Nonlinear Spatiotemporal Dynamics Research Articles

Spatiotemporal pattern formation in parametrically driven two-dimensional Bose–Einstein condensates

We investigate spatiotemporal periodic patterns in harmonically trapped Bose–Einstein condensates (BECs) driven by a periodic modulation of the interaction. Resonant with the breathing mode, we show the emergence of a square lattice pattern containing two orthonormal stripes. We study the time evolutions of the lattice patterns for different driving strengths and dissipations. We find that its spatial periodicity and temporal oscillating frequency match the Bogoliubov dispersion, which is the intrinsic property of the system and relevant to the parametric amplification of elementary excitations. In the circumstances of strong driving strength and low dissipation, we further observe the triad interaction and the resulting superlattice state, which are well explained by the nonlinear amplitude equation for superimposed stripes. These results shed light on unexplored nonlinear spatiotemporal dynamics of two-dimensional patterns in harmonically trapped BECs that can pave the way for engineering exotic patterns by state-of-the-art experiments.

Enhancing Deep Learning-Based City-Wide Traffic Prediction Pipelines Through Complexity Analysis

Deep learning models can effectively capture the non-linear spatiotemporal dynamics of city-wide traffic forecasting. Evidence of varying deep learning model performance between different cities, different prediction horizons, different scales, specific city regions, and during particular hours of the day abounds in the literature on deep learning-based traffic prediction, yet a unified metric to quantify the complexity of different prediction tasks does not exist. This paper proposes two metrics—model complexity (MC) and intrinsic complexity (IC). While MC quantifies the effective complexity of deep learning models for city-wide traffic prediction tasks, the IC quantifies the underlying complexity of the prediction task. Being an effective complexity metric, MC depends on the model and the data. The IC depends only on the data and is invariant to the model being used. Both metrics are validated through systematic experimentation using traffic volume data from three cities. Finally, we demonstrate how these metrics can improve the workflows for deep learning-based data-driven traffic prediction pipelines and deployment by reducing the hyperparameter search scope and comparing the effectiveness of different design pathways.

Neural downscaling for complex systems: from large-scale to small-scale by neural operator

Researchers have long been working on interpreting and predicting the dynamics of complex systems in various fields. Conventional methods including full-scale simulations and reduced-order models are used to solve related problems, but often yield unsatisfactory results in terms of precision and computational efficiency. This is primarily due to the challenges posed by simulating small-scale dynamics. An alternative consideration is to leverage well-represented and readily accessible large-scale dynamics to assist small-scale modelling. This paper proposes a novel methodology, named as Neural Downscaling (ND), that effectively captures small-scale dynamics within a complementary subspace from corresponding large-scale dynamics well-represented in a low-dimensional space. The framework of ND integrates neural operator techniques with the principles of inertial manifold and nonlinear Galerkin theory. The effectiveness and efficiency of ND are demonstrated in the complex systems governed by Kuramoto–Sivashinsky and Navier–Stokes equations. Being an unprecedentedly deterministic model targeted at general small-scale dynamics, ND sheds light on the intricate spatiotemporal nonlinear dynamics of complex systems and reveals how small-scale dynamics are interacting with large-scale dynamics.

Magneto-Acoustic Field-Induced Unstable Interface of Magnetic Microswarm.

Research on the interfacial instability of two-phase systems can help in gaining a better understanding of various hydrodynamic instabilities in nature. However, owing to the nonlinear and complex spatiotemporal dynamics of the unstable interface, the instability is challenging to control and suppress. This paper presents a novel interfacial instability of the magnetic microswarm induced by the competition between the destabilizing effect of magnetic field and the stabilizing effect of acoustic field. The physics underlying this novel phenomenon is discussed by analyzing the contributions of the external fields. Unlike previous studies, this study demonstrates that the instability is independent of the interfacial force or diffusion effect and can persist without dissipation over time. The manipulation of the unstable interface is further achieved by adjusting the configuration of the magneto-acoustic system. This approach can be used in thermal encoding metamaterials and has great potential applications in systems where the instability is detrimental.

