Low-dimensional Dynamics Research Articles

Verification of chaotic behavior in an experimental loudspeaker

The dynamics of an experimental electrodynamic loudspeaker is studied by using the tools of chaos theory and time series analysis. Delay time, embedding dimension, fractal dimension, and other empirical quantities are determined from experimental data. Particular attention is paid to issues of stationarity in a system in order to identify sources of uncertainty. Lyapunov exponents and fractal dimension are measured using several independent techniques. Results are compared in order to establish independent confirmation of low dimensional dynamics and a positive dominant Lyapunov exponent. We thus show that the loudspeaker may function as a chaotic system suitable for low dimensional modeling and the application of chaos control techniques.

Low dimensional behavior of large systems of globally coupled oscillators

It is shown that, in the infinite size limit, certain systems of globally coupled phase oscillators display low dimensional dynamics. In particular, we derive an explicit finite set of nonlinear ordinary differential equations for the macroscopic evolution of the systems considered. For example, an exact, closed form solution for the nonlinear time evolution of the Kuramoto problem with a Lorentzian oscillator frequency distribution function is obtained. Low dimensional behavior is also demonstrated for several prototypical extensions of the Kuramoto model, and time-delayed coupling is also considered.

Detection of low-dimensional chaos in wind time series

In the present work we investigated the existence of low-dimensional deterministic chaos in wind time series. The time series were obtained from the New Anchialos (Greece) Air Base measurement station. In a first place we used the raw data without any noise filtering. Characteristic times were extracted using power spectrum and average mutual information function. The estimation of invariant measures, such as the correlation dimension and Lyapunov exponents indicate the possible existence of a low-dimensional attractor. After noise removal with the use of the local projective method the analysis indicates in a more clear way the existence of a low-dimensional attractor. In addition, the null hypothesis was tested for the dynamical characteristics of the wind time series by using the surrogate data test and the corresponding results provide significant evidence for the existence of low-dimensional chaotic dynamics underlying the wind time series.

Chaos and plasticity in superconductor vortices: Low-dimensional dynamics

We present results of numerical simulations for driven vortex lattices in the presence of a random disorder at zero temperature. We show that the plastic dynamics of vortices display dissipative chaos. Intermittency ``routes to chaos'' have clearly been identified below the differential resistance peak. The peak region is characterized by positive Lyapunov exponents, which are characteristic of chaos, and low frequency broad-band noise. Furthermore, we find a low fractal dimension of the strange attractor, which suggests that only a few dynamical variables are sufficient to model the complex plastic dynamics of vortices.

Experimental evidence of low-dimensional chaotic convection dynamics in a toroidal magnetized plasma

In a toroidal plasma confined by a purely toroidal magnetic field with a weak vertical field superimposed a system of convection cells are generated spontaneously, interacting with a background electron density gradient. The dynamics of this interaction is low-dimensional, chaotic, and consistent with solutions of the Lorenz equations in the diffusionless limit.

Controlling Engine System: a Low-Dimensional Dynamics in a Spark Ignition Engine of a Motorcycle

We analyze a time series of the combustion pressure in the idle state, measured from a spark ignition engine of a motorcycle. It is clarified that the engine system can be described by a lowdimensional deterministic dynamics perturbed by some stochastic process.We also propose a method to stabilize the chaotic behaviour of engine’s data by adopting the Pyragas’ method. We actually use this method in a computer experiment for the control of combustion pressure data to demonstrate the efficiency of the proposed method. As a result of the experiment, we eliminate the fluctuations in the combustion pressure data and obtain a periodic orbit.

Searching chaos and coherent structures in the atmospheric turbulence above the Amazon forest

In this work, the possible chaotic nature of the atmospheric turbulence above a densely forested area in the Amazon region is investigated. To this end, we use high-resolution temperature data obtained during a micrometeorological measurement campaign in the Brazilian Amazonia. Estimates of the correlation dimension (D(2)=3.50+/-0.05) and of the largest Lyapunov exponent (lambda(1)=0.050+/-0.002) suggest the existence of chaos in the atmospheric boundary layer. Our findings indicate that this low-dimensional chaotic dynamics is associated with the presence of the coherent structures within the boundary layer right above the canopy top and not with the atmospheric turbulence per se, as previously claimed.

Low-dimensional chaos and wave turbulence in plasmas

We investigated drift-wave turbulence in the plasma edge of a small tokamak by considering solutions of the Hasegawa-Mima equation involving three interacting modes in Fourier space. The resulting low-dimensional dynamics presented periodic as well as chaotic evolution of the Fourier-mode amplitudes, and we performed the control of chaotic behaviour through the application of a fourth resonant wave of small amplitude.

