High-dimensional Phase Space Research Articles

Stable chaos and delayed onset of statisticality in unimolecular dissociation reactions

Statistical models provide a powerful and useful class of approximations for calculating reaction rates by bypassing the need for detailed, and often difficult, dynamical considerations. Such approaches invariably invoke specific assumptions about the extent of intramolecular vibrational energy flow in the system. However, the nature of the transition to the statistical regime as a function of the molecular parameters is far from being completely understood. Here, we use tools from nonlinear dynamics to study the transition to statisticality in a model unimolecular reaction by explicitly visualizing the high dimensional classical phase space. We identify generic features in the phase space involving the intersection of two or more independent anharmonic resonances and show that the presence of correlated, but chaotic, intramolecular dynamics near such junctions leads to nonstatisticality. Interestingly, akin to the stability of asteroids in the Solar System, molecules can stay protected from dissociation at the junctions for several picoseconds due to the phenomenon of stable chaos.

Synchronization in coupled oscillators with multiplex interactions

The study of synchronizations in coupled oscillators is very important for understanding the occurrence of self-organized behaviors in complex systems. In the traditional Kuramoto model that has been extensively applied to the study of synchronous dynamics of coupled oscillators, the interaction function among oscillators is pairwise. The multiplex interaction mechanism that describes triple or multiple coupling functions has been a research focus in recent years. When the multiplex coupling dominates the interactions among oscillators, the phase oscillator systems can exhibit the typical abrupt desynchronization transitions. In this paper, we extensively investigate the synchronous dynamics of the Kuramoto model with mean-field triple couplings. We find that the abrupt desynchronization transition is irreversible, i.e. the system may experience a discontinuous transition from coherent state to incoherent state as the coupling strength deceases adiabatically, while the reversed transition cannot occur by adiabatically increasing the coupling. Moreover, the coherent state strongly depends on initial conditions. The dynamical mechanism of this irreversibility is theoretically studied by using the self-consistency approach. The neutral stability of ordered state is also explained through analyzing the linear-stability of the incoherent state. Further studies indicate that the system may experience a cascade of desynchronized standing-wave transitions when the width of the distribution function of natural frequencies of oscillators is changed. At the critical coupling, the motion of coupled oscillators in high-dimensional phase space becomes unstable through the saddle-node bifurcation and collapses into a stable low-dimensional invariant torus, which corresponds to the standing-wave state. The above conclusions and analyses are further extended to the case of multi-peak natural-frequency distributions. The results in this work reveal various collective synchronous states and the mechanism of the transitions among these macroscopic states brought by multiplex coupling. This also conduces to the in-depth understanding of transitions among collective states in other complex systems.

A Highly Parallel Multilevel Newton--Krylov--Schwarz Method with Subspace-Based Coarsening and Partition-Based Balancing for the Multigroup Neutron Transport Equation on Three-Dimensional Unstructured Meshes

The multigroup neutron transport equation is crucial for studying the motion of neutrons and their interaction with materials. Numerical simulation of the multigroup neutron transport equation is computationally challenging because the equation is defined on a high-dimensional phase space, the computational spatial domain is complex, and the materials are heterogeneous. A scalable parallel solver is required to address such a challenge. In this paper, we study a highly parallel Newton--Krylov--Schwarz (NKS) method consisting of a Newton-based eigenvalue solver, a Krylov subspace method, and a novel multilevel Schwarz preconditioner. The multilevel method is one of the most popular preconditioners for accelerating neutron transport calculations, but the construction of coarse spaces can be expensive and often unscalable when a large number of processors is used. We propose a novel matrix coarsening algorithm in which a multilevel hierarchy is constructed using a single-component matrix instead of the full matrix of the neutron transport equation. This new coarsening algorithm is referred to as “subspace-based coarsening.” Above 8,000 processors, we show a 13x enhancement in multilevel preconditioner setup time when using the subspace-based coarsening method. A partition-based balancing strategy is studied to enhance the parallel efficiency of the NKS algorithm by equalizing the work for each processor. A hierarchical mesh partitioning algorithm is employed to generate a large number of submeshes while minimizing off-node communication. We demonstrate that the proposed algorithm is scalable with more than 10,000 processors for a realistic application on three-dimensional unstructured meshes with a few billion degrees of freedom. Neutron transport calculations using the improved NKS algorithm are twice as fast as those based on the unmodified NKS solver when over 8,000 processors are employed.

