High-dimensional Nonlinear Systems Research Articles

Relative periodic orbits form the backbone of turbulent pipe flow

The chaotic dynamics of low-dimensional systems, such as Lorenz or Rössler flows, is guided by the infinity of periodic orbits embedded in their strange attractors. Whether this is also the case for the infinite-dimensional dynamics of Navier–Stokes equations has long been speculated, and is a topic of ongoing study. Periodic and relative periodic solutions have been shown to be involved in transitions to turbulence. Their relevance to turbulent dynamics – specifically, whether periodic orbits play the same role in high-dimensional nonlinear systems like the Navier–Stokes equations as they do in lower-dimensional systems – is the focus of the present investigation. We perform here a detailed study of pipe flow relative periodic orbits with energies and mean dissipations close to turbulent values. We outline several approaches to reduction of the translational symmetry of the system. We study pipe flow in a minimal computational cell at $Re=2500$, and report a library of invariant solutions found with the aid of the method of slices. Detailed study of the unstable manifolds of a sample of these solutions is consistent with the picture that relative periodic orbits are embedded in the chaotic saddle and that they guide the turbulent dynamics.

Efficient statistically accurate algorithms for the Fokker–Planck equation in large dimensions

Solving the Fokker–Planck equation for high-dimensional complex turbulent dynamical systems is an important and practical issue. However, most traditional methods suffer from the curse of dimensionality and have difficulties in capturing the fat tailed highly intermittent probability density functions (PDFs) of complex systems in turbulence, neuroscience and excitable media. In this article, efficient statistically accurate algorithms are developed for solving both the transient and the equilibrium solutions of Fokker–Planck equations associated with high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures. The algorithms involve a hybrid strategy that requires only a small number of ensembles. Here, a conditional Gaussian mixture in a high-dimensional subspace via an extremely efficient parametric method is combined with a judicious non-parametric Gaussian kernel density estimation in the remaining low-dimensional subspace. Particularly, the parametric method provides closed analytical formulae for determining the conditional Gaussian distributions in the high-dimensional subspace and is therefore computationally efficient and accurate. The full non-Gaussian PDF of the system is then given by a Gaussian mixture. Different from traditional particle methods, each conditional Gaussian distribution here covers a significant portion of the high-dimensional PDF. Therefore a small number of ensembles is sufficient to recover the full PDF, which overcomes the curse of dimensionality. Notably, the mixture distribution has significant skill in capturing the transient behavior with fat tails of the high-dimensional non-Gaussian PDFs, and this facilitates the algorithms in accurately describing the intermittency and extreme events in complex turbulent systems. It is shown in a stringent set of test problems that the method only requires an order of O(100) ensembles to successfully recover the highly non-Gaussian transient PDFs in up to 6 dimensions with only small errors.

Learning dynamic models for open loop predictive control of soft robotic manipulators

The soft capabilities of biological appendages like the arms of Octopus vulgaris and elephants’ trunks have inspired roboticists to develop their robotic equivalents. Although there have been considerable efforts to replicate their morphology and behavior patterns, we are still lagging behind in replicating the dexterity and efficiency of these biological systems. This is mostly due to the lack of development and application of dynamic controllers on these robots which could exploit the morphological properties that a soft-bodied manipulator possesses. The complexity of these high-dimensional nonlinear systems has deterred the application of traditional model-based approaches. This paper provides a machine learning-based approach for the development of dynamic models for a soft robotic manipulator and a trajectory optimization method for predictive control of the manipulator in task space. To the best of our knowledge this is the first demonstration of a learned dynamic model and a derived task space controller for a soft robotic manipulator. The validation of the controller is carried out on an octopus-inspired soft manipulator simulation derived from a piecewise constant strain approximation and then experimentally on a pneumatically actuated soft manipulator. The results indicate that such an approach is promising for developing fast and accurate dynamic models for soft robotic manipulators while being applicable on a wide range of soft manipulators.

