High-dimensional Phase Space Research Articles

Manifold Learning with Self-Organizing Mapping for Feature Extraction of Nonlinear Faults in Rotating Machinery

A new method for extracting the low-dimensional feature automatically with self-organization mapping manifold is proposed for the detection of rotating mechanical nonlinear faults (such as rubbing, pedestal looseness). Under the phase space reconstructed by single vibration signal, the self-organization mapping (SOM) with expectation maximization iteration algorithm is used to divide the local neighborhoods adaptively without manual intervention. After that, the local tangent space alignment algorithm is adopted to compress the high-dimensional phase space into low-dimensional feature space. The proposed method takes advantages of the manifold learning in low-dimensional feature extraction and adaptive neighborhood construction of SOM and can extract intrinsic fault features of interest in two dimensional projection space. To evaluate the performance of the proposed method, the Lorenz system was simulated and rotation machinery with nonlinear faults was obtained for test purposes. Compared with the holospectrum approaches, the results reveal that the proposed method is superior in identifying faults and effective for rotating machinery condition monitoring.

Approximating high-dimensional dynamics by barycentric coordinates with linear programming.

The increasing development of novel methods and techniques facilitates the measurement of high-dimensional time series but challenges our ability for accurate modeling and predictions. The use of a general mathematical model requires the inclusion of many parameters, which are difficult to be fitted for relatively short high-dimensional time series observed. Here, we propose a novel method to accurately model a high-dimensional time series. Our method extends the barycentric coordinates to high-dimensional phase space by employing linear programming, and allowing the approximation errors explicitly. The extension helps to produce free-running time-series predictions that preserve typical topological, dynamical, and/or geometric characteristics of the underlying attractors more accurately than the radial basis function model that is widely used. The method can be broadly applied, from helping to improve weather forecasting, to creating electronic instruments that sound more natural, and to comprehensively understanding complex biological data.

Fractional Delay Time Embedding of EEG Signals into High Dimensional Phase Space

We propose a method for reconstruction of the high-dimensional phase space of the electroencephalography (EEG) signal. The method is based on the selection of positive trajectories from the phase space, distance-adaptive sampling of the negative trajectories from the phase space, classification of trajectories in the phase space, and reconstruction of a fuzzy state of the signal for classification of EEG signals. DOI: http://dx.doi.org/10.5755/j01.eee.20.8.8441

Duality between the trigonometricBCnSutherland system and a completed rational Ruijsenaars–Schneider–van Diejen system

We present a new case of duality between integrable many-body systems, where two systems live on the action-angle phase spaces of each other in such a way that the action variables of each system serve as the particle positions of the other one. Our investigation utilizes an idea that was exploited previously to provide group-theoretic interpretation for several dualities discovered originally by Ruijsenaars. In the group-theoretic framework, one applies Hamiltonian reduction to two Abelian Poisson algebras of invariants on a higher dimensional phase space and identifies their reductions as action and position variables of two integrable systems living on two different models of the single reduced phase space. Taking the cotangent bundle of U(2n) as the upstairs space, we demonstrate how this mechanism leads to a new dual pair involving the BCn trigonometric Sutherland system. Thereby, we generalize earlier results pertaining to the An trigonometric Sutherland system as well as a recent work by Pusztai on the hyperbolic BCn Sutherland system.

Visualization of protein folding funnels in lattice models.

Protein folding occurs in a very high dimensional phase space with an exponentially large number of states, and according to the energy landscape theory it exhibits a topology resembling a funnel. In this statistical approach, the folding mechanism is unveiled by describing the local minima in an effective one-dimensional representation. Other approaches based on potential energy landscapes address the hierarchical structure of local energy minima through disconnectivity graphs. In this paper, we introduce a metric to describe the distance between any two conformations, which also allows us to go beyond the one-dimensional representation and visualize the folding funnel in 2D and 3D. In this way it is possible to assess the folding process in detail, e.g., by identifying the connectivity between conformations and establishing the paths to reach the native state, in addition to regions where trapping may occur. Unlike the disconnectivity maps method, which is based on the kinetic connections between states, our methodology is based on structural similarities inferred from the new metric. The method was developed in a 27-mer protein lattice model, folded into a 3×3×3 cube. Five sequences were studied and distinct funnels were generated in an analysis restricted to conformations from the transition-state to the native configuration. Consistent with the expected results from the energy landscape theory, folding routes can be visualized to probe different regions of the phase space, as well as determine the difficulty in folding of the distinct sequences. Changes in the landscape due to mutations were visualized, with the comparison between wild and mutated local minima in a single map, which serves to identify different trapping regions. The extension of this approach to more realistic models and its use in combination with other approaches are discussed.