Application of Recurrence Quantification Analysis for Non-Linear Dynamical Systems

The Recurrence plots (RPs) have been introduced in several different scientific and medical disciplines. The main purpose of recurrence plot is used to of identify the higher dimensional phase space trajectories. RPs are purely graphically representation which have been designed for the detection of hidden dynamical patterns and non-linearity present in the data, the evaluation of error which is caused by observational noise can be done by Recurrence Quantification Analysis (RQA). RQA method is initially used to minimize the error present in the given signals. RQA method is a basically a technique for the analysis of nonlinear data to quantify the number and duration of a dynamical systems. The recurrence plot is used for time series domain for multidimensional signal also. Recurrence is the property of non-stationary and dynamical system to characteristics the time series analysis in phase space trajectories. Recurrence Quantification Analysis is used to derive from recurrence plots, which are based upon distances matrices of time series.

Application of Recurrence Quantification Analysis for Non-Linear Dynamical Systems

The Recurrence plots (RPs) have been introduced in several different scientific and medical disciplines. The main purpose of recurrence plot is used to of identify the higher dimensional phase space trajectories. RPs are purely graphically representation which have been designed for the detection of hidden dynamical patterns and non-linearity present in the data, the evaluation of error which is caused by observational noise can be done by Recurrence Quantification Analysis (RQA). RQA method is initially used to minimize the error present in the given signals. RQA method is a basically a technique for the analysis of nonlinear data to quantify the number and duration of a dynamical systems. The recurrence plot is used for time series domain for multidimensional signal also. Recurrence is the property of non-stationary and dynamical system to characteristics the time series analysis in phase space trajectories. Recurrence Quantification Analysis is used to derive from recurrence plots, which are based upon distances matrices of time series.

Instantaneous control of interacting particle systems in the mean-field limit

Controlling large particle systems in collective dynamics by a few agents is a subject of high practical importance, e.g., in evacuation dynamics. In this paper we study an instantaneous control approach to steer an interacting particle system into a certain spatial region by repulsive forces from a few external agents, which might be interpreted as shepherd dogs leading sheep to their home. We introduce an appropriate mathematical model and the corresponding optimization problem. In particular, we are interested in the interaction of numerous particles, which can be approximated by a mean-field equation. Due to the high-dimensional phase space this will require a tailored optimization strategy. The arising control problems are solved using adjoint information to compute the descent directions. Numerical results on the microscopic and the macroscopic level indicate the convergence of optimal controls and optimal states in the mean-field limit, i.e., for an increasing number of particles.

Classification of chaotic time series with deep learning

We use standard deep neural networks to classify univariate time series generated by discrete and continuous dynamical systems based on their chaotic or non-chaotic behaviour. Our approach to circumvent the lack of precise models for some of the most challenging real-life applications is to train different neural networks on a data set from a dynamical system with a basic or low-dimensional phase space and then use these networks to classify univariate time series of a dynamical system with more intricate or high-dimensional phase space. We illustrate this generalisation approach using the logistic map, the sine-circle map, the Lorenz system, and the Kuramoto–Sivashinsky equation. We observe that a convolutional neural network without batch normalisation layers outperforms state-of-the-art neural networks for time series classification and is able to generalise and classify time series as chaotic or not with high accuracy.

Stability of exoplanetary systems retrieved from scalar time series

Abstract We propose a novel method applied to extrasolar planetary dynamics to describe the system stability. The observations in this field serve the measurements mainly of radial velocity, transit time, and/or celestial position. These scalar time series are used to build up the high-dimensional phase space trajectory representing the dynamical evolution of planetary motion. The framework of nonlinear time series analysis and Poincaré recurrences allows us to transform the obtained univariate signals into complex networks whose topology carries the dynamical properties of the underlying system. The network-based analysis is able to distinguish the regular and chaotic behaviour not only for synthetic inputs but also for noisy and irregularly sampled real world observations. The proposed scheme does not require neither n-body integration nor best fitting planetary model to perform the stability investigation, therefore, the computation time can be reduced drastically compared to those of the standard numerical methods.

Distinguishing Biomolecular Pathways and Metastable States.

