a Windows 95 environment. Using the OPA program, a typical pavement section was analyzed under different traffic volumes and material properties. The optimum design strategy that produces the minimum total pavement cost in each case was determined. The initial construction cost, overlay cost, highway user cost, and total pavement cost were also calculated. The methodology developed during this research should lead to more cost-effective pavements for agencies adopting the recommended analysis methods.