Multiple states in highly turbulent Taylor-Couette flow.

The ubiquity of turbulent flows in nature and technology makes it of utmost importance to fundamentally understand turbulence. Kolmogorov's 1941 paradigm suggests that for strongly turbulent flows with many degrees of freedom and large fluctuations, there would only be one turbulent state as the large fluctuations would explore the entire higher dimensional phase space. Here we report the first conclusive evidence of multiple turbulent states for large Reynolds number, Re = O(10(6)) (Taylor number Ta = O(10(12))) Taylor-Couette flow in the regime of ultimate turbulence, by probing the phase space spanned by the rotation rates of the inner and outer cylinder. The manifestation of multiple turbulent states is exemplified by providing combined global torque- and local-velocity measurements. This result verifies the notion that bifurcations can occur in high-dimensional flows (that is, very large Re) and questions Kolmogorov's paradigm.

Compressible generalized hybrid Monte Carlo

One of the most demanding calculations is to generate random samples from a specified probability distribution (usually with an unknown normalizing prefactor) in a high-dimensional configuration space. One often has to resort to using a Markov chain Monte Carlo method, which converges only in the limit to the prescribed distribution. Such methods typically inch through configuration space step by step, with acceptance of a step based on a Metropolis(-Hastings) criterion. An acceptance rate of 100% is possible in principle by embedding configuration space in a higher dimensional phase space and using ordinary differential equations. In practice, numerical integrators must be used, lowering the acceptance rate. This is the essence of hybrid Monte Carlo methods. Presented is a general framework for constructing such methods under relaxed conditions: the only geometric property needed is (weakened) reversibility; volume preservation is not needed. The possibilities are illustrated by deriving a couple of explicit hybrid Monte Carlo methods, one based on barrier-lowering variable-metric dynamics and another based on isokinetic dynamics.

The dynamics of growing islets and transmission of schistosomiasis japonica in the Yangtze River.

We formulate and analyze a system of ordinary differential equations for the transmission of schistosomiasis japonica on the islets in the Yangtze River, China. The impact of growing islets on the spread of schistosomiasis is investigated by the bifurcation analysis. Using the projection technique developed by Hassard, Kazarinoff and Wan, the normal form of the cusp bifurcation of codimension 2 is derived to overcome the technical difficulties in studying the existence, stability, and bifurcation of the multiple endemic equilibria in high-dimensional phase space. We show that the model can also undergo transcritical bifurcations, saddle-node bifurcations, a pitchfork bifurcation, and Hopf bifurcations. The bifurcation diagrams and epidemiological interpretations are given. We conclude that when the islet reaches a critical size, the transmission cycle of the schistosomiasis japonica between wild rats Rattus norvegicus and snails Oncomelania hupensis could be established, which serves as a possible source of schistosomiasis transmission along the Yangtze River.

Improved mass measurement using the boundary of many-body phase space

We show that mass measurements for new particles appearing in decay chains can be improved by determining the boundary of the available phase space in its full dimensionality rather than by using one-dimensional kinematic features for each stage of the cascade decay. This is demonstrated for the case of one particle decaying to three visible and one invisible particles in a two-stage cascade, but our methods also apply to a more general set of decay topologies. We show that not only mass differences but also the overall scale of masses can be determined with high precision without having to rely on cross section information. The improvement arises from the properties of the higher-dimensional phase space itself, independent of the matrix element for the decay, and it is not weakened by the presence of intermediate on-shell particles in the cascade. Our results are particularly significant for the case of low signal statistics, a distinct possibility for new physics searches in the near future.

Perpetual extraction of work from a nonequilibrium dynamical system under Markovian feedback control

By treating both control parameters and dynamical variables as probabilistic variables, we develop a succinct theory of perpetual extraction of work from a generic classical nonequilibrium system subject to a heat bath via repeated measurements under a Markovian feedback control. It is demonstrated that a problem for perpetual extraction of work in a nonequilibrium system is reduced to a problem of Markov chain in the higher-dimensional phase space. We derive a version of the detailed fluctuation theorem, which was originally derived for classical nonequilibrium systems by Horowitz and Vaikuntanathan [Phys. Rev. E 82, 061120 (2010)], in a form suitable for the analyses of perpetual extraction of work. Since our theory is formulated for generic dynamics of probability distribution function in phase space, its application to a physical system is straightforward. As simple applications of the theory, two exactly solvable models are analyzed. The one is a nonequilibrium two-state system and the other is a particle confined to a one-dimensional harmonic potential in thermal equilibrium. For the former example, it is demonstrated that the observer on the transitory steps to the stationary state can lose energy and that work larger than that achieved in the stationary state can be extracted. For the latter example, it is demonstrated that the optimal protocol for the extraction of work via repeated measurements can differ from that via a single measurement. The validity of our version of the detailed fluctuation theorem, which determines the upper bound of the expected work in the stationary state, is also confirmed for both examples. These observations provide useful insights into exploration for realistic modeling of a machine that extracts work from its environment.