Protein folding occurs in a high dimensional phase space, and the representation of the associated energy landscape is nontrivial. A widely applied approach to studying folding landscapes is to describe the dynamics along a small number of reaction coordinates. However, other strategies involve more elaborate analysis of the complex phase space. There have been many attempts to obtain a more detailed representation of all available conformations for a given system. In this work, we address this problem using a metric based on internal distances between amino acids to describe the differences between any two conformations. Using an effective projection method, we are able to go beyond the typical one-dimensional representation and provide intuitive two dimensional visualizations of the landscape. We refer to this method as the energy landscape visualization method (ELViM). We have applied this methodology using a Cα structure-based model to study the folding of two well-known proteins: SH3 domain and protein-A. Our visualization method yields a detailed description of the folding process, making possible the identification of transition state regions, and establishing the paths that lead to the native state. For SH3, we have analyzed structural differences in the distribution of folding routes. The competition between the native and mirror structures in protein A is also discussed. Finally, the method is applied to study conformational changes in the protein elongation factor thermally unstable. Distinct features of ELViM are that it does not require or assume a reaction coordinate, and it does not require analysis of kinetic aspects of the system.

Dynamical thermalization in time-dependent billiards.

Numerical experiments of the statistical evolution of an ensemble of noninteracting particles in a time-dependent billiard with inelastic collisions reveals the existence of three statistical regimes for the evolution of the speed ensemble, namely, diffusion plateau, normal growth/exponential decay, and stagnation. These regimes are linked numerically to the transition from Gauss-like to Boltzmann-like speed distributions. Furthermore, the different evolution regimes are obtained analytically through velocity-space diffusion analysis. From these calculations, the asymptotic root mean square of speed, initial plateau, and the growth/decay rates for an intermediate number of collisions are determined in terms of the system parameters. The analytical calculations match the numerical experiments and point to a dynamical mechanism for "thermalization," where inelastic collisions and a high-dimensional phase space lead to a bounded diffusion in the velocity space toward a stationary distribution function with a kind of "reservoir temperature" determined by the boundary oscillation amplitude and the restitution coefficient.

A novel weak fault signal detection approach for a rolling bearing using variational mode decomposition and phase space parallel factor analysis

Considering the difficulty of fault feature extraction from a rolling bearing under strong background noise, we present a novel approach based on variational mode decomposition (VMD) and phase space parallel factor analysis, for detecting the weak fault signal of a rolling bearing. In this scheme, the VMD method is first adopted to decompose the raw vibration signal into several intrinsic mode components. The selected intrinsic mode function component with maximal kurtosis is subsequently embedded into the high dimensional phase space by phase space reconstruction. Then, the independent components are estimated in the high dimensional phase space by parallel factor analysis. In addition, a new criterion combining kurtosis and feature energy factor (FEF) is proposed for the selection of the time delay and embedding dimension. Finally, the optimal independent component with the largest FEF is selected for envelope spectrum analysis, and the faint fault characteristic frequencies of the vibration signal can be extracted. The feasibility of the proposed scheme is demonstrated through simulation and experimental vibration signals. Results indicate that the proposed method has better capability in detecting a weak fault signal of a rolling bearing, compared with VMD-fast spectral kurtogram and phase space independent component analysis.

Finding normally hyperbolic invariant manifolds in two and three degrees of freedom with Hénon-Heiles-type potential.

We present a method based on a Lagrangian descriptor for revealing the high-dimensional phase space structures that are of interest in nonlinear Hamiltonian systems with index-1 saddle. These phase space structures include a normally hyperbolic invariant manifold and its stable and unstable manifolds, which act as codimension-1 barriers to phase space transport. In this article, finding the invariant manifolds in high-dimensional phase space will constitute identifying coordinates on these invariant manifolds. The method of Lagrangian descriptor is demonstrated by applying to classical two and three degrees of freedom Hamiltonian systems which have implications for myriad applications in chemistry, engineering, and physics.