Spatiotemporal nonlinear dynamics and chaos in a mechanical Duffing-type system

This paper investigates spatiotemporal nonlinear dynamics and chaos in a dissipative mechanical Duffing-type system subjected to external stimulus. A nonlinear wave equation with cubic nonlinearity governs the system dynamics. A perturbation description is employed to build mathematical tools that represent different aspects of system dynamics, from local to global behaviors. Lyapunov exponents are defined from the different perturbations allowing the evaluation of local, convective and mean exponents. Different dynamical regimes are investigated considering homogeneous and heterogeneous spatial stimuli. Distinct dynamical responses are observed including periodic, quasi-periodic and chaotic behaviors. A novel concept of chaotic wave is employed to explain the spatial transport of chaos through the media considering heterogeneous conditions. Chaotic wave velocity is measured by the convective Lyapunov exponents.

Transient Propagation of the Invasion Front in the Homogeneous Landscape and in the Presence of a Road

Understanding the propagation of invasive plants at the beginning of invasive spread is important as it can help practitioners eradicate harmful species more efficiently. In our work the propagation regime of the invasive plant species is studied at the short-time scale before a travelling wave is established and advances into space at a constant speed. The integro-difference framework has been employed to deal with a stage-structured population, and a short-distance dispersal mode has been considered in the homogeneous environment and when a road presents in the landscape. It is explained in the paper how nonlinear spatio-temporal dynamics arise in a transient regime where the propagation speed depends on the detection threshold population density. Furthermore, we investigate the question of whether the transient dynamics become different when the homogeneous landscape is transformed into the heterogeneous one. It is shown in the paper how invasion slows down in a transient regime in the presence of a road.

Enhanced recovery caused by nonlinear dynamics in the wake of a floating offshore wind turbine

An experimental study in a wind tunnel is presented to explore the wake of a floating wind turbine subjected to harmonic side-to-side and fore–aft motions under laminar inflow conditions. The wake recovery is analysed as a function of the frequency of motion $f_p$ , expressed by the rotor-based Strouhal number, $St = f_p D / U_{\infty }$ ( $D$ is the rotor diameter, $U_{\infty }$ the inflow wind speed). Our findings indicate that both directions of motion accelerate the transition to the far-wake compared with the fixed turbine. The experimental outcomes confirm the computational fluid dynamics results of Li et al. (J. Fluid Mech., vol. 934, 2022, p. A29) showing that sideways motions lead to faster wake recovery, especially for $St \in [0.2, 0.6]$ . Additionally, we find that fore–aft motions also lead to better recovery for $St \in [0.3, 0.9]$ . The recovery is closely linked to nonlinear spatiotemporal dynamics found in the shear layer region of the wake. For both directions of motion and $St \in [0.2, 0.55]$ , the noisy wake dynamics lock in to the frequency of the motion. In this synchronised-like state, sideways motions result in large coherent structures of meandering, and fore–aft movements induce coherent pulsing of the wake. For fore–aft motion and $St \in [0.55, 0.9]$ , the wake shows a more complex quasiperiodic dynamic, namely, a self-generated meandering mode emerges, which interacts nonlinearly with the excitation frequency $St$ , as evidenced by the occurrence of mixing components. The coherent structures grow nonlinearly, enhance wake mixing and accelerate the transition to the far-wake, which, once reached, exhibits universal behaviour.

Spatiotemporal nonlinear dynamics in multimode fiber laser based on carbon nanotubes

Spatiotemporal nonlinear dynamics in multimode fiber laser based on carbon nanotubes

DEFM: Delay-embedding-based forecast machine for time series forecasting by spatiotemporal information transformation.

Making accurate forecasts for a complex system is a challenge in various practical applications. The major difficulty in solving such a problem concerns nonlinear spatiotemporal dynamics with time-varying characteristics. Takens' delay embedding theory provides a way to transform high-dimensional spatial information into temporal information. In this work, by combining delay embedding theory and deep learning techniques, we propose a novel framework, delay-embedding-based forecast Machine (DEFM), to predict the future values of a target variable in a self-supervised and multistep-ahead manner based on high-dimensional observations. With a three-module spatiotemporal architecture, the DEFM leverages deep neural networks to effectively extract both the spatially and temporally associated information from the observed time series even with time-varying parameters or additive noise. The DEFM can accurately predict future information by transforming spatiotemporal information to the delay embeddings of a target variable. The efficacy and precision of the DEFM are substantiated through applications in three spatiotemporally chaotic systems: a 90-dimensional (90D) coupled Lorenz system, the Lorenz 96 system, and the Kuramoto-Sivashinsky equation with inhomogeneity. Additionally, the performance of the DEFM is evaluated on six real-world datasets spanning various fields. Comparative experiments with five prediction methods illustrate the superiority and robustness of the DEFM and show the great potential of the DEFM in temporal information mining and forecasting.

