Elucidating motion patterns in sperm cell motion with dynamic mode decomposition

Recent years have seen increased emphasis on mathematical model reduction using modal decomposition techniques of high dimensional flow field data from experiments as well as numerical simulations. These tools decode the complex unsteady flow-field into several modes. Different tools highlight different flow dynamics. In the experimental community, Proper Orthogonal Decomposition (POD) has been the most commonly used technique, ranking modes by their relative energy content, thereby losing temporal information. However, many dynamics are not highlighted by the most energetic structures. In transitional flows for example, growth of flow structures is a more important indicator. The Dynamic Mode Decomposition (DMD) technique highlighted by Schmid1 achieves this by ranking modes by the most dynamically varying flow features. In this work, we use DMD and POD to analyze flow past a SD7003 airfoil undergoing periodic plunging motion, which is representative of a wide variety of phenomena present in MAVs and UAVs.The DMD modes highlight the dynamically varying nature of highly transient low Reynolds number flows and identify the specific flow structures associated with the plunging motion.A stability analysis of the computed DMD modes is performed and unstable flow structures are identified. Flow structures which are common to the dominant modes in both POD and DMD are identified using the two techniques in conjunction with each other. Further, the original flow field is reconstructed from the DMD modes and their individual modal behavior is analyzed to understand the net contribution of the mode to the total flow field.Finally, DMD modes in local regions are used to decide optimum probe placement locations, which capture majority of the flow dynamics based on sparse available time signals.

This study investigates the dynamic behavior of a flexible cable with heterogeneous stiffness using a data-driven approach. The study aims to develop accurate models that describe intricate structures with rigid or flexible components. To achieve this, reflective markers were attached to the cable at equal spacing and the motion was manually excited and captured using an 8 camera setup and OptiTrack’s Motive software. Cable displacement data at marker locations were used as initial conditions for various Dynamic Mode Decomposition (DMD) models. The performance of the data-driven cable model is compared against the performance of the DMD modeling approach, fitting the dynamics of single- and multidegree-of-freedom systems with added white noise. In this work, the authors have considered using time delays and Wavelets-based DMD. The study found that the wavelet-based DMD (WDMD) model was the most accurate method for reconstructing the response of the cable in the test cases. Researchers suggest that this data-driven approach can be applied to predict the dynamic behavior of nonlinear systems, with potential applications in civil engineering, aerospace, and robotics. Overall, this study presents a promising approach for developing accurate models of complex structures with rigid or flexible components. The findings of this study may be valuable in designing structures that can withstand dynamic loads and vibrations.

This study investigates the dynamic behavior of a flexible cable with heterogeneous stiffness using a data-driven approach. The study aims to develop accurate models describing intricate structures with rigid or flexible components. To achieve this, reflective markers were attached to the cable at equal spacing, and the motion was manually excited and captured using an 8-camera setup and OptiTrack's Motive software. The cable displacement data at the marker locations were used as initial conditions for various Dynamic Mode Decomposition (DMD) models. The performance of the data- driven cable model is compared against the performance of the DMD modeling approach, fitting the dynamics of single- and multi-degree of freedom systems with added white noise. In this work, authors have considered using time delays and Wavelets-based DMD. The study found that the Wavelet-based DMD (WDMD) model was the most accurate method for reconstructing the response of the cable in the test cases. The researchers suggest that this data-driven approach can be applied to predict the dynamic behavior of non-linear systems, with potential applications in civil engineering, aerospace, and robotics. Overall, this study presents a promising approach to developing accurate models of complex structures with rigid or flexible components. The findings of this study can be valuable for designing structures that can withstand dynamic loads and vibrations.