Sparsity enabled cluster reduced-order models for control

Characterizing and controlling nonlinear, multi-scale phenomena are central goals in science and engineering. Cluster-based reduced-order modeling (CROM) was introduced to exploit the underlying low-dimensional dynamics of complex systems. CROM builds a data-driven discretization of the Perron–Frobenius operator, resulting in a probabilistic model for ensembles of trajectories. A key advantage of CROM is that it embeds nonlinear dynamics in a linear framework, which enables the application of standard linear techniques to the nonlinear system. CROM is typically computed on high-dimensional data; however, access to and computations on this full-state data limit the online implementation of CROM for prediction and control. Here, we address this key challenge by identifying a small subset of critical measurements to learn an efficient CROM, referred to as sparsity-enabled CROM. In particular, we leverage compressive measurements to faithfully embed the cluster geometry and preserve the probabilistic dynamics. Further, we show how to identify fewer optimized sensor locations tailored to a specific problem that outperform random measurements. Both of these sparsity-enabled sensing strategies significantly reduce the burden of data acquisition and processing for low-latency in-time estimation and control. We illustrate this unsupervised learning approach on three different high-dimensional nonlinear dynamical systems from fluids with increasing complexity, with one application in flow control. Sparsity-enabled CROM is a critical facilitator for real-time implementation on high-dimensional systems where full-state information may be inaccessible.

State estimation comparison for a high-dimensional nonlinear system by particle-based filtering methods

The sequential filtering scheme provides a suitable framework for estimating and tracking geophysical states of systems as new data become available online. Mathematical foundations of sequential Bayesian filtering are reviewed with emphasis on practical issues for both particle filters and Kalman-based filters. In this study, we further investigate the study of Kim (2005) such that the sequential Importance resampling method (SIR), Ensemble Kalman Filter (EnKF), and the Maximum Entropy Filter (MEF) are tested in a relatively high dimensional ocean model that conceptually represents the Atlantic thermohaline circulation. The model exhibits large-amplitude transitions between strong (thermo-dominated) and weak (salinity-dominated) circulations that represent climate states between ice-age and normal climate.The performance of the particle-based schemes is compared with the convergent results from SIR based on measurement errors, observation locations, and particle sizes in various sets of twin experiments. The sensitivity analysis shows strength and weakness of each filtering method when applied to multimodal non-linear systems. As the number of particles is increased, SIR achieves the convergent results that are mathematically optimal solutions. EnKF shows suboptimal results regardless of sample sizes, and MEF achieves the optimal solution even with a small sample size. Both EnKF, and MEF produces robust results with a relatively small sample size or increased measurement locations. Small measurement errors or short intervals of observations (or, more frequent observations) significantly improve the performances of SIR and EnKF, and MEF still show robust results even with a relatively small sample size or sparse measurement locations when the system experiences the transition between one region to the other region.

Explicit time integration of eddy current problems using a selective matrix update strategy

PurposeDiscretizing the magnetic vector potential formulation of eddy current problems in space results in an infinitely stiff differential algebraic equation system that is integrated in time using implicit time integration methods. Applying a generalized Schur complement to the differential algebraic equation system yields an ordinary differential equation (ODE) system. This ODE system can be integrated in time using explicit time integration schemes by which the solution of high-dimensional nonlinear algebraic systems of equations is avoided. The purpose of this paper is to further investigate the explicit time integration of eddy current problems.Design/methodology/approachThe resulting magnetoquasistatic Schur complement ODE system is integrated in time using the explicit Euler method taking into account the Courant–Friedrich–Levy (CFL) stability criterion. The maximum stable CFL time step can be rather small for magnetoquasistatic field problems owing to its proportionality to the smallest edge length in the mesh. Ferromagnetic materials require updating the reluctivity matrix in nonlinear material in every time step. Because of the small time-step size, it is proposed to only selectively update the reluctivity matrix, keeping it constant for as many time steps as possible.FindingsNumerical simulations of the TEAM 10 benchmark problem show that the proposed selective update strategy decreases computation time while maintaining good accuracy for different dynamics of the source current excitation.Originality/valueThe explicit time integration of the Schur complement vector potential formulation of the eddy current problem is accelerated by updating the reluctivity matrix selectively. A strategy for this is proposed and investigated.

