Phase Space Projections Research Articles

Dimension reduction, exact recovery, and error estimates for sparse reconstruction in phase space

An important theme in modern inverse problems is the reconstruction of time-dependent data from only finitely many measurements. To obtain satisfactory reconstruction results in this setting it is essential to strongly exploit temporal consistency between the different measurement times. The strongest consistency can be achieved by reconstructing data directly in phase space, the space of positions and velocities. However, this space is usually too high-dimensional for feasible computations. We introduce a novel dimension reduction technique, based on projections of phase space onto lower-dimensional subspaces, which provably circumvents this curse of dimensionality: Indeed, in the exemplary framework of superresolution we prove that known exact reconstruction results stay true after dimension reduction, and we additionally prove new error estimates of reconstructions from noisy data in optimal transport metrics which are of the same quality as one would obtain in the non-dimension-reduced case.

Revisiting “Non-Thermal” Batch Microwave Oven Inactivation of Microorganisms

Over the last few decades there has been active discussion concerning the mechanisms involved in “non-thermal” microwave-assisted inactivation of microorganisms. This work presents a novel non-invasive acoustic measurement of a domestic microwave oven cavity-magnetron operating at fo = 2.45 ± 0.05 GHz (λo ~ 12.2 cm) that is modulated in the time-domain (0 to 2 minutes). The measurements reveal the cavity-magnetron cathode filament cold-start warm-up period and the pulse width modulation periods (time-on time-off and base-time period, where time-on minus base-time = duty cycle). The waveform information is used to reconstruct historical microwave “non-thermal” homogeneous microorganism inactivation experiments: where tap-water is used to mimic the microorganism suspension; and ice, crushed ice, and ice slurry mixture are used as the cooling media. The experiments are described using text, diagrams, and photographs. Four key experimental parameters are indentified that influence the suspension time-dependent temperature profile. First, where the selected process time > the time-base, the cavity-magnetron continuous wave rated power should be used for each second of microwave illumination. Second, external crushed ice and ice slurry baths induce different cooling profiles due to difference in their heat absorption rates. In addition external baths may shield the suspension resulting in a retarding of the time-dependent heating profile. Third, internal cooling systems dictate that the suspension is directly exposed to microwave illumination due to the absence of surrounding ice volume. Fourth, four separated water dummy-loads isolate and control thermal heat transfer (conduction) to and from the suspension, thereby diverting a portion of the microwave power away from the suspension. Energy phase-space projections were used to compare the “non-thermal” energy densities of 0.03 to 0.1 kJ·m-1 at 800 W with reported thermal microwave-assisted microorganism inactivation energy densities of 0.5 to 5 kJ·m-1 at 1050 ± 50 W. Estimations of the “non-thermal” microwave-assisted root mean square of the electric field strength are found to be in the range of 22 to 41.2 V·m-1 for 800 W.

Local dimensionality and inverse persistence analysis of atmospheric turbulence in the stable boundary layer.

The dynamics across different scales in the stable atmospheric boundary layer has been investigated by means of two metrics, based on instantaneous fractal dimensions and grounded in dynamical systems theory. The wind velocity fluctuations obtained from data collected during the Cooperative Atmosphere-Surface Exchange Study-1999 experiment were analyzed to provide a local (in terms of scales) and an instantaneous (in terms of time) description of the fractal properties and predictability of the system. By analyzing the phase-space projections of the continuous turbulent, intermittent, and radiative regimes, a progressive transformation, characterized by the emergence of multiple low-dimensional clusters embedded in a high-dimensional shell and a two-lobe mirror symmetrical structure of the inverse persistence, have been found. The phase space becomes increasingly complex and anisotropic as the turbulent fluctuations become uncorrelated. The phase space is characterized by a three-dimensional structure for the continuous turbulent samples in a range of scales compatible with the inertial subrange, where the phase-space-filling turbulent fluctuations dominate the dynamics, and is low dimensional in the other regimes. Moreover, lower-dimensional structures present a stronger persistence than the higher-dimensional structures. Eventually, all samples recover a three-dimensional structure and higher persistence level at large scales, far from the inertial subrange. The two metrics obtained in the analysis can be considered as proxies for the decorrelation time and the local anisotropy in the turbulent flow.

Predicting beam transmission using 2-dimensional phase space projections of hadron accelerators

We present a method to compress the 2D transverse phase space projections from a hadron accelerator and use that information to predict the beam transmission. This method assumes that obtaining at least three projections of the 4D transverse phase space is possible and that an accurate simulation model is available for the beamline. Using a simulated model, we show that—a computer can train a convolutional autoencoder to reduce phase-space information which can later be used to predict the beam transmission. Finally, we argue that although using projections from a realistic nonlinear distribution produces less accurate results, the method still generalizes well.