Finding NHIM: Identifying high dimensional phase space structures in reaction dynamics using Lagrangian descriptors

Phase space structures such as dividing surfaces, normally hyperbolic invariant manifolds, and their stable and unstable manifolds have been an integral part of computing quantitative results such as the transition fraction, rate of stability basin erosion in multi-stable mechanical systems, and rate constants in chemical reactions. Thus, methods that can reveal the geometry of these invariant manifolds in high dimensional phase space (4 or more dimensions) need to be benchmarked by comparing with known results. In this study we assess the capability of one such method called Lagrangian descriptor for revealing the types of high dimensional phase space structures associated with an index-1 saddle in Hamiltonian systems. The Lagrangian descriptor based approach is applied to two and three degree-of-freedom quadratic Hamiltonian systems where the high dimensional phase space structures are known, that is as closed-form analytical expressions. This leads to a direct comparison of features in the Lagrangian descriptor contour maps and the phase space structures’ intersection with an isoenergetic two-dimensional surface, and hence provides a verification of the method.

Collapse of hierarchical phase space and mixing rates in Hamiltonian systems

We employ statistical properties of Poincaré recurrences to investigate dynamical behaviors of coupled Hamiltonian maps with a mixed phase space, where sticking to regular (or resonant) islands degrades chaotic properties of the system. In particular we investigate two and three coupled Chirikov–Taylor standard mappings (SM) and choose nonlinearity parameters corresponding to a mixed phase space. Our first set of findings are: (i) the collapse of hierarchical phase space depends of the coupling intensity, (ii) the penetration of resonance islands generates a trapping regime responsible for a long intermediate algebraic decay which is suppressed as coupling increases; and (iii) the projection of recurrence-time statistics (RTS) onto two-dimensional planes of the high-dimensional phase space of coupled maps is useful to understand different dynamical features, when coupling strengths are varied. We also estimate the asymptotic polynomial decay exponent of RTS γ for both cases, and elaborate an indirect way to estimate the decay exponent of correlations χ by using large deviations theory applied to the probability distributions of finite-time largest Lyapunov exponents (FTLLEs). In the asymptotic regime both methods yield γ∼1.20 (2 SM) and γ∼1.10 (3 SM): the corresponding power-law exponents for time correlation decay χ is given by the known relationship χ=γ−1. As higher-dimensional Hamiltonian systems (with mixed phase space) share the crucial property that resonance islands are no more forbidden domains in phase space, the results obtained here for two and three coupled maps should be extended to even higher-dimensional systems.

Intermittent stickiness synchronization

This work uses the statistical properties of finite-time Lyapunov exponents (FTLEs) to investigate the intermittent stickiness synchronization (ISS) observed in the mixed phase space of high-dimensional Hamiltonian systems. Full stickiness synchronization (SS) occurs when all FTLEs from a chaotic trajectory tend to zero for arbitrarily long time windows. This behavior is a consequence of the sticky motion close to regular structures which live in the high-dimensional phase space and affects all unstable directions proportionally by the same amount, generating a kind of collective motion. Partial SS occurs when at least one FTLE approaches zero. Thus, distinct degrees of partial SS may occur, depending on the values of nonlinearity and coupling parameters, on the dimension of the phase space, and on the number of positive FTLEs. Through filtering procedures used to precisely characterize the sticky motion, we are able to compute the algebraic decay exponents of the ISS and to obtain remarkable evidence about the existence of a universal behavior related to the decay of time correlations encoded in such exponents. In addition we show that even though the probability of finding full SS is small compared to partial SSs, the full SS may appear for very long times due to the slow algebraic decay of time correlations in mixed phase space. In this sense, observations of very late intermittence between chaotic motion and full SS become rare events.

Quantitative analysis of financial system fragility based on manifold curvature

Manifold curvature is an intrinsic attribute of a dynamical system. By analyzing curvature characteristics of embedding attractor manifold, resistance and resilience of financial dynamic system are investigated in the face of disturbances, thus system fragility is discussed quantitatively from a systemic perspective. First, financial time series data are reconstructed into a high-dimensional phase space, topologically equivalent to its original dynamical system. Then, manifold learning is employed to extract attractor manifold embedding in financial dynamic system. Last, by numerical calculation, the manifold curvature is obtained and further visualized with spatial mapping. In empirical study, Shanghai Composite Index and Dow Jones Industrial Index are adopted as experimental data, and then a more comprehensive understanding about financial systems fragility is obtained by exploring the dynamical properties related to manifold curvature.

Patient-Specific Seizure Detection Method using Hybrid Classifier with Optimized Electrodes.