Near field characteristics of swirling flow past a sudden expansion

The effect of swirl on the flow characteristics of an axisymmetric sudden expansion chamber with an expansion ratio of 2.5 was examined experimentally. Particle image velocimetry (PIV) was employed to capture the instantaneous flow field. The experiments were carried out for three swirl numbers of 0, 0.17 and 0.65 and a Reynolds number of 10,000. The measured time-mean and fluctuating velocities downstream of the expansion showed that the introduction of weak swirl does not affect the flow significantly whereas stronger swirl ( S = 0.65 ) results in the formation of a central recirculation region, a typical manifestation of vortex breakdown, which alters the character of the flow. Proper orthogonal decomposition (POD) was employed to further characterise the flow by decomposing it into eigenmodes and identifying dominant structures and their spatial distribution. Both velocity-based and vorticity-based POD were implemented. The analysis indicated that the first two eigenmodes are the most energetic ones and are associated with the instability of the jet, whereas higher modes are associated with rolling vortices along the shear layer and smaller scale motion. Phase portraits of the time-dependent amplitude coefficients also indicated the existence of low-dimensional dynamics in the swirling flow for swirl numbers well below the critical ones for vortex breakdown. This low dimensionality at low swirl numbers might be due to a precessing motion of the flow.

Compact dynamical model of brain activity

A compact physiologically based mean-field formulation of brain dynamics is proposed to model observed brain activity and electroencephalographic (EEG) signals. In contrast to existing formulations, which are more detailed and complicated, our model is described by a single second-order delay differential equation that encapsulates salient aspects of the physiology. The model captures essential features of activity mediated by fast corticocortical connections and delayed feedbacks via extracortical pathways and external stimuli. In the linear regime, these features can be simply expressed by three coefficients derived from the properties of these physiological pathways and explicit nonlinear approximations are also derived. This compact model successfully reproduces the main features of experimental EEG's and the predictions of previous models, including resonance peaks in EEG spectra and nonlinear dynamics. As an illustration, key features of the dynamics of epileptic seizures are shown to be reproduced by the model. Due to its compact form, the model will facilitate insight into nonlinear brain dynamics via standard nonlinear techniques and will guide analysis and investigation of more complex models. It is thus a useful tool for analyzing complex brain activity, especially when it exhibits low-dimensional dynamics.

Manipulating the edge of instability

We investigate the integration of visual and tactile sensory input for dynamic manipulation. Our experimental data and computational modeling reveal that time-delays are as critical to task-optimal multisensory integration as sensorimotor noise. Our focus is a dynamic manipulation task “at the edge of instability.” Mathematical bifurcation theory predicts that this system will exhibit well-classified low-dimensional dynamics in this regime. The task was using the thumbpad to compress a slender spring prone to buckling as far as possible, just shy of slipping. As expected from bifurcation theory, principal components analysis gives a projection of the data onto a low dimensional subspace that captures 91–97% of its variance. In this subspace, we formulate a low-order model for the brain+hand+spring dynamics based on known mechanical and neurophysiological properties of the system. By systematically occluding vision and anesthetically blocking thumbpad sensation in 12 consenting subjects, we found that vision contributed to dynamic manipulation only when thumbpad sensation was absent. The reduced ability of the model system to compress the spring with absent sensory channels closely resembled the experimental results. Moreover, we found that the model reproduced the contextual usefulness of vision only if we took account of time-delays. Our results shed light on critical features of dynamic manipulation distinct from those of static pinch, as well as the mechanism likely responsible for loss of manual dexterity and increased reliance on vision when age or neuromuscular disease increase noisiness and/or time-delays during sensorimotor integration.

Low Dimension Dynamics in the EPRB Experiment with Random Variable Analyzers

The Einstein–Podolsky–Rosen–Bohm (EPRB) experiment performed with random variable and spatially separated analyzers is a milestone test in the controversy between Objective Local Theories (OLT) and Quantum Mechanics (QM). Only a few OLT are still possible. Some of the surviving OLT (specifically, the so called non-ergodic theories) would be undetectable in the averaged statistical values, but they may leave their trace in the time dynamics. For, while QM predicts random processes, the OLT of this kind predict the existence of regularities that may be revealed as a low dimensional object in the phase space. We perform a numerical analysis of the time-resolved data recorded in that experiment to unveil any hypothetical low dimensional dynamics that may be present. We find no consistent indication of such dynamics except for one data file, the longest of all in the real time. The possible causes of these dynamics are discussed.