Spectral-temporal-spatial customization via modulating multimodal nonlinear pulse propagation

Multimode fibers (MMFs) are gaining renewed interest for nonlinear effects due to their high-dimensional spatiotemporal nonlinear dynamics and scalability for high power. High-brightness MMF sources with effective control of the nonlinear processes would offer possibilities in many areas from high-power fiber lasers, to bioimaging and chemical sensing, and to intriguing physics phenomena. Here we present a simple yet effective way of controlling nonlinear effects at high peak power levels. This is achieved by leveraging not only the spatial but also the temporal degrees of freedom during multimodal nonlinear pulse propagation in step-index MMFs, using a programmable fiber shaper that introduces time-dependent disorders. We achieve high tunability in MMF output fields, resulting in a broadband high-peak-power source. Its potential as a nonlinear imaging source is further demonstrated through widely tunable two-photon and three-photon microscopy. These demonstrations provide possibilities for technology advances in nonlinear optics, bioimaging, spectroscopy, optical computing, and material processing.

Wavelength-tunable quasi-unidirectional spatiotemporal mode-locked solitons in an isolator-free Er-doped fiber laser

In this letter, we have demonstrated the realization of the wavelength-tunable spatiotemporal mode-locked (STML) operation in an Er-doped partial multimode fiber laser without the intracavity isolator. The laser adopts an all-fiber structure and the saturable absorption effect is attributed to nonlinear multimode interference (NL-MMI) from the single mode fiber-graded index multimode fiber-single mode fiber (SMF-GIMF-SMF) design. The STML laser can operate in the quasi-unidirectional regime with an emission strength difference of larger than 30 dB for the two counter-propagating lasers. And the switching between the dominating operating directions in the clockwise (CW) or counter-clockwise (CCW) directions can be easily achieved by adjusting the MM polarization controller (PC). Both STML conventional solitons (CSs) and the STML soliton molecules are obtained with a similar tunable wavelength from ∼1.56 μm to ∼1.58 μm for the opposite operating directions. Furthermore, the spatial dynamics of the STML soliton are also investigated in detail. The experimental results are of great help for us to explore nonlinear spatiotemporal dynamics and provide potential applications in fields of all-optical signal processing, fiber sensing and information coding.

Buildup of multiple spatiotemporal nonlinear dynamics in an all-fiber multimode laser

Ultrafast lasers based on multimode fibers have attracted extensive attention owing to the large mode-field area and nonlinear tolerance. The high spatial degree of freedom of multimode fibers is significant for spatiotemporal pulses locked both in transverse and longitudinal modes, where the energy of output pulses can be remarkably improved. Herein, the 1.5-μm all-fiber spatiotemporal mode-locked laser was realized based on carbon nanotubes as a saturable absorber. Moreover, by tuning the polarization controller and the pump power carefully, the output wavelengths can be ranged from 1529 to 1565 nm based on the multimode interference filter. In addition, Q-switched mode-locking and spatiotemporal mode-locked dual combs were also observed by further adjusting the polarization controller. Such a kind of an all-fiber multimode laser offers a crucial insight into the spatiotemporal nonlinear dynamics, which is of great significance in scientific research and practical applications.

Time-space separation-based data driven method for monitoring distributed parameter process with sparse and noisy sensor data

In industrial applications, real-time monitoring of the distributed parameter processes is a difficult task, due to the infinite-dimensional nature and the nonlinear spatiotemporal dynamics of distributed parameter systems (DPSs). In addition, the limited number of sensors and the existence of measurement noise may result in failure for detecting DPS dynamics. In this study, a data-driven model is proposed to reconstruct the spatiotemporal states in DPSs using sparse and noisy sensor data based on the time/space separation theory. The proposed reconstruction model maps the high-dimensional DPS states into low-dimensional spatial features using an autoencoder-based spatial model, and then predicts temporal evolution of the spatial features based on the long short-term memory network (LSTM). In addition, in order to improve robustness of the model, a random noise layer is developed and is integrated with the reconstruction model to extract robust spatiotemporal features. The performance of the proposed model is illustrated through applications in two case studies including a thermal process of lithium-ion battery and a fluid process in oceanography. The effects of sensor number and noise intensity on the proposed spatiotemporal model for monitoring DPSs are quantitatively investigated. In addition, the proposed spatiotemporal model is compared with proper orthogonal decomposition-based model. Research results demonstrate that the proposed spatiotemporal model is able to produce accurate and robust reconstruction results for monitoring distributed parameter processes with sparse and noisy sensor data.

