Low-dimensional Dynamical Systems Research Articles

Inferences on the source of long-period seismicity at Campi Flegrei from polarization analysis and reconstruction of the asymptotic dynamics

In the present paper, we analyse long-period (LP) events and seismic noise recorded at Campi Flegrei during the uplift episode of 2004–2006. The results of a detailed polarization analysis indicate that the large number of LPs detected during the seismic crisis were accompanied by a sustained activity consisting of very low-energy volcanic signals. These weak volcanic signals, which are usually absent in ambient noise recorded in the studied area, and LP events have similar properties, namely they are radially polarized towards the same LP source. The large dataset analysed allows us to study the statistics of the polarization, leading to the extraction of the average properties of the volcanic signals. An estimate of the complexity of the system can be provided by the degrees of freedom necessary to describe the asymptotic dynamics in a reconstructed phase space. This analysis shows that both LPs and low-energy signals can be described by a low-dimensional dynamical system, while ambient noise unrelated to volcanic activity is higher dimensional. Based on these observations, we interpret the phenomena observed during the seismic crisis in the framework of the theory of self-oscillating systems, in which LPs and low-energy signals represent self-oscillations generated by a persistent hydrothermal source.

Dynamical responses of oscillating yeast cell suspensions to periodic forcing.

Fermenting S. cerevisiae cells suspensions undergo synchronized metabolic oscillations whose amplitude and frequency depends on the population density. The asymptotic population-level dynamics is qualitatively and quantitatively accounted for by mathematical models describing the intracellular glycolytic oscillations and the coupling through the external medium. Such dynamical systems often display low-dimensional dynamics for the parameters that quantitatively fit the experimental observations. However, the high dimensionality of the chemical space and the possible existence of intermingled time scales can potentially support more complex dynamics, in particular when the system is perturbed by external forcings. Here, we explore experimentally and by means of numerical models the response of cellular populations to different kinds of periodic forcing: resonant forcing of diluted, nonoscillating populations and strongly nonresonant forcing of populations in oscillating regimes with different dynamical features. We show that, in all cases, low-dimensional semi-quantitative models reproduce the observed dynamics. Both a nonlinear analysis of the experimental time series and the models indicate that complex-looking time series correspond to quasiperiodic and not to chaotic regimes. The fact that low-dimensional dynamical systems are able to reproduce the response of biological populations in different regimes of external periodic forcing supports the use of theoretical models for inquiring the dynamical behaviour of collectively oscillating cells.

Regional dynamical modeling and control of flow problems under unmeasurable and non-constant viscosity

In this paper a novel approach to the modeling and control of flow problems is considered. The main extension over existing methods is the ability to handle a local region of interest, and the capability to deal with the fluid viscosity being non-constant and unmeasurable. For the modeling part, first a number of snapshots of the fluid flow process at different viscosity values are obtained by computational fluid dynamics simulations of the Navier–Stokes equations governing the flow. Wavelet transform, thresholding and reconstruction are applied to these snapshots and it is seen that the flow process can be represented with acceptable accuracy using only the approximation coefficients of the wavelet transform. The support of the basis functions are selected to tightly cover the desired region of interest so that the flow dynamics outside the desired region do not affect the model significantly. Subspace system identification methods are used to fit a low-dimensional dynamical system model to the approximation coefficients, which yields a set of linear time invariant models, one for each breakpoint viscosity. A single uncertain model is built to capture all of the models in this set, and a robust controller is designed for this uncertain model using D-K iteration. The technique is illustrated on a sample flow configuration on a square domain where the input affects the system through the boundary conditions.