In this paper the EEG signal is analyzed by reconstructing the time series EEG signal in High dimensional Phase Space. The computational complexity in higher dimension is reduced by Principal Component Analysis for the High dimensional Phase Space output. Poincare sectioning is done for the first and second Principal Components (PCs). The intersection points of PCs and the Poincare section are collected and used for features calculation. Two layer of classification is done using SVM as first layer and Naive Bayes as second layer. The proposed methodology is evaluated using the CHB-MIT database for 23 subjects. The results are obtained using different channel combinations of EEG signal and highest of 95.63% accuracy, 95.7% sensitivity and 96.55% specificity is obtained for 12 electrode combinations which include electrodes from parietal and occipital lobes. This infers that most of the subjects have dysfunction in hearing (controlled by parietal) and vision (controlled by occipital) during the time of seizure. This GUI has channel selection option and seizure detection for every channel (23) for every 1s.

The ADHD effect on the high-dimensional phase space trajectories of EEG signals

Abstract Attention-deficit/hyperactivity disorder (ADHD) as a behavioral challenge, which affects the people's learning and experiences, is one of the disorders, which leads to reducing the complexity of brain processes and human behaviors. Nevertheless, recent studies often focused on the effects of this disorder in the frequency content of single and multichannel EEG segments and only a few studies that employed the approximate entropy for estimating this reduction. In this study, we provide a different view of this reduction by focusing on the texture of patterns appeared on the auto-recurrence plots obtained from the phase space trajectories reconstructed from the EEG signals recorded under the open-eyes and closed-eyes resting conditions. The outcomes of this analysis generally indicated a significant difference in the texture of recurrence plots, which its reason was the increase of recurrence, parallel and similar behaviors in the trajectories. Evaluating the features extracted from these recurrence plots in the studied children without and with ADHD using the sequential forward selection (SFS) algorithm also provided a remarkable accuracy (90.95% for the testing sets), which is a confirmation on changing the texture of recurrence plots relevant to the EEG signals of ADHD children. Nevertheless, evaluating these results and the results of previous researches with each other represented that the volume of statistical population is an important factor for reducing the rate of separability in the classifiers developed by an EEG segment. Therefore, these findings generally proved that although the ADHD averagely leads to the complexity reduction of EEG processes, the classifiers developed by just an EEG segment cannot be applicable in clinical conditions.

High-Order Accurate Conservative Finite Difference Methods for Vlasov Equations in 2D+2V

This manuscript discusses discretization of the Vlasov--Poisson system in 2D+2V phase space using high-order accurate conservative finite difference algorithms. One challenge confronting direct kinetic simulation is the significant computational cost associated with the high-dimensional phase space description. In the present work we advocate the use of high-order accurate schemes as a mechanism to reduce the computational cost required to deliver a given level of error in the computed solution. We pursue a discretely conservative finite difference formulation of the governing equations, and discuss fourth- and sixth-order accurate schemes. In addition, we employ a minimally dissipative nonlinear scheme based on the well-known WENO (weighted essentially nonoscillatory) approach. Verification of the full formulation is performed using the method of manufactured solutions. Results are also presented for the physically relevant scenarios of Landau damping, and growth of transverse instabilities from an imposed plane wave.

Poincaré's section analysis for PPG-based automatic emotion recognition

Given the importance and abundant use of emotional recognition in the human–computer interfaces and multimedia applications, several methods for designing such systems have been reported in the literature. However, the performance of the photoplethysmography (PPG) signal in these devices has not been sufficiently studied. In addition, details of the phase space geometry of this signal have not been investigated in the emotional states up to now. In connection with this issue, Poincaré's sections can quantify the geometric patterns of the trajectory in the high-dimensional phase space. Therefore, this work attempted to propose an automatic emotion recognizer which can characterize the PPG signals during the exposure of emotional music-video. Our focus was to detect and classify dynamical behaviors of the PPG trajectories in three emotional classes of love, hate, and fun using Poincaré's section measures. To this effect, the 2D phase space of PPG was firstly reconstructed. Then, forming the Poincaré's sections in different angles, some geometric indices were extracted. Finally, using support vector machine, the PPG measures were classified into emotional states. Our results showed that basin geometry of the PPG phase states was significantly different in different emotional states. The maximum accuracy rates of 96.67% and 91.11% were achieved for a binary and multi-class classification scheme, respectively. The proposed framework was fast and operated using the single-sensor signal. In conclusion, the dynamical Poincaré's section indices of PPG signals during three emotional states paved the way for designing an online emotion recognition system.