Stochastic inequalities and applications to dynamics analysis of a novel SIVS epidemic model with jumps

This paper proposes a new nonlinear stochastic SIVS epidemic model with double epidemic hypothesis and Lévy jumps. The main purpose of this paper is to investigate the threshold dynamics of the stochastic SIVS epidemic model. By using the technique of a series of stochastic inequalities, we obtain sufficient conditions for the persistence in mean and extinction of the stochastic system and the threshold which governs the extinction and the spread of the epidemic diseases. Finally, this paper describes the results of numerical simulations investigating the dynamical effects of stochastic disturbance. Our results significantly improve and generalize the corresponding results in recent literatures. The developed theoretical methods and stochastic inequalities technique can be used to investigate the high-dimensional nonlinear stochastic differential systems.

Generalized Lie symmetry approach for fractional order systems of differential equations. III

The generalized Lie symmetry technique is proposed for the derivation of point symmetries for systems of fractional differential equations with an arbitrary number of independent as well as dependent variables. The efficiency of the method is illustrated by its application to three higher dimensional nonlinear systems of fractional order partial differential equations consisting of the (2 + 1)-dimensional asymmetric Nizhnik-Novikov-Veselov system, (3 + 1)-dimensional Burgers system, and (3 + 1)-dimensional Navier-Stokes equations. With the help of derived Lie point symmetries, the corresponding invariant solutions transform each of the considered systems into a system of lower-dimensional fractional partial differential equations.

Variable Selection and Identification of High-Dimensional Nonparametric Additive Nonlinear Systems

This paper considers variable selection and identification of dynamic additive nonlinear systems via kernel-based nonparametric approaches, where the number of variables and additive functions may be large. Variable selection aims to find which additive functions contribute and which do not. The proposed variable selection consists of two successive steps. At the first step, one estimates each additive function by kernel-based nonparametric identification approaches without suffering from the curse of dimensionality. At the second step, a nonnegative garrote estimator is applied to identify which additive functions are nonzero by utilizing the obtained nonparametric estimates of each function. Under weak conditions, the nonparametric estimates of each additive function can achieve the same asymptotic properties as for 1D nonparametric identification based on kernel functions. It is also established that the nonnegative garrote estimator turns a consistent estimate for each additive function into a consistent variable selection with probability one as the number of samples tends to infinity. Two simulation examples are presented to verify the effectiveness of the variable selection and identification approaches proposed in the paper.

Output-Tracking Quantized Explicit Nonlinear Model Predictive Control Using Multiclass Support Vector Machines

In applications involving digital control, the set of admissible control actions is finite/quantized. Coupled with state constraints and fast dynamics, explicit model predictive control (EMPC) provides an attractive control formalism. However, the design of data-driven EMPCs with finite admissible control sets is a challenging and relatively unexplored problem. In this paper, a systematic data-driven method is proposed for the design of quantized EMPCs (Q-EMPCs) for time-varying output tracking in nonlinear systems. The design involves: 1) sampling the admissible state space using low-discrepancy sequences to provide scalability to higher dimensional nonlinear systems; 2) at each sampled data point, solving for optimal quantized model predictive control actions and determining feasibility of the intrinsic mixed-integer nonlinear programming problem; and 3) constructing the Q-EMPC control surface using multiclass support vector machines (MC-SVMs). In particular, four widely used MC-SVM algorithms are employed to construct the proposed data-driven Q-EMPC. Extensive testing and comparison among the different MC-SVM algorithms is performed on 2-D and 5-D benchmark examples to demonstrate the effectiveness and scalability of the proposed methodology.

An efficient response identification strategy for nonlinear structures subject to nonstationary generated seismic excitations

ABSTRACTThe Monte Carlo method is computationally demanding. Consequently, applying that method to high-dimensional nonlinear systems is practically not feasible. Therefore, a new strategy is proposed in this article, which enables the evaluation of the response statistics of high-dimensional structures subject to random seismic excitations in a nonlinear reduced low-order subspace. It is shown that the new model order reduction method provides the possibility to overcome the computational issue of large-scale Monte Carlo simulations. It provides reliable approximations of the response statistics, when compared with those of the non-reduced Monte Carlo method.