The Nonadditive Entropy Sq and Its Applications in Physics and Elsewhere: Some Remarks

The nonadditive entropy Sq has been introduced in 1988 focusing on a generalization of Boltzmann–Gibbs (BG) statistical mechanics. The aim was to cover a (possibly wide) class of systems among those very many which violate hypothesis such as ergodicity, under which the BG theory is expected to be valid. It is now known that Sq has a large applicability; more specifically speaking, even outside Hamiltonian systems and their thermodynamical approach. In the present paper we review and comment some relevant aspects of this entropy, namely (i) Additivity versus extensivity; (ii) Probability distributions that constitute attractors in the sense of Central Limit Theorems; (iii) The analysis of paradigmatic low-dimensional nonlinear dynamical systems near the edge of chaos; and (iv) The analysis of paradigmatic long-range-interacting many-body classical Hamiltonian systems. Finally, we exhibit recent as well as typical predictions, verifications and applications of these concepts in natural, artificial, and social systems, as shown through theoretical, experimental, observational and computational results.

Improved dimensionally-reduced visual cortical network using stochastic noise modeling

In this paper, we extend our framework for constructing low-dimensional dynamical system models of large-scale neuronal networks of mammalian primary visual cortex. Our dimensional reduction procedure consists of performing a suitable linear change of variables and then systematically truncating the new set of equations. The extended framework includes modeling the effect of neglected modes as a stochastic process. By parametrizing and including stochasticity in one of two ways we show that we can improve the systems-level characterization of our dimensionally reduced neuronal network model. We examined orientation selectivity maps calculated from the firing rate distribution of large-scale simulations and stochastic dimensionally reduced models and found that by using stochastic processes to model the neglected modes, we were able to better reproduce the mean and variance of firing rates in the original large-scale simulations while still accurately predicting the orientation preference distribution.

Boundary collapse in models of shear-flow transition

We explore two low-dimensional dynamical systems modeling transition to turbulence in shear flows to try to understand the nature of the boundary ∂B of the basin of attraction B of the stable, laminar point at the origin of coordinates. Components of ∂B are found to exist in two types: one (the ‘strong’ type) separating B from a complementary set where orbits never relaminarize, and a second (the ‘weak’ type) separating B into two parts locally but not globally. For a boundary of weak type, orbits on each side relaminarize but may be distinguished from one another by features such as orbital complexity and time to relaminarize. The basin boundary may be of a single type, or may be a union of components of different types. The models are parametrized and may transform from one type to another at a critical parameter value. In the models studied here the change from purely strong type to a union of the two types occurs via the collapse of two sheets of a strong boundary into a single sheet. This is accompanied, at the critical value of the parameter, by the appearance of a homoclinic orbit and the subsequent occurrence of a periodic orbit on the strong part of the boundary.

Updating response sensitivity models of nonlinear vibrating structures using particle filters

The problem of updating response gradients with respect to chosen system parameters based on spatially sparse measurements is considered. The measurement noise and imperfections in mathematical modeling are treated as Gaussian white noise processes. The system states are augmented by response gradients with respect to system parameters and an extended set of equations in the state space is formulated. These equations are cast in the form of Ito’s stochastic differential equations and measured data are assimilated into this model using Monte Carlo based Bayesian filtering tools. Illustrative examples include a few low dimensional dynamical systems with cubic and hereditary nonlinearities.

Properties of an Arithmetic Code for Geodesic Flows

Topological analysis of chaotic dynamical systems emerged in the nineties as a powerful tool in the study of strange attractors in low-dimensional dynamical systems. It is based on identifying the stretching and squeezing mechanisms responsible for creating a strange attractor and organize all the unstable periodic orbits in this attractor. This method is concerned with the manifold generated by the chaotic system. Furthermore, as a mathematical object, the manifolds have a well studied geometric and algebraic structure, particularly for the case of compact surfaces. Intending to use this structure in the analysis and application of chaotic systems through their topological characteristics, we determine properties of geodesic codes for compact surfaces necessary for the construction of encoders from the symbolic sequences of experimental data generated by the unstable periodic orbits of the strange attractor (related to the behavior changes of the system with the variation of control parameters) to the geodesic code sequences, which permits to use the surface structure to study the system orbits.

Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems

This paper first summarizes the theory of quasi-periodic bifurcations for dissipative dynamical systems. Then it presents algorithms for the computation and continuation of invariant circles and of their bifurcations. Finally several applications are given for quasiperiodic bifurcations of Hopf, saddle-node and period-doubling type.