A novel hybrid deep learning model with ARIMA Conv-LSTM networks and shuffle attention layer for short-term traffic flow prediction

Traffic flow prediction requires learning of nonlinear spatio-temporal dynamics which becomes challenging due to its inherent nonlinearity and stochasticity. Addressing this shortfall, we propose a new hybrid deep learning model based on an attention mechanism that uses multi-layered hybrid architectures to extract spatial–temporal, nonlinear characteristics. Firstly, by designing the autoregressive integral moving average (ARIMA) model, trends and linear regression are extracted; then, integration of convolutional neural network (CNN) and long short-term memory (LSTM) networks leads to better understanding of the model's correlations, serving for more accurate traffic prediction. Secondly, we develop a shuffle attention-based (SA) Conv-LSTM module to determine significance of flow sequences by allocating various weights. Thirdly, to effectively analyse short-term temporal dependencies, we utilise bidirectional LSTM (Bi-LSTM) components to capture periodic features. Experimental results illustrate that our Shuffle Attention ARIMA Conv-LSTM (SAACL) model provides better prediction than other comparable methods, particularly for short-term forecasting, using PeMS datasets.

Investigation of High‐Power Spatiotemporal Mode‐Locking with High Beam Quality

AbstractThe developed spatiotemporal mode‐locked (STML) laser has emerged as an effective platform for investigating high‐dimensional nonlinear spatiotemporal dynamics. Additionally, it offers a novel avenue for the design of fiber oscillators capable of operating at high power levels. At present, the focus of STML laser research primarily revolves around this aspect. However, considering practical applications, there is a strong desire to conceive a simple all‐fiber STML oscillator with both high beam quality and high power. In this study, an all‐fiber, high‐power STML oscillator based on multimode fibers is constructed and investigated. By manipulating the pump power and the polarization state, the laser could operate in a dissipative soliton or noise‐like pulse regime, both with average output powers of two operation states reaching the Watt level. Simultaneously, high‐quality output beam profiles are observed in both operation states. To the best of knowledge, this is the first demonstration of high‐power spatiotemporal mode‐locking with high beam quality. The study holds great benefits for advancing the investigations of compact all‐fiber STML lasers which deliver high‐power outputs with superior beam quality and ultimately propels the application of STML lasers. Furthermore, this study contributes to the understanding of the behavior of STML lasers with high power.

Radial Basis Function-Based Differential Quadrature Approach to Study Reaction–Diffusion of Ca2+ in T Lymphocyte

T lymphocytes have a primary role in both health and disease. Extracellular and intracellular signals determine whether a T-cell activates different cells, divides, or begins apoptosis. The reaction–diffusion process of Ca2+ ions is critical for the initiation, sustenance, and termination of the immunological function of T cell. A nonlinear spatio-temporal dynamics of Ca2+ in T cells is modeled incorporating parameters Sarco/endoplasmic reticulum Ca2+-ATPase (SERCA) pump, Ryanodine receptor, source amplitude, and buffers. A numerical meshless approach using multiquadric radial basis functions (MQRBF), differential quadrature, and Runge–Kutta method is developed for the solution. The results obtained here give better insights of calcium dynamics in T cells.