Nonlinear time series analysis of food intake in the dab and the rainbow trout

Evidence suggests that dab and rainbow trout are able to quickly adjust their food intake to an appropriate level when offered novel diets. In addition day-to-day and meal-to-meal food intake varies greatly and meal timing is plastic. Why this is the case is not clear: Food intake in fish is influenced by many factors, however the hierarchy and mechanisms by which these interact is not yet fully understood. A model of food intake may be helpful to understand these phenomena; to determine model type it is necessary to understand the qualitative nature of food intake. Food intake can be regarded as an autoregressive (AR) time series, as the amount of food eaten at time t will be influenced by previous meals, and this allows food intake to be considered using time series analyses. Here, time series data were analysed using nonlinear techniques to obtain qualitative information from which evidence for the hierarchy of mechanisms controlling food intake may be drawn. Time series were obtained for a group of dab and individuals and a group of rainbow trout for analysis. Surrogate data sets were generated to test several null hypotheses describing linear processes and all proved significantly different to the real data, suggesting nonlinear dynamics. Examination of topography and recurrence diagrams suggested that all series were deterministic and non-stationary. The point correlation dimension (PD2i) suggested low-dimensional dynamics. Our findings suggest therefore that any model of appetite should create output that is deterministic, non-stationary, low-dimensional and having nonlinear dynamics.

Evidence for low dimensional chaos in sunspot cycles

Sunspot cycles are widely used for investigating solar activity. In 1953 Bracewell argued that it is sometimes desirable to introduce the inversion of the magnetic field polarity, and that can be done with a sign change at the beginning of each cycle. It will be shown in this paper that, for topological reasons, this so-called Bracewell index is inappropriate and that the symmetry must be introduced in a more rigorous way by a coordinate transformation. The resulting symmetric dynamics is then favourably compared with a symmetrized phase portrait reconstructed from the z -variable of the Rossler system. Such a link with this latter variable – which is known to be a poor observable of the underlying dynamics – could explain the general difficulty encountered in finding evidence of low-dimensional dynamics in sunspot data.

Technology Development of Estimation Method for Complex Gas-liquid Two-phase Flow in Nuclear Reactors

Numerical simulation of gas-liquid two-phase flow is important for design of nuclear power systems. Conventional codes require experiments from laboratory scale to mock-up to obtain database for constitutive equations and verification. The aim of this study is to provide new simulation technique to substitute for a part of costly experiments. The extended two-fluid model developed by Hitachi, Ltd. was employed to treat inhomogeneous and intermittent two-phase flow behavior, which had been considered through constitutive equations. Validity of the new simulation code for complex two-phase flow was proved by comparison between calculated and experimental void fraction signals from two-phase flows in circular tubes, CCFL of a thick plate, and natural circulation of aerated pool in double cylinders. For measurement of instantaneous, multi-dimensional void fraction distributions, wire-mesh sensor techniques were employed for all test setups. It was found that the two-phase flow has higher orders of dynamics than initially expected, leading to development of robust analysis methods. The comparison was successful for well-structured flow patterns as slug flow, since the large diameter bubbles show low dimensional dynamics. It is also noted that an optimal delay time reconstruction method and a point correlation dimension method together were useful to increase the reliability of the analysis.

Nonlinear dynamics of homeothermic temperature control in skunk cabbage,Symplocarpus foetidus

Certain primitive plants undergo orchestrated temperature control during flowering. Skunk cabbage, Symplocarpus foetidus, has been demonstrated to maintain an internal temperature of around 20 degrees C even when the ambient temperature drops below freezing. However, it is not clear whether a unique algorithm controls the homeothermic behavior of S. foetidus, or whether such an algorithm might exhibit linear or nonlinear thermoregulatory dynamics. Here we report the underlying dynamics of temperature control in S. foetidus using nonlinear forecasting, attractor and correlation dimension analyses. It was shown that thermoregulation in S. foetidus was governed by low-dimensional chaotic dynamics, the geometry of which showed a strange attractor named the "Zazen attractor." Our data suggest that the chaotic thermoregulation in S. foetidus is inherent and that it is an adaptive response to the natural environment.