Lagrangian Dynamics and Models of the Velocity Gradient Tensor in Turbulent Flows

Many fundamental and intrinsic properties of small-scale motions in turbulence can be described using the velocity gradient tensor. This tensor encodes interesting geometric and statistical information such as the alignment of vorticity with respect to the strain-rate eigenvectors, rate of deformation and shapes of fluid material volumes, non-Gaussian statistics, and intermittency. In the inertial range of turbulence, similar properties can be described using the coarse-grained or filtered velocity gradient tensor. In this article we review various models that aim at understanding these phenomena using a small number of ordinary differential equations, written either as a low-dimensional dynamical system or as a set of stochastic differential equations. Typically these describe the Lagrangian evolution of the velocity gradient tensor elements following fluid particles and require models for the pressure Hessian and viscous effects. Sample results from various models are shown, and open challenges are highlighted.

Scalable Bayesian Reduced-Order Models for Simulating High-Dimensional Multiscale Dynamical Systems

While existing mathematical descriptions can accurately account for phenomena at microscopic scales (e.g., molecular dynamics), these are often high-dimensional, stochastic, and their applicability over macroscopic time scales of physical interest is computationally infeasible or impractical. In complex systems, with limited physical insight on the coherent behavior of their constituents, the only available information is data obtained from simulations of the trajectories of huge numbers of degrees of freedom over microscopic time scales. The analysis of these large amounts of data hinges upon the ability to efficiently extract meaningful latent properties and to discover reduced, predictive descriptions. This paper discusses a Bayesian approach to deriving probabilistic coarse-grained models that simultaneously addresses the problems of identifying appropriate reduced coordinates and the effective dynamics in this lower-dimensional representation. At the core of the models proposed lie simple, low-dimensional dynamical systems which serve as the building blocks of the global model. These approximate the latent generating sources and parametrize the reduced-order dynamics. On their own, each of these simple models would be unable to explain and predict the various complexities encountered in multiscale dynamics of physical interest. Similar to the way one would synthesize opinions from various experts in order to reach a conclusion, we propose probabilistic models that combine the predictions of all these building blocks in order to obtain an integrated model that provides a good global approximation. We discuss parallelizable, online inference and learning algorithms that employ sequential Monte Carlo samplers and scale linearly with the dimensionality of the observed dynamics. We propose a Bayesian adaptive time-integration scheme that utilizes probabilistic predictive estimates and enables rigorous concurrent simulation over macroscopic time scales. The data-driven perspective advocated assimilates computational and experimental data and thus can materialize data-model fusion. It can deal with applications that lack a mathematical description and where only observational data is available. Furthermore, it makes nonintrusive use of existing computational models.

Dynamical origin of complex motor patterns

Behavior emerges as the nervous system generates motor patterns in charge of driving a peripheral biomechanical device. For several cases in the animal kingdom, it has been identified that the motor patterns used in order to accomplish a diversity of tasks are the different solutions of a simple, low dimensional nonlinear dynamical system. Yet, motor patterns emerge from the interaction of an enormous number of individual dynamical units. In this work, we study the dynamics of the average activity of a large set of coupled excitable units which are periodically forced. We show that low dimensional, yet non trivial dynamics emerges. As a case study, we analyze the air sac pressure patterns used by domestic canaries during song, which consists of a succession of repetitions of different syllable types. We show that the pressure patterns used to generate different syllables can be approximated by the solutions of the investigated model. In this way, we are capable of integrating different description scales of our problem.

Reliability models for existing structures based on dynamic state estimation and data based asymptotic extreme value analysis