On the chaotic nature of random telegraph noise in unipolar RRAM memristor devices

Random telegraph noise (RTN) owns its very name to its assumed stochastic nature. In this paper, we follow up previous works that questioned this stochastic nature, and we investigate this assumption using experimentally measured noise coming from properly biased Ni/HfO2 unipolar Resistive RAM memristor nanodevices. We have used established, well–known tools from nonlinear theory to examine the current–noise temporal series. Evaluation results show that this series appears to exhibit not a stochastic, but a deterministic chaotic behavior, also demostrating interesting fractal characteristics in 2D and 3D phase space projections. The presented results clearly advocate for a strong component of complex (chaotic) fluctuation of deterministic origin, instead of a typical (fully stochastic) RTN. This result could pave the path for an enhanced understanding of the mechanisms behind RTN emergence, as well as improve its noise models.

Determining the global manifold structure of a continuous-time heterodimensional cycle

<p style='text-indent:20px;'>A heterodimensional cycle consists of two saddle periodic orbits with unstable manifolds of different dimensions and a pair of connecting orbits between them. Recent theoretical work on chaotic dynamics beyond the uniformly hyperbolic setting has shown that heterodimensional cycles may occur robustly in diffeomorphisms of dimension at least three. We consider the first explicit example of a heterodimensional cycle in the continuous-time setting, which has been identified by Zhang, Krauskopf and Kirk [<i>Discr. Contin. Dynam. Syst. A</i> <b>32</b>(8) 2825-2851 (2012)] in a four-dimensional vector-field model of intracellular calcium dynamics.</p><p style='text-indent:20px;'>We show here how a boundary-value problem set-up can be employed to determine the organization of the dynamics in a neighborhood in phase space of this heterodimensional cycle, which consists of a single connecting orbit of codimension one and an entire cylinder of structurally stable connecting orbits between two saddle periodic orbits. More specifically, we compute the relevant stable and unstable manifolds, which we visualize in different projections of phase space and as intersection sets with a suitable three-dimensional Poincaré section. In this way, we show that, locally near the intersection set of the heterodimensional cycle, the manifolds interact as described by the theory for three-dimensional diffeomorphisms. On the other hand, their global structure is more intricate, which is due to the fact that it is not possible to find a Poincaré section that is transverse to the flow everywhere. More generally, our results show that advanced numerical continuation techniques enable one to investigate how abstract concepts â€" such as that of a heterodimensional cycle of a diffeomorphism â€" arise and manifest themselves in explicit continuous-time systems from applications.</p>

Could MT2 be a singularity variable?

The algebraic singularity method is a framework for analyzing collider events with missing energy. It provides a way to draw out a set of singularity variables that can catch singular features originating from the projection of full phase space onto the observable phase space of measured particle momenta. It is a promising approach applicable to various physics processes with missing energy but still requires more studies for use in practice. Meanwhile, in the double-sided decay topology with an invisible particle on each side, the MT2 variable has been known to be a useful collider observable for measuring particle masses from missing energy events or setting signal regions of collider searches. We investigate the relation between the two different types of kinematic variables in double-sided decay topology. We find that the singularity variables contain the MT2 variable in many cases, although the former is not a strict superset of the latter.

Chaotic dynamics of size‐dependent curvilinear Euler–Bernoulli beam resonators (MEMS) in a stationary thermal field

AbstractIn this work the chaotic dynamics of flexible curvilinear Euler–Bernoulli micro‐beams embedded into a stationary temperature field is investigated. The temperature field is modelled based on a the Duhamel–Neumann theory and is free from the restrictions on the temperature field distribution along beam thickness. The von Kármán geometric strain–stress relations are employed. The governing nonlinear PDEs are yielded by the Hamilton principle with an account of the modified couple stress theory. The finite dimension problem is truncated to a finite system of nonlinear ODEs using the finite difference method (FDM) and then the Cauchy problem is solved with a help of the Runge–Kutta method. Action of the 2D thermal field is defined by solution to the heat transfer PDE which is also solved by FDM of the second order of accuracy. The so‐called charts of vibration regimes (amplitude‐frequency planes) are constructed. In particular, novel features of nonlinear (chaotic) dynamics versus the change of the magnitude of the size (length) dependent parameter are reported.The carried out numerical analysis is supported by the monitoring of frequency power spectra based on the fast Fourier transform (FFT), phase space projections, Poincaré maps and LLEs (largest Lyapunov exponents). We have also analyzed system chaotic dynamics of the classical and size‐dependent beam models versus series of values of the two control parameters, i.e. beam curvature and its size‐dependent length. Moreover, we have detected and illustrated the novel scenarios of transition form regular to chaotic vibrations of the studied beams, governed by non‐linear PDEs.