Mapping nonlinear brain dynamics by phase space embedding with fMRI data

The human brain is a complex neurobiological system exhibiting complex nonlinear spatiotemporal dynamics. While functional magnetic resonance imaging (fMRI) has been widely used to study brain activity, whole-brain nonlinear dynamics in fMRI data have not been extensively examined. The present study applied phase space embedding on resting-state fMRI data and characterized their phase space dynamics with the sum of lengths of portrait edges (SE) in the reconstructed phase portrait. The effects of repetition time (TR), bandpass filtering and the added noise power of BOLD signals on the optimal embedding parameters (embedding time delay τ and embedding dimension m) were examined with experimental or simulated fMRI data. Our results show that τ and m vary with the three acquisition parameters. The present method was applied to the autism spectrum disorder dataset from Autism Imaging Data Exchange I to demonstrate its capability in the characterization of abnormal brain dynamics. The resultant SE maps were statistically compared between patients and controls, and the significant differences in SE were fed into a support vector machine (SVM) for classification. A significant increase in SE in the default mode network (DMN) and salience network (SN), as well as the visual network, was found in autistic patients. With the SE features of these regions, our SVM classifier achieved superior accuracy (74.55% with 10-folds cross validation) compared with prior studies, indicating that phase space embedding and SE mapping are promising in characterizing the nonlinear dynamics of the BOLD signal and might be useful for brain biomarker discovery in clinical psychiatry.

Dynamics of Chapman-Jouguet pulsating detonations with Chain-Branching kinetics: Fickett’s analogue and Euler equations

The present study examines the spatiotemporal nonlinear dynamics of detonations over a wide range of reaction time scales away from the neutral stability region. This is addressed by one-dimensional numerical simulations with chain-branching kinetics. Fickett’s detonation analogue and Euler’s equations were used as evolution equations. A shock-fitting solver is used to reduce CPU time. Up to four thousand five hundred simulations have been carried out. Detailed bifurcation diagrams have been generated to explore the detonation dynamics. For long/intermediate reaction time scales, away from the neutral boundary, the traditional period-doubling cascade to chaos is seen. For square wave detonations, away from the neutral stability, almost periodic oscillations are recorded. This result might have implications for the existence of a characteristic length scale, the cell size, on typical cellular detonations which have a short reaction length.

Physics-constrained deep learning of nonlinear normal modes of spatiotemporal fluid flow dynamics

In this study, we present a physics-constrained deep learning method to discover and visualize from data the invariant nonlinear normal modes (NNMs) which contain the spatiotemporal dynamics of the fluid flow potentially containing strong nonlinearity. Specifically, we develop a NNM-physics-constrained convolutional autoencoder (NNM-CNN-AE) integrated with a multi-temporal-step dynamics prediction block to learn the nonlinear modal transformation, the NNMs containing the spatiotemporal dynamics of the flow, and reduced-order reconstruction and long-time future-state prediction of the flow fields, simultaneously. In test cases, we apply the developed method to analyze different flow regimes past a cylinder, including laminar flows with low Reynolds number in transient and steady states (RD = 100) and high Reynolds number flow (RD = 1000), respectively. The results indicate that the identified NNMs are able to reveal the nonlinear spatiotemporal dynamics of these flows, and the NNMs-based reduced-order modeling consistently achieves better accuracy with orders of magnitudes smaller errors in construction and prediction of the nonlinear velocity and vorticity fields, compared to the linear proper orthogonal decomposition (POD) method and the Koopman-constrained-CNN-AE using the same number or dimension of modes. We perform an analysis of the modal energy distribution of NNMs and find that compared to POD modes, the few fundamental NNMs capture a very high level of total energy of the flow, which is advantageous for reduced-order modeling and representation of the complex flows. Finally, we discuss the potentials and limitations of the presented method.

Spatiotemporal kernel-local-embedding modeling approach for nonlinear distributed parameter systems

In actual distributed parameter systems (DPSs), each spatial point has nonlinear energy transfer with its local neighbor points. Also, this energy transfer is affected by the past states. However, these properties are often ignored by most of modeling methods, which causes these methods ineffective in modeling of DPSs. Aiming for this problem, a spatiotemporal kernel-local-embedding (STKLLE) Modeling approach is proposed here to reconstruct the nonlinear spatiotemporal dynamics of DPSs. First, in order to present the complex dynamics on space, a STKLLE strategy is developed to extract space basis functions (SBFs). On the one hand, this STKLLE method represents the energy transfer relation with its neighboring points and maintains this local relation in model. On the other hand, it considers the influence of the adjacent past states to the current state. Then, using T–S fuzzy algorithm, a temporal model is designed to represent the temporal dynamics of DPSs in each sampling period. Integrating these SBFs and the temporal fuzzy model, a spatiotemporal model is constructed to well present and predict the nonlinear spatiotemporal dynamics in DPSs. Through the actual experiment on Lithium-ion batteries and heating oven, the effectiveness of this proposed model is detailly verified, and quantitative comparisons with several data-driven modeling algorithms are further carried out to demonstrate the model efficiency.