The problem of time variant reliability analysis of existing structures subjected to stationary random dynamic excitations is considered. The study assumes that samples of dynamic response of the structure, under the action of external excitations, have been measured at a set of sparse points on the structure. The utilization of these measurements in updating reliability models, postulated prior to making any measurements, is considered. This is achieved by using dynamic state estimation methods which combine results from Markov process theory and Bayes’ theorem. The uncertainties present in measurements as well as in the postulated model for the structural behaviour are accounted for. The samples of external excitations are taken to emanate from known stochastic models and allowance is made for ability (or lack of it) to measure the applied excitations. The future reliability of the structure is modeled using expected structural response conditioned on all the measurements made. This expected response is shown to have a time varying mean and a random component that can be treated as being weakly stationary. For linear systems, an approximate analytical solution for the problem of reliability model updating is obtained by combining theories of discrete Kalman filter and level crossing statistics. For the case of nonlinear systems, the problem is tackled by combining particle filtering strategies with data based extreme value analysis. In all these studies, the governing stochastic differential equations are discretized using the strong forms of Ito–Taylor’s discretization schemes. The possibility of using conditional simulation strategies, when applied external actions are measured, is also considered. The proposed procedures are exemplified by considering the reliability analysis of a few low-dimensional dynamical systems based on synthetically generated measurement data. The performance of the procedures developed is also assessed based on a limited amount of pertinent Monte Carlo simulations.

Boom-Bust Cycles: Leveraging, Complex Securities, and Asset Prices

It appears that many boom-bust cycles are driven by linkages between the credit market and asset prices. Recently, in addition to credit, bonds and stocks, new complex securities have been developed, e.g., MBS, CDO and CDS, that act as instruments of financial intermediation and evaluation of credit risk. We study their role in the recent financial market meltdown and how they exacerbate leverage cycles. We first introduce a baseline model that allows for financial market boom-bust cycles. Then we extend the model to demonstrate the magnifying effects arising from the pricing of the new financial instruments. Finally, we summarize the mechanism of leverage cycles in a low dimensional dynamic system. We also spell out some implication for monetary policy.

Dimensionally-reduced visual cortical network model predicts network response and connects system- and cellular-level descriptions

Systems-level neurophysiological data reveal coherent activity that is distributed across large regions of cortex. This activity is often thought of as an emergent property of recurrently connected networks. The fact that this activity is coherent means that populations of neurons may be thought of as the carriers of information, not individual neurons. Therefore, systems-level descriptions of functional activity in the network often find their simplest form as combinations of the underlying neuronal variables. In this paper, we provide a general framework for constructing low-dimensional dynamical systems that capture the essential systems-level information contained in large-scale networks of neurons. We demonstrate that these dimensionally-reduced models are capable of predicting the response to previously un-encountered input and that the coupling between systems-level variables can be used to reconstruct cellular-level functional connectivities. Furthermore, we show that these models may be constructed even in the absence of complete information about the underlying network.

Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition

The proper orthogonal decomposition (POD) is the prevailing method for basis generation in the model reduction of fluids. A serious limitation of this method, however, is that it is empirical. In other words, this basis accurately represents the flow data used to generate it, but may not be accurate when applied ‘off-design’. Thus, the reduced-order model may lose accuracy for flow parameters (e.g. Reynolds number, initial or boundary conditions and forcing parameters) different from those used to generate the POD basis and generally does. This paper investigates the use of sensitivity analysis in the basis selection step to partially address this limitation. We examine two strategies that use the sensitivity of the POD modes with respect to the problem parameters. Numerical experiments performed on the flow past a square cylinder over a range of Reynolds numbers demonstrate the effectiveness of these strategies. The newly derived bases allow for a more accurate representation of the flows when exploring the parameter space. Expanding the POD basis built at one state with its sensitivity leads to low-dimensional dynamical systems having attractors that approximate fairly well the attractor of the full-order Navier–Stokes equations for large parameter changes.

The von Kármán Sodium experiment: Turbulent dynamical dynamos

The von Kármán Sodium (VKS) experiment studies dynamo action in the flow generated inside a cylinder filled with liquid sodium by the rotation of coaxial impellers (the von Kármán geometry). We first report observations related to the self-generation of a stationary dynamo when the flow forcing is Rπ-symmetric, i.e., when the impellers rotate in opposite directions at equal angular velocities. The bifurcation is found to be supercritical with a neutral mode whose geometry is predominantly axisymmetric. We then report the different dynamical dynamo regimes observed when the flow forcing is not symmetric, including magnetic field reversals. We finally show that these dynamics display characteristic features of low dimensional dynamical systems despite the high degree of turbulence in the flow.