Banach-Valued Multilinear Singular Integrals with Modulation Invariance

Abstract We prove that the class of trilinear multiplier forms with singularity over a one-dimensional subspace, including the bilinear Hilbert transform, admits bounded $L^p$-extension to triples of intermediate $\operatorname{UMD}$ spaces. No other assumption, for instance of Rademacher maximal function type, is made on the triple of $\operatorname{UMD}$ spaces. Among the novelties in our analysis is an extension of the phase-space projection technique to the $\textrm{UMD}$-valued setting. This is then employed to obtain appropriate single-tree estimates by appealing to the $\textrm{UMD}$-valued bound for bilinear Calderón–Zygmund operators recently obtained by the same authors.

Phase Space Projection of Dynamical Systems

AbstractDynamical systems are commonly used to describe the state of time‐dependent systems. In many engineering and control problems, the state space is high‐dimensional making it difficult to analyze and visualize the behavior of the system for varying input conditions. We present a novel dimensionality reduction technique that is tailored to high‐dimensional dynamical systems. In contrast to standard general purpose dimensionality reduction algorithms, we use energy minimization to preserve properties of the flow in the high‐dimensional space. Once the projection operator is optimized, further high‐dimensional trajectories are projected easily. Our 3D projection maintains a number of useful flow properties, such as critical points and flow maps, and is optimized to match geometric characteristics of the high‐dimensional input, as well as optional user constraints. We apply our method to trajectories traced in the phase spaces of second‐order dynamical systems, including finite‐sized objects in fluids, the circular restricted three‐body problem and a damped double pendulum. We compare the projections with standard visualization techniques, such as PCA, t‐SNE and UMAP, and visualize the dynamical systems with multiple coordinated views interactively, featuring a spatial embedding, projection to subspaces, our dimensionality reduction and a seed point exploration tool.

Dynamics of laminar and transitional flows over slip surfaces: effects on the laminar–turbulent separatrix

The effect of slip surfaces on the laminar-turbulent separatrix of plane Poiseuille flow is studied by direct numerical simulation. Turbulence lifetimes, the likelihood that turbulence is sustained, is investigated for transitional flows with various slip lengths. Slip surfaces decrease the likelihood of sustained turbulence compared to the no-slip case, and likelihood is further decreased as slip length is increased. A deterministic analysis of the effects of slip surfaces on transition to turbulence is performed using nonlinear traveling wave solutions to the Navier-Stokes equations, also known as exact coherent solutions. Two solution families, dubbed P3 and P4, are used since their lower-branch solutions are embedded on the boundary of the basin of attraction of laminar and turbulent flows (Park & Graham 2015). Additionally, they exhibit distinct flow structures -- the P3 and P4 are denoted as core mode and critical layer mode, respectively. Distinct effects of slip surfaces on the solutions are observed by the skin friction evolution, linear growth rate, and phase-space projection of transitional trajectories. The slip surface modifies transition dynamics little for the core mode, but considerably for the critical layer mode. Most importantly, the slip surface promotes different transition dynamics -- early and bypass-like transition for the core mode and delayed and H-/K-type-like transition for the critical layer mode. Based on spatiotemporal and quadrant analyses, it is found that slip surfaces promote the prevalence of strong wall-toward motions (sweep-like events) near vortex cores close to the channel centre, inducing an early transition, while sustained ejection events are present in the region of the ${\Lambda}$-shaped vortex cores close to the critical layer, resulting in a delayed transition.

Phase-space dynamics of opposition control in wall-bounded turbulent flows

We investigate the nonlinear phase-space dynamics of plane Couette flow and plane Poiseuille flow under the action of opposition control at low Reynolds numbers in domains close to the minimal unit. In Couette flow, the effect of the control is analysed by focussing on a pair of non-trivial equilibrium solutions. It is found that the control only slightly modifies the statistics, turbulent skin friction and phase-space projection of the lower-branch equilibrium solution, which, in this case, is in fact identical to the edge state. On the other hand, the upper-branch equilibrium solution and mean turbulent state are modified considerably when the control is applied. In phase space, they gradually approach the lower-branch equilibrium solution on increasing the control amplitude, and this results in an elevation of the critical Reynolds number at which the equilibrium solutions first occur via a saddle-node bifurcation. It is also found that the upper-branch equilibrium solution is stabilised by the control. In Poiseuille flow, we study an unstable periodic orbit on the edge state and find that it, too, is modified very little by opposition control. We again observe that the turbulent state gradually approaches the edge state in phase space as the control amplitude is increased. In both flows, we find that the control significantly reduces the fluctuating strength of the turbulent state in phase space. However, the reduced distance between the turbulent trajectory and the edge state yields a significant reduction in turbulence lifetimes for both Couette and Poiseuille flow. This demonstrates that opposition control greatly increases the probability of the trajectory escaping from the turbulent state, which takes the form of a chaotic saddle.