Testing for nonlinearity and low-dimensional dynamics in the slow solar wind

We focus on nonlinearity and possible deterministic behavior of the low-speed solar wind. We analyze time series of velocities of the streams including Alfvénic fluctuations measured by the Helios spacecraft in the inner heliosphere. Because of nonlinear behavior of that flow we use the average-mutual-information and false-nearest-neighbors methods. The fraction of false-nearest-neighbors drops to zero at dimension equal to four for both the radial velocity and Alfvénic velocity, independent of the distance from the Sun. The question of possible colored noise in the solar wind is also discussed. The obtained results clearly indicate that a low-dimensional attractor should exist. One can therefore hope to infer information about some complex nonlinear phenomena from the geometrical properties of the attractor or by solving a set of four nonlinear ordinary differential equations describing the system. One can expect that the attractor should contain information about the dynamic variations of the coronal streamers or it could represent a structure of the time sequence of near-Sun coronal fine-stream tubes.

Noise-induced synchronization of uncoupled nonlinear systems

We derive a recursion formulae of transition probability of the noise-induced synchronization arising in a pair of identical uncoupled logistic maps linked by common noisy excitation only. The formulae has a delta-type stationary solution which represents the perfect synchronization with probability 1. The stationary solution maintains under chaotic bifurcation while the escape times to reach the perfect synchronization increase in the chaotic region. The escape times analysis implies existence of lower dimensional dynamics around the perfect synchronization. We also provide a physical implementation of the synchronization.

Mechanisms for the Development of Locally Low-Dimensional Atmospheric Dynamics

Abstract The complexity of atmospheric instabilities is investigated by a combination of numerical experiments and diagnostic tools that do not require the assumption of linear error dynamics. These tools include the well-established analysis of the local energetics of the atmospheric flow and the recently introduced ensemble dimension (E dimension). The E dimension is a local measure that varies in both space and time and quantifies the distribution of the variance between phase space directions for an ensemble of nonlinear model solutions over a geographically localized region. The E dimension is maximal, that is, equal to the number of ensemble members (k), when the variance is equally distributed between k phase space directions. The more unevenly distributed the variance, the lower the E dimension. Numerical experiments with the state-of-the-art operational Global Forecast System (GFS) of the National Centers for Environmental Prediction (NCEP) at a reduced resolution are carried out to investigate the spatiotemporal evolution of the E dimension. This evolution is characterized by an initial transient phase in which coherent regions of low dimensionality develop through a rapid local decay of the E dimension. The typical duration of the transient is between 12 and 48 h depending on the flow; after the initial transient, the E dimension gradually increases with time. The main goal of this study is to identify processes that contribute to transient local low-dimensional behavior. Case studies are presented to show that local baroclinic and barotropic instabilities, downstream development of upper-tropospheric wave packets, phase shifts of finite amplitude waves, anticyclonic wave breaking, and some combinations of these processes can all play crucial roles in lowering the E dimension. The practical implication of the results is that a wide range of synoptic-scale weather events may exist whose prediction can be significantly improved in the short and early medium range by enhancing the prediction of only a few local phase space directions. This potential is demonstrated by a reexamination of the targeted weather observations missions from the 2000 Winter Storm Reconnaissance (WSR00) program.

Whole Cell Stochastic Model Reproduces the Irregularities Found in the Membrane Potential of Bursting Neurons

Irregular intrinsic behavior of neurons seems ubiquitous in the nervous system. Even in circuits specialized to provide periodic and reliable patterns to control the repetitive activity of muscles, such as the pyloric central pattern generator (CPG) of the crustacean stomatogastric ganglion (STG), many bursting motor neurons present irregular activity when deprived from synaptic inputs. Moreover, many authors attribute to these irregularities the role of providing flexibility and adaptation capabilities to oscillatory neural networks such as CPGs. These irregular behaviors, related to nonlinear and chaotic properties of the cells, pose serious challenges to developing deterministic Hodgkin-Huxley-type (HH-type) conductance models. Only a few deterministic HH-type models based on experimental conductance values were able to show such nonlinear properties, but most of these models are based on slow oscillatory dynamics of the cytosolic calcium concentration that were never found experimentally in STG neurons. Based on an up-to-date single-compartment deterministic HH-type model of a STG neuron, we developed a stochastic HH-type model based on the microscopic Markovian states that an ion channel can achieve. We used tools from nonlinear analysis to show that the stochastic model is able to express the same kind of irregularities, sensitivity to initial conditions, and low dimensional dynamics found in the neurons isolated from the STG. Without including any nonrealistic dynamics in our whole cell stochastic model, we show that the nontrivial dynamics of the membrane potential naturally emerge from the interplay between the microscopic probabilistic character of the ion channels and the nonlinear interactions among these elements. Moreover, the experimental irregular behavior is reproduced by the stochastic model for the same parameters for which the membrane potential of the original deterministic model exhibits periodic oscillations.