Feature extraction of kernel regress reconstruction for fault diagnosis based on self-organizing manifold learning

The feature space extracted from vibration signals with various faults is often nonlinear and of high dimension. Currently, nonlinear dimensionality reduction methods are available for extracting low-dimensional embeddings, such as manifold learning. However, these methods are all based on manual intervention, which have some shortages in stability, and suppressing the disturbance noise. To extract features automatically, a manifold learning method with self-organization mapping is introduced for the first time. Under the non-uniform sample distribution reconstructed by the phase space, the expectation maximization(EM) iteration algorithm is used to divide the local neighborhoods adaptively without manual intervention. After that, the local tangent space alignment(LTSA) algorithm is adopted to compress the high-dimensional phase space into a more truthful low-dimensional representation. Finally, the signal is reconstructed by the kernel regression. Several typical states include the Lorenz system, engine fault with piston pin defect, and bearing fault with outer-race defect are analyzed. Compared with the LTSA and continuous wavelet transform, the results show that the background noise can be fully restrained and the entire periodic repetition of impact components is well separated and identified. A new way to automatically and precisely extract the impulsive components from mechanical signals is proposed.

Crisis, unstable dimension variability, and bifurcations in a system with high-dimensional phase space: Coupled sine circle maps

The phenomenon of crisis in systems evolving in high-dimensional phase space can show unexpected and interesting features. We study this phenomenon in the context of a system of coupled sine circle maps. We establish that the origins of this crisis lie in a tangent bifurcation in high dimensions, and identify the routes that lead to the crisis. Interestingly, multiple routes to crisis are seen depending on the initial conditions of the system, due to the high dimensionality of the space in which the system evolves. The statistical behavior seen in the phase diagram of the system is also seen to change due to the dynamical phenomenon of crisis, which leads to transitions from nonspreading to spreading behavior across an infection line in the phase diagram. Unstable dimension variability is seen in the neighborhood of the infection line. We characterize this crisis and unstable dimension variability using dynamical characterizers, such as finite-time Lyapunov exponents and their distributions. The phase diagram also contains regimes of spatiotemporal intermittency and spatial intermittency, where the statistical quantities scale as power laws. We discuss the signatures of these regimes in the dynamic characterizers, and correlate them with the statistical characterizers and bifurcation behavior. We find that it is necessary to look at both types of correlators together to build up an accurate picture of the behavior of the system.

Bianchi IX model: Reducing phase space

The mathematical structure of higher-dimensional physical phase spaces of the nondiagonal Bianchi IX model is analyzed in the neighborhood of the cosmological singularity by using dynamical system methods. Critical points of the Hamiltonian equations appear at infinities and are of a nonhyperbolic type, which is a generic feature of the considered singular dynamics. The reduction of the kinematical symplectic 2-form to the constraint surface enables the determination of the physical Hamiltonian. This procedure lowers the dimensionality of the dynamics arena. The presented analysis of the phase space is based on canonical transformations. We test our method for the specific subspace of the physical phase space. The obtained results encourage further examination of the dynamics within our approach.

From strong chaos via weak chaos to regular behaviour: Optimal interplay between chaos and order

We investigate the interplay between chaotic and integrable Hamiltonian systems. In detail, a fully connected four-site lattice system associated with the discrete nonlinear Schrödinger equation is studied. On an embedded two-site segment (dimer) of the four-site system (tetramer) the coupling element between its two sites is time-periodically modified by an external driving term rendering the dimer dynamics chaotic, along with delocalisation of initially single-site excitations. Starting from an isolated dimer system the strength of the coupling to the remaining two sites of the tetramer is treated as a control parameter. It is striking that when the dimer interacts globally with the remaining two sites, thus constituting a fully connected tetramer, a non-trivial dependence of the degree of localisation on the strength of the coupling is found. There even exist ranges of optimal coupling strengths for which the driven tetramer dynamics becomes not only regular but also restores complete single-site localisation. We relate the re-establishment of complete localisation with transitions from permanent chaos via regular transients to permanent stable motion on a torus in the higher-dimensional phase space. In conclusion, increasing the dimension of a system can have profound effects on the character of the dynamics in higher-dimensional mixed phase spaces such that even full stabilisation of motion can be accomplished.