PRIMARY PROCESSING OF TELEMETRIC INFORMATION USING DYNAMIC MODELS OF PARAMETER CHANGE AND PARTIAL NONLINEAR FILTRATION

Introduction: At the stage of preliminary processing of telemetric information about the technical state of on-board spacecraft equipment, there is a need to detect changes in telemetry parameters which are important from the viewpoint of detecting and fixing emergencies. The solution of this problem is complicated by nonlinear character of change in the majority of on-board equipment functioning parameters and by various perturbations which cause changes in the analyzed telemetric parameters. Purpose: Developing a method of telemetry data processing in order to increase the reliability of heterogeneous telemetry information from spacecrafts under real perturbation and nonlinear changes in their parameters, based on filtering the time series of telemetric parameter values. Results: A method of telemetry information processing is proposed, based on dynamic models of change in telemetric parameters of on-board spacecraft equipment in phase space projections and partial nonlinear filtering of the analyzed time series, using stochastic approximation of state vector distribution probability density function. We discuss the advantages of primary processing of telemetric parameters of spacecraft airborne equipment in multidimensional phase space, provided by the possibility of selecting the most significant phase portrait projections from the viewpoint of filtering results. The developed algorithm of non-linear partial filtration allows you to adapt to the perturbation parameters. The validity of this method has been demonstrated for Gaussian and non-Gaussian perturbations at conditions under which the stability of linear filtration methods is violated. Practical relevance: The proposed approach is unified for processing of dissimilar telemetry parameters, providing an increase in estimation reliability of both functional and signal telemetric parameters of spacecrafts under real operating conditions.

Focused conformational sampling in proteins.

A detailed understanding of the conformational dynamics of biological molecules is difficult to obtain by experimental techniques due to resolution limitations in both time and space. Computer simulations avoid these in theory but are often too short to sample rare events reliably. Here we show that the progress index-guided sampling (PIGS) protocol can be used to enhance the sampling of rare events in selected parts of biomolecules without perturbing the remainder of the system. The method is very easy to use as it only requires as essential input a set of several features representing the parts of interest sufficiently. In this feature space, new states are discovered by spontaneous fluctuations alone and in unsupervised fashion. Because there are no energetic biases acting on phase space variables or projections thereof, the trajectories PIGS generates can be analyzed directly in the framework of transition networks. We demonstrate the possibility and usefulness of such focused explorations of biomolecules with two loops that are part of the binding sites of bromodomains, a family of epigenetic "reader" modules. This real-life application uncovers states that are structurally and kinetically far away from the initial crystallographic structures and are also metastable. Representative conformations are intended to be used in future high-throughput virtual screening campaigns.

Reduction of Classical Measurement Noise via Quantum-Dense Metrology.

Quantum-dense metrology constitutes a special case of quantum metrology in which two orthogonal phase space projections of a signal are simultaneously sensed beyond the shot-noise limit. Previously, it was shown that the additional sensing channel that is provided by quantum-dense metrology contains information that can be used to identify and to discard corrupted segments from the measurement data. Here, we propose and demonstrate a new method in which this information is used for improving the sensitivity without discarding any measurement segments. Our measurement reached sub-shot-noise performance, although initially strong classical noise polluted the data. The new method has high potential for improving the noise spectral density of gravitational-wave detectors at signal frequencies of high astrophysical relevance.

Event structure investigation of AuAu interactions from HIJING model using fractal dimensions

This work presents an implementation of fractal geometry methods in the study of event structure for AuAu interactions at collision energies √sNN = 9.2, 62 and 200 GeV for different interaction dynamics. The events are generated by using the HIJING model. It is shown, that the fractal dimension of events in phase space projections rapidity-transverse momentum (y - pt) and azimuthal angle-transverse momentum (φ - pt) are sensitive to the interaction dynamics.