Mammalian spermatozoa gain competence to fertilize an oocyte as they travel through the female reproductive tract. This process is accompanied by an elevation of sperm intracellular calcium and a membrane hyperpolarization. The latter is evoked by K+ efflux; however, the molecular identity of the potassium channel of human spermatozoa (hKSper) is unknown. Here, we characterize hKSper, reporting that it is regulated by intracellular calcium but is insensitive to intracellular alkalinization. We also show that human KSper is inhibited by charybdotoxin, iberiotoxin, and paxilline, while mouse KSper is insensitive to these compounds. Such unique properties suggest that the Slo1 ion channel is the molecular determinant for hKSper. We show that Slo1 is localized to the sperm flagellum and is inhibited by progesterone. Inhibition of hKSper by progesterone may depolarize the spermatozoon to open the calcium channel CatSper, thus raising [Ca2+] to produce hyperactivation and allowing sperm to fertilize an oocyte.DOI: http://dx.doi.org/10.7554/eLife.01009.001

Low-dimensional coherent patterns underlie the behavior of many complex dynamical systems. We propose a novel dynamic mode decomposition (DMD) framework to discover reduced-order models of complex physics from spatiotemporal data. This algorithm combines time-delay embedding and reduced-order spectral projections to transform the initial dynamics into a latent space where DMD can efficiently resolve their evolution in space and time. Here, the efficiency and accuracy of this approach, which we call time-augmented, space-contracted DMD (TASC-DMD), is demonstrated in several benchmark tests, and is then utilized to compactly characterize human brain deformation. Sparse identification of nonlinear dynamics (SINDy) is then employed to discover a parsimonious model for brain deformation in TASC coordinates. This integrated algorithm (TASC-SINDy) is trained on 4D strain data from in vivo tagged magnetic resonance imaging (tagged MRI) in 36 human subjects from a cohort of 45 subjects. A unique and sparse set of governing equations describing the temporal dynamics of the brain was discovered using only three generic modes. The TASC-SINDy model achieved exceptional dimensionality reduction and accurately predicted dynamic strain fields for the nine test subjects not used for training. This data-driven approach can systematically unravel dynamics and improve predictions in many complex physical systems.

Dynamic mode decomposition (DMD) is a recently developed tool for the analysis of the behavior of complex dynamical systems. In this paper, we will propose an extension of DMD that exploits low-rank tensor decompositions of potentially high-dimensional data sets to compute the corresponding DMD modes and eigenvalues. The goal is to reduce the computational complexity and also the amount of memory required to store the data in order to mitigate the curse of dimensionality. The efficiency of these tensor-based methods will be illustrated with the aid of several different fluid dynamics problems such as the von Kármán vortex street and the simulation of two merging vortices.

Fault diagnosis of multivariate signals is a significant area of engineering research. Feature information is collected from multiple dimensions. This improves the accuracy and robustness of mechanical fault diagnosis. Dynamic mode decomposition (DMD) is a method that extracts the basic structure and mode of motion, describing the behavior of dynamic systems. It decomposes a high-dimensional nonlinear dynamic system into single-frequency DMD modes. With the advent of the big data era, the interest in tensor analysis has increased. Tensor train (TT) decomposition represents the tensor structure through a series of low-rank tensor products. To extract the fault features of multivariate signals, a low-rank TT DMD (LRTTDMD) model is proposed in this article. The multivariate signals are first converted into a data tensor using phase space reconstruction. Subsequently, the data tensor is transformed into TT format using TT decomposition while ensuring a certain level of error. The tensor operation is introduced into the original DMD to process the data tensor of TT format and extract the fault feature. The construction of tensor low-rank extraction framework enhances the robustness of the algorithm to noise. Finally, the effectiveness and feasibility of LRTTDMD are validated using simulation signals and laboratory self-test bearing fault signals.