A statistically accurate modified quasilinear Gaussian closure for uncertainty quantification in turbulent dynamical systems

We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation.

Simulating rare events using a weighted ensemble-based string method.

We introduce an extension to the weighted ensemble (WE) path sampling method to restrict sampling to a one-dimensional path through a high dimensional phase space. Our method, which is based on the finite-temperature string method, permits efficient sampling of both equilibrium and non-equilibrium systems. Sampling obtained from the WE method guides the adaptive refinement of a Voronoi tessellation of order parameter space, whose generating points, upon convergence, coincide with the principle reaction pathway. We demonstrate the application of this method to several simple, two-dimensional models of driven Brownian motion and to the conformational change of the nitrogen regulatory protein C receiver domain using an elastic network model. The simplicity of the two-dimensional models allows us to directly compare the efficiency of the WE method to conventional brute force simulations and other path sampling algorithms, while the example of protein conformational change demonstrates how the method can be used to efficiently study transitions in the space of many collective variables.

Gating of maxi channels observed from pseudo-phase portraits

Phase space has been used to visualize and analyze the dynamic behavior of stochastic and chaotic systems. We applied this concept to maxi channels recorded from excised inside-out patches of in situ interstitial cells of Cajal. Pseudo-phase portraits of channel current were fairly homogeneous from patch to patch. They showed three main peaks, α, β, and γ, in increasing conductance. These represented single or near aggregated states. The α-peak was the closed state. The β-peak was small, consisting of a single conductance state, or in some cases two (a doublet). The β-peak state(s) had a long lifetime and displayed a characteristic behavior of frequent short transitions to γ but not to α. It was always preceded by a short series of α/γ-transitions. The γ-peak was the largest and consisted of a large number of conductance states with fast state transitions, sometimes to the extent of causing a diffusive-type behavior. Phase portraits allowed us to construct a provisional gating scheme for the maxi channel and suggest that further analysis of recordings in higher dimensional phase space and with related techniques may be promising.

Photonic Nonlinear Transient Computing with Multiple-Delay Wavelength Dynamics

We report on the experimental demonstration of a hybrid optoelectronic neuromorphic computer based on a complex nonlinear wavelength dynamics including multiple delayed feedbacks with randomly defined weights. This neuromorphic approach is based on a new paradigm of a brain-inspired computational unit, intrinsically differing from Turing machines. This recent paradigm consists in expanding the input information to be processed into a higher dimensional phase space, through the nonlinear transient response of a complex dynamics excited by the input information. The computed output is then extracted via a linear separation of the transient trajectory in the complex phase space. The hyperplane separation is derived from a learning phase consisting of the resolution of a regression problem. The processing capability originates from the nonlinear transient, resulting in nonlinear transient computing. The computational performance is successfully evaluated on a standard benchmark test, namely, a spoken digit recognition task.

Driven coupled Morse oscillators: visualizing the phase space and characterizing the transport

Recent experimental and theoretical studies indicate that intramolecular energy redistribution (IVR) is nonstatistical on intermediate timescales even in fairly large molecules. Therefore, it is interesting to revisit the old topic of IVR versus quantum control and one expects that a classical-quantum perspective is appropriate to gain valuable insights into the issue. However, understanding classical phase space transport in driven systems is a prerequisite for such a correspondence based approach and is a challenging task for systems with more than two degrees of freedom. In this work we undertake a detailed study of the classical dynamics of a minimal model system – two kinetically coupled Morse oscillators in the presence of a monochromatic laser field. Using the technique of wavelet transforms a representation of the high dimensional phase space, the resonance network or Arnold web, is constructed and analysed. The key structures in phase space which regulate the dissociation dynamics are identified. Furthermore, we show that the web is non-uniform with the classical dynamics exhibiting extensive stickiness, resulting in anomalous transport. Our work also shows that pairwise irrational barriers might be crucial even in higher dimensional systems.

On higher-dimensional contrast structure of singularly perturbed Dirichlet problem

In this paper, we address the existence and asymptotic analysis of higher-dimensional contrast structure of singularly perturbed Dirichlet problem. Based on the existence, an asymptotical analysis of a steplike contrast structure (i.e., an internal transition layer solution) is studied by the boundary function method via a proposed smooth connection. In the framework of this paper, we propose a first integral condition, under which the existence of a heteroclinic orbit connecting two equilibrium points is ensured in a higher-dimensional fast phase space. Then, the step-like contrast structure is constructed, and the internal transition time is determined. Meanwhile, the uniformly valid asymptotical expansion of such an available step-like contrast structure is obtained. Finally, an example is presented to illustrate the result.