Low dimensional description of pedestrian-induced oscillation of the Millennium Bridge

When it opened to pedestrian traffic in the year 2000, London’s Millennium Bridge exhibited an unwanted, large side-to-side oscillation, which was apparently due to a resonance between the stepping frequency of walkers and one of the bridge modes. Models for this event, and similar events on other bridges, have been proposed. The model most directly addressing the synchronization mechanism of individual walkers and the resulting global response of the bridge-pedestrian system is the one developed by Eckhardt et al. [Phys. Rev. E 75, 021110 (2007)]. This model treats individual walkers with a phase oscillator description and is inherently high dimensional with system dimensionality (N+2), where N is the number of walkers. In the present work we use a method proposed by Ott and Antonsen [Chaos 18, 037113 (2008)] to reduce the model of Eckhardt et al. to a low dimensional dynamical system, and we employ this reduced description to study the global dynamics of the bridge/pedestrian interaction. More generally, this treatment serves as an interesting example of the possibility of low dimensional macroscopic behavior in large systems of coupled oscillators.

Spatiotemporal perspective on the decay of turbulence in wall-bounded flows

By use of a reduced model focusing on the in-plane dependence of plane Couette flow, it is shown that the turbulent-->laminar relaxation process can be understood as a nucleation problem similar to that occurring at a thermodynamic first-order phase transition. The approach, apt to deal with the large extension of the system considered, challenges the current interpretation in terms of chaotic transients typical of temporal chaos. The study of the distribution of the sizes of laminar domains embedded in turbulent flow proves that an abrupt transition from sustained spatiotemporal chaos to laminar flow can take place at some given value of the Reynolds number Rlow, whether or not the local chaos lifetime, as envisioned within low-dimensional dynamical systems theory, diverges at finite R beyond Rlow.

Control of single neuron activity by global network dynamics in auditory cortex

Event Abstract Back to Event Control of single neuron activity by global network dynamics in auditory cortex The electrical activity of neurons reflects an interplay of sensory input and the brain's own internal dynamics. In sensory cortices, neuronal firing correlates well with characteristics of sensory stimuli; on the other hand, each cortical neuron receives the vast majority of its inputs from other cortical neurons, suggesting that spiking is largely controlled by the activity of the local network. This viewpoint is supported by simultaneous recordings of multiple cells, which show synchronized patterns of spontaneous activity beyond those imposed by the structure of the sensory stimulus. In prior work, we found that the global dynamics of population activity in rat auditory cortex is well approximated by a low-dimensional self-exciting system model, and that the variations in population dynamics with cortical state can be captured by changes in the parameters of this model. Here we investigate the relationship of electrophysiologically identified single neurons to global population dynamics in this context. In the model, population activity is characterized by a pair of 'mean field' variables, v and w, that reflect the average population firing rate and integrated recent past activity, respectively. To predict the spike times of individual neurons from population activity, we use an approach analogous to the computation of hippocampal "place fields", by predicting the mean firing probability of each neuron as a function of position in the v-w plane, which we term the 'activity field.' We find that activity fields for different neurons are quite localized in distinct regions of the v-w plane. We conclude that, while the mean activity of a column is well approximated by a simple low-dimensional dynamical system, individual neurons show widely different relationships to this dynamics. For any neuron, activity fields computed from spontaneous and sensory-evoked activity were similar, further suggesting that the structure of sensory responses reflects the same network dynamics that shape cortical spontaneous activity. This work is supported by NIH Grants MH073245 and R01DC009947. CC was also supported by a Courant Instructorship. Conference: Computational and systems neuroscience 2009, Salt Lake City, UT, United States, 26 Feb - 3 Mar, 2009. Presentation Type: Poster Presentation Topic: Poster Presentations Citation: (2009). Control of single neuron activity by global network dynamics in auditory cortex. Front. Syst. Neurosci. Conference Abstract: Computational and systems neuroscience 2009. doi: 10.3389/conf.neuro.06.2009.03.152 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 02 Feb 2009; Published Online: 02 Feb 2009. Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Google Google Scholar PubMed Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.