A time resolved Particle Image Velocimetry (TR-PIV) system has been employed to investigate a laminar separation bubble which is induced by a strong adverse pressure gradient typical of Ultra-High-Lift lo pressure turbine blades. Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) are described and applied within the paper. These techniques allow reducing the degrees of freedom of complex systems producing a low order model ranked by the energy content (POD) or by the modal contribution to the dynamics of the system itself (DMD), useful to highlight the dominant dynamics. The time-space evolution of the laminar separation bubble is characterized by rollup vortices shed in the surround of the bubble maximum displacement as a consequence of the Kelvin-Helmholtz instability process as well as by a low frequency motion of the separated shear layer. The decomposition techniques proposed allow the identification of these coherent structures and the characterization of their modal properties (e.g. temporal frequency, spatial wavelength and growth rate). The POD separates the different dynamics that induce velocity fluctuations at different frequencies and wavelength looking at their contribution to the overall kinetic energy. The DMD provides complementary information: the unstable spatial frequencies are identified with their growth (or decay) rates. DMD modes associated with the Kelvin-Helmholtz instability and the corresponding vortex shedding phenomenon clearly dominate the unsteady behavior of the laminar separation bubble, being characterized by the highest growth rate. Modes with longer wavelength describe the low frequency motion of the laminar separation bubble and are neutrally stable. Results reported in the paper prove the ability of the present methods in extracting the dominant dynamics from a large dataset, providing robust and rapid tools for the in dept analysis of transition and separation processes.

A hydrocyclone is a centrifugal separation device with self‐induced rotating flow field, and the air core oscillation is a pivotal phenomenon profoundly impacting its efficiency and energy consumption. This study employs computational fluid dynamics (CFD) and dynamic mode decomposition (DMD) to analyze the periodic fluctuations of the air core in a hydrocyclone. The results reveal a pair of conjugate modes extracted by the DMD method closely matches the principal frequency of the quasi‐periodic flow. Notably, Modes 2 and 3 reflect the oscillating motion of the air core and the intense rotating airflow with the four‐helix vortex structure. They have intense rotation, stretching, and shear, which will disrupt the stability of the flow field of the hydrocyclone and affect its performance. These findings provide new insights into the periodic motion within the hydrocyclone, and the established methodology can be applied to the modal analysis of other rotational flow fields.

This study examines the predictive performance, preprocessing impact, computational feasibility, and robustness of data-driven models in simulating absorber behaviour in carbon capture systems under real-world conditions. Five algorithms — Dynamic Mode Decomposition (simplified and full), Autoregressive Integrated Moving Average, Random Forest, Support Vector Regression, and Long Short-Term Memory networks — were evaluated across three preprocessing scenarios: Robust Principal Component Analysis with and without interpolation, and unprocessed data. Results show that preprocessing generally improves accuracy, with RPCA-based approaches outperforming untreated datasets across most horizons, although its impact on robustness under noise remains limited. Robustness analysis was conducted on the three best-performing models — DMD, ARIMA, and LSTM — revealing distinct behaviours. Dynamic Mode Decomposition was the most computationally efficient, providing near-instantaneous training and prediction, and maintained acceptable performance under noise. ARIMA exhibited strong robustness and predictive capacity, with minimal performance degradation across noise levels. In contrast, Long Short-Term Memory networks, while effective for long-term forecasting, displayed high computational costs and significant sensitivity to stochastic training effects. These limitations resulted in inconsistent performance across noise levels, even under low perturbations. The study highlights trade-offs between accuracy, feasibility, and robustness, stressing the importance of aligning model choice with deployment constraints. While black-box methods offer strong predictions, their sensitivity to randomness and computational demands hinder practical use. Robust and reproducible approaches like DMD balance efficiency and reliability, making them well-suited for industrial carbon capture applications. • Benchmarks data-driven models on real-world CO 2 capture absorber dynamics. • RPCA preprocessing is shown to significantly enhance model accuracy and robustness. • For real-time use, DMD balances computational efficiency and predictive reliability. • Inherent stochasticity in LSTM training limits its industrial reproducibility.