Carrier scenarios in optically injected quantum-dot semiconductor lasers

A model for optical injection locking in quantum dot (QD) lasers is stated where the electron and hole dynamics are treated separately and their scenarios in the ground state (GS), excited state (ES) in the QDs and in the wetting layer (WL) are examined. A decline in the GS electron occupation probability, and then, a left returned spiral behavior is shown in their phase space projections at high injection ratio. The GS occupation probabilities for electrons and holes are reduced drastically with increasing injection ratio. The injection map is also plotted.

A phase space denoising method for chaotic maps

The spectra of chaotic maps are much wider than those of chaotic flows, and their overlapped regions with Gaussian white noise are much larger, thus the denoising method for chaotic flows is unsuitable for chaotic maps. Within a semi-blind analysing framework, the parameter estimating problem for chaotic systems can be boiled down to a least square evaluating procedure. In this paper we start with estimating the evolution parameters of chaotic maps by using a least square fitting method. After that, phase space reconstruction and projection operation are employed to get noise suppression for the observed data. The simulation results indicate that the proposed algorithm surpasses the extended Kalman filter (EKF) and the unscented Kalman filter (UKF) in denoising, as well as maintaining the characteristic quantities of chaotic maps.

Extracting Fetal Electrocardiogram from Being Pregnancy Based on Nonlinear Projection

Fetal heart rate extraction from the abdominal ECG is of great importance due to the information that carries in assessing appropriately the fetus well‐being during pregnancy. In this paper, we describe a method to suppress the maternal signal and noise contamination to discover the fetal signal in a single‐lead fetal ECG recordings. We use a locally linear phase space projection technique which has been used for noise reduction in deterministically chaotic signals. Henceforth, this method is capable of extracting fetal signal even when noise and fetal component are of comparable amplitude. The result is much better if the noise is much smaller (P wave and T wave can be discovered).

A network model of a multi-item working memory task based on competitive reverberating neural activity

One of the greatest challenges for computational neuroscientists is to model the neural underpinnings of complex, ecologically relevant behavior. However, the construction of models for the neural dynamics which allow such behavior is very difficult, even when restricted to a single aspect of a task. Such a model needs to reproduce both the behavioral and the neuronal data, and should yet be simple enough to provide insight into the neural dynamics. Furthermore, to sufficiently constrain a model, electrophysiological recordings of awake, behaving animals are necessary, which are both challenging to obtain and to analyze. Recently, researchers have investigated simultaneous multi-unit activity in the anterior cingulate cortex in rats during a spatial working memory and decision making task [1]. The animals were moving in an eight arm radial maze, foraging for food which was located at the end of each arm. In order to obtain food as fast as possible, the rats had to keep a memory of the arms they visited before, so they could avoid entering them again. The study reported clearly dissociable patterns of population activity that were specific for the different phases of the task [1]. Further analysis of this data showed that in a low-dimensional phase space projection, the multi-unit activity described recurrent, orbit-like trajectories which also distinguish between the different arms that the rats previously entered. Here, we present a neural network model based on these observations which provides a possible neural basis for the multiple-item working memory needed in this task. The network consists of eight pools of neurons, which are connected in a circular fashion, such that activity continuously reverberates between states corresponding to ea ch of the eight arms. To encode whether a given arm was visited before or not without interrupting the flow of activity, each pool is subdivided into mutually inhibiting populations of neurons, which are equivalent in their projections from and to neighboring pools. One of these sub-populations is associated with a “new” arm, while the other one correspondingly encodes a previously visited, “old” arm. Finally, sensory stimulation is simulated by activating the “new” populations in the beginning of the trial, and the “old” population of a given arm at the time it is entered. In order to perform the necessary computation, each pool must exhibit and maintain the appropriate bias between the two subpopulations, i.e. either the new or the old population must be active while the other is silent. This bias is initially induced by the sensory stimulation. After the stimulus is gone, the memory is maintained by the reverberating activity itself: Activity decays with a time constant such that there is still some fraction of the peak firing rate left by the time the activity wave returns to that pool. Thus, the population which is more active at this time will be enhanced faster and stronger by the wave, and the bias is fortified because of the mutual inhibition. In terms of dynamics, the network gives rise to persisting working memory representations of the visited arms if the interplay between slowly decaying activity in each pool (which carries the selective bias), and reverberating activity forms a stable limit cycle in the phase space of firing rates. To confirm this quantitatively, we reduced the model further to a system of two pools and conducted a formal stability analysis on the resulting dynamics. In this way, we derived stability conditions on the parameters such that both selectivity between the new and old pools, and continuously reverberating activity is maintained. These conditions generalize to an arbitrary number of pools, and can thus be used to constrain an optimization process to fit the model to the trajectories in the neuronal data.