Flutter of high-dimension nonlinear system for a FGM truncated conical shell

ABSTRACTThe nonlinear flutters of a truncated conical shell, which is subjected to aerodynamic pressure and aerodynamic heating, are researched. Material properties with gradient features along the radial direction depend on the temperature. The supersonic aerodynamic force is obtained by applying the first-order piston theory, including the correction factor for curvature. The temperature in the external surface of the functionally graded material truncated conical shell rises as a result of viscous aerodynamic heating, and the temperature distribution along the thickness can be described by polynomial series. Hamilton's principle is utilized to obtain the nonlinear partial differential equilibrium equation of the system. Using Galerkin's method, a high-dimensional nonlinear system can be derived. Without considering the parts of nonlinear terms and the external forcing excitation, the flutter boundaries are obtained by solving the eigenvalues problem. The influences of ratios of top radius to thickness, semi-vertex angle, and volume fraction index on nonlinear dynamic characteristics of functionally graded material truncated conical shell are studied in detail by the fourth-order Runge–Kutta algorithm.

Initial-state dependence of the route to chaos of an external-cavity laser

External-cavity semiconductor lasers (ECLs), consisting of a laser diode in front of a mirror to reflect light back into the laser diode, are among the most important dynamical systems because of their ultrafast dynamics, their tunability, and the numerous existing applications. The dynamics of an ECL is greatly influenced by the existence and stability of numerous modes of the external cavity. In such high-dimensional nonlinear systems, numerous attractors, located around various modes of the external cavity, can coexist in phase space for a given set of parameters, a phenomenon called generalized multistability. In this work, we propose a procedure that allows one to select experimentally different modes of the external cavity as different initial states. We use this procedure to reveal experimentally generalized multistability in an ECL through the demonstration that different routes to chaos exist in an ECL, depending on the initial state selected. In particular, we show that the famous quasiperiodic route to chaos is only observed for specific choices of initial conditions.

State estimation and prediction using clustered particle filters

Particle filtering is an essential tool to improve uncertain model predictions by incorporating noisy observational data from complex systems including non-Gaussian features. A class of particle filters, clustered particle filters, is introduced for high-dimensional nonlinear systems, which uses relatively few particles compared with the standard particle filter. The clustered particle filter captures non-Gaussian features of the true signal, which are typical in complex nonlinear dynamical systems such as geophysical systems. The method is also robust in the difficult regime of high-quality sparse and infrequent observations. The key features of the clustered particle filtering are coarse-grained localization through the clustering of the state variables and particle adjustment to stabilize the method; each observation affects only neighbor state variables through clustering and particles are adjusted to prevent particle collapse due to high-quality observations. The clustered particle filter is tested for the 40-dimensional Lorenz 96 model with several dynamical regimes including strongly non-Gaussian statistics. The clustered particle filter shows robust skill in both achieving accurate filter results and capturing non-Gaussian statistics of the true signal. It is further extended to multiscale data assimilation, which provides the large-scale estimation by combining a cheap reduced-order forecast model and mixed observations of the large- and small-scale variables. This approach enables the use of a larger number of particles due to the computational savings in the forecast model. The multiscale clustered particle filter is tested for one-dimensional dispersive wave turbulence using a forecast model with model errors.

A Variational Bayesian Multiple Particle Filtering Scheme for Large-Dimensional Systems

This paper considers the Bayesian filtering problem in high-dimensional nonlinear state-space systems. In such systems, classical particle filters (PFs) are impractical due to the prohibitive number of required particles to obtain reasonable performances. One approach that has been introduced to overcome this problem is the concept of multiple PFs (MPFs), where the state-space is split into low-dimensional subspaces and then a separate PF is applied to each subspace. Remarkable performances of MPF-like filters motivated our investigation here into a new strategy that combines the variational Bayesian approach to split the state-space with random sampling techniques, to derive a new computationally efficient MPF. The propagation of each particle in the prediction step of the resulting filter requires generating only a single particle in contrast with standard MPFs, for which a set of (children) particles is required. We present simulation results to evaluate the behavior of the proposed filter and compare its performances against standard PF and a MPF.