Dynamic Mode Decomposition (DMD) has become a widely adopted technique for identifying dominant patterns that encapsulate the fundamental behavior of complex systems using observed data. Despite its popularity, DMD’s effectiveness is highly contingent upon the structure and fidelity of the input time series. To enhance the stability and generalization of the DMD framework, we incorporate two foundational conditions: Persistence of Excitation (PE) and the Granger Causality Test (GCT). By leveraging a Hankel matrix-based embedding, we extend the dynamic system’s representation by integrating additional latent variables, thereby enriching the modal subspace. The PE condition offers guidance on the minimum trajectory length necessary for accurate estimation of system dynamics, while GCT serves as a tool for determining the suitable model complexity by uncovering directional dependencies among variables. Although PE provides a basis for constructing approximate linear dynamics, it is the GCT-filtered causal structures that support trustworthy long-term forecasting. We apply this improved method to the task of Coherency Identification (CI) within multi-machine power systems (MMPS), which is vital for analyzing transient stability. Empirical results based on a 22-bus, six-generator test network illustrate how PE effectively captures key dynamic modes and confirm the importance of GCT in achieving model reliability and predictive performance.

Model reduction and analysis of deep dynamic stall on a plunging airfoil

The main objective of this book is to investigate the behavior of nonlinear unstable dynamic economic systems. In particular, we are interested in the behavior of dynamic systems near critical points. In the preceding examples, we showed that dramatic changes and chaotic economic behavior occur when linear stability is lost due to small shifts of parameters. However, analysis of nonlinear phenomena often involves sophisticated methods, especially when the problem is defined in many dimensions. It is rather desirable to develop some methods to reduce a high-dimensional problem to a lower one. The Haken slaving principle may be used to eliminate some of the variables when the system is close to points where linear stability is lost, so that the macroscopic behavior of the system is governed by very few degrees of freedom only. We can also find similar ideas in the center manifold theorem, although the slaving principle is more general. In this chapter, we are also interested in the role of adjustment speeds of economic variables and the time scale of the study in economic analysis.KeywordsEconomic AnalysisHopf BifurcationSingular PerturbationInvariant ManifoldEconomic VariableThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Vibration induced by dry friction is ubiquitous in various engineering fields. To explore the vibration characteristics for further studies and/or controls, it is of great theoretical and practical significances to investigate the non-linear dynamic behaviors of the friction systems. This study considers the slight vibration of a two-degree-of-freedom non-linear dry friction excitation system. The differential equations of system motion are established according to Newton’s law of motion. Moreover, the system’s non-linear dynamic is studied when the block velocity is always less than the friction surface velocity. The results indicate that the linearized matrix of the vibration system has a pair of purely imaginary eigenvalues for some critical values of the relevant parameters. The Poincaré-Birkhoff normal forms are utilized to simplify the motion equation under the non-resonant assumption to obtain a simplified equation with only the resonant terms. Furthermore, the truncated part of the simplified equation is analyzed in the case of only linear terms degeneration. Finally, numerical simulations reflect some qualitative conclusions about the system’s local dynamic properties, including equilibrium point, periodic motion, torus motion, and their stability.

In many insects, sperm cells are produced in bundles with their heads being held together by a glycoprotein matrix secreted by a cyst cell. Mature sperm cells in the seminal vesicles are usually free, but in sawflies and several other insects, such structures (spermatodesmata) remain intact and sperm cells may be ejaculated as bundles. Here we report the occurrence of spermatodesmata in mature males of the ant Lasius pallitarsis. Microscopic investigations of the abdominal contents of males immediately prior to their nuptial flights showed that the anterior ends of numerous sperm cells were embedded in an oval-shaped 20 by 30 micrometer extracellular fibrous cap. Individual sperm ranged in length from 55 to 75 micrometers with an average overall length of 65 micrometers. The bulb-shaped heads of the sperm were relatively small, only about 1.5 micrometers in length and about 1.1 micrometers in diameter. The diameter of the sperm tails was approximately 1 micrometer. Observations of live preparations of the spermatodesmata showed increasingly active undulating wave-like movement of the sperm tails as the slide preparations aged. This appears to be the first case of sperm bundles being present in the seminal vesicles of mature ant males – males that are immediately poised to complete their nuptial mating flight.