An EnKF-based scheme to optimize hyper-parameters and features for SVM classifier

The quality of models built by machine learning algorithms mostly depends on the careful tuning of hyper-parameters and feature weights. This paper introduces a novel scheme to optimize hyper-parameters and features by using the Ensemble Kalman Filter (EnKF), which is an iterative optimization algorithm used for high-dimensional nonlinear systems. We build a framework for applying the EnKF method on parameter optimization problems. We propose ensemble evolution to converge to the global optimum. We also optimize the EnKF calculation for large datasets by using the computationally efficient UR decomposition. To demonstrate the performance of our proposed design, we apply our approach for the tuning problem of Support Vector Machines. Experimental results show that the better global optima can be identified by our approach with acceptable computation cost compared to three state-of-the-art Bayesian optimization methods (SMAC, TPE and SPEARMINT).

Finite time blowup for high dimensional nonlinear wave systems with bounded smooth nonlinearity

ABSTRACTWe consider the global regularity problem for nonlinear wave systems on Minkowski spacetime ℝ1+d with d'Alembertian , where the field u: ℝ1+d → ℝm is vector valued, and the nonlinearity f: ℝm → ℝm is a smooth function with f(0) = 0 and all derivatives bounded; the higher dimensional sine-Gordon equation □u = sin u is a model example of this class of nonlinear wave system. For dimensions d ≤ 9, it follows from the work of Heinz, Pecher, Brenner, and von Wahl that one has smooth solutions to this equation for any smooth choice of initial data. Perhaps surprisingly, we show that this result is almost sharp, in the sense that for any d ≥ 11, there exists an m (in fact we can take m = 2) and a nonlinearity f: ℝm → ℝm with all derivatives bounded, for which the above equation admits solutions that blow up in finite time. The intermediate case d = 10 remains open.

REVIEW ON NONLINEAR VIBRATION AND MODELING OF LARGE SPAN CABLE-STAYED BRIDGE

As a hot research topic, the nonlinear dynamics of cable-stayed bridge has been highlighted in the field of mechanics, structure and bridge. With the development of new materials, such as CFRP cable, and new construction technology, the main span of cable-stayed bridge has been enlarged, which make this kind of bridge more competitive in bridge engineering. However, the increase of its span and the application of new materials make the bridge structure become lighter and more flexible, so that the nonlinear vibration of the structure becomes more prominent than ever, which may endanger the safety of the bridge. Based on our research of the nonlinear dynamics of cable-stayed bridge in recent years, the research progress in recent 10 years on the nonlinear dynamics and modeling of large span cable-stayed bridge is reviewed in detail. The nonlinear dynamic models, theories and modeling and solving methods for cable, beam, cable-beam and cable-stayed bridges are discussed. The results show that the cable stayed bridge has rich nonlinear dynamic behavior due to the coupling problem of the multi flexible cable, the large span beam, and the complexity of the environmental load. At the same time, due to the bottleneck problem of high dimensional nonlinear system, the study on the nonlinear dynamic behavior of the whole cable-stayed bridge becomes very complicated. Finally, based on the future development trend of cable-stayed bridge, some new ideas on nonlinear dynamics of cable-stayed bridge is proposed and future research directions are discussed.

A Tractable Fault Detection and Isolation Approach for Nonlinear Systems With Probabilistic Performance

This article presents a novel perspective along with a scalable methodology to design a fault detection and isolation (FDI) filter for high dimensional nonlinear systems. Previous approaches on FDI problems are either confined to linear systems or they are only applicable to low dimensional dynamics with specific structures. In contrast, shifting attention from the system dynamics to the disturbance inputs, we propose a relaxed design perspective to train a linear residual generator given some statistical information about the disturbance patterns. That is, we propose an optimization-based approach to robustify the filter with respect to finitely many signatures of the nonlinearity. We then invoke recent results in randomized optimization to provide theoretical guarantees for the performance of the proposed filer. Finally, motivated by a cyber-physical attack emanating from the vulnerabilities introduced by the interaction between IT infrastructure and power system, we deploy the developed theoretical results to detect such an intrusion before the functionality of the power system is disrupted.

Variable selection in nonlinear non-parametric system\\ identification

This paper considers a problem of variable selection for a high dimensional nonlinear non-parametric system. Different algorithms are proposed for cases when the number of observed data increasing to infinity, being finite, and the nonlinear system being additive. The theoretical properties of the proposed algorithms are obtained, which are validated by simulation examples. The algorithms find the relationship between the input and output variables, and further the inter-dependence of input variables so that the importance of the input variables can be established.