Finite-time Lyapunov Exponents Research Articles

Lagrangian approach for analysis of acoustic energy transport in open cavity flows

The energy transport in aero-acoustics is investigated in the Lagrangian frame. First, based on finite-time Lyapunov exponent (FTLE) and momentum potential theory, a Lagrangian approach is proposed to identify transport barriers of acoustic energy. Specifically, the method, named relative flux gradient (RFG), is presented in detail. Then, to verify the method, it is applied to analytical fields, showing that it could reveal the wavefronts and energy transport barriers depending on the time interval of computation. Moreover, RFG is applied to analyze a simulated flow field of an open cavity flow, and the results are compared with the Lagrangian coherent structures identified by FTLE, demonstrating great similarity. Importantly, the differences between the structures are further analyzed, illustrating several transport channels that correspond to the Rossiter mode and showing a complex interaction between acoustic and vorticity modes. Finally, the relationship between the identified transport barriers and the acoustic behaviors in Eulerian frame is studied in detail. The results show that the transport barriers identified by RFG significantly impact the orbits in phase space, and in particular, RFG has the potential to illustrate and analyze the transport of acoustic energy in complex flow fields in a quantitative way: one method for direct analysis of acoustic phenomena in complex flow regions.

Vector form intrinsic finite element analysis for nonlinear parametric resonances of planar beam structures

The parametric resonance of a beam structure may lead to a disastrous consequence. To assess the safety of a structure, structural engineers need a straightforward and practical method to calculate the large-amplitude nonlinear parametric resonance responses of the beam structure. This paper proposes a newly developed vector form intrinsic finite element (VFIFE) calculation scheme to analyse the nonlinear parametric resonance of planar beam structures. By taking into account the geometric stiffness effect of the internal axial force for the frame element, a new relationship between internal nodal forces and deformation components is first established based on the VFIFE principle and the parametric resonance mechanism of frame structures. A numerical example of a portal frame is presented to demonstrate the efficacy of this new calculation scheme. A parametric resonance test of the cantilever beam is conducted to verify the applicability of the VFIFE method. The numerical results show that the proposed scheme can well simulate the nonlinear phenomena of parametric resonances. The numerical predictions are consistent with the experimental observations. A filtering-based finite-time Lyapunov exponent approach is presented to determine the stability boundary of the structure. The numerical stability boundary of parametric resonance agrees well with the experimental boundary. Possible causes of deviations between the numerical and experimental results are discussed. The proposed VFIFE scheme is a simple and effective means of analysing the stability and nonlinear responses of the parametric resonance of planar beam structures.

Lagrangian and Eulerian perspectives of turbulent transport mechanisms in a lateral cavity

The dynamics of turbulent flows past lateral cavities is relevant for multiple environmental applications. In rivers and coastal environments, these lateral recirculating regions constitute surface storage zones, where large-scale turbulent coherent structures control the transport and fate of contaminants. Mass transport in these flows is typically represented by one-dimensional first-order equations that predict the evolution of the spatially integrated concentration between the cavity and the main channel. These models, however, cannot represent the long-term evolution of the concentration or incorporate memory effects induced by turbulence. In this investigation, we carry out large-eddy simulations (LES) of the open-channel flow with a lateral square cavity of Mignot et al. (Phys. Fluids, vol. 28, issue 4, 2016, 045104). The model is coupled with an advection–diffusion equation and a Lagrangian particle model to investigate the transport mechanisms in the cavity and across the interface. From the simulations we provide quantitative comparisons of the physical processes from both perspectives, and investigate the effects of turbulent coherent structures on residence times and trajectories from finite-time Lyapunov exponents. From the Lagrangian results, we identify general spatial distributions of time scales in the cavity associated with the dynamics of coherent structures, providing new insights into the mechanisms that drive the global transport. We also show that an upscaled model informed by LES and based on a fractional derivative captures the evolution of concentration, and the exchange between the cavity and the main channel, providing accurate predictions of mass transport and reproducing the temporal dependence observed at larger scales.

Generating Multiwing Hidden Chaotic Attractors With Only Stable Node-Foci: Analysis, Implementation, and Application

Based on Shil'nikov criterion, i.e., extending the number of unstable saddle-focus -type equilibria of chaotic system, a rich and varied multiwing chaos with complicated topology and remarkable pseudorandomness has been coined. However, it is a challenging task to construct multiwing chaos with only stable node-focus -type equilibria as the Shil'nikov criterion is not applicable in the circumstances. To eliminate this difficulty and enrich the multiwing chaos, we construct a simple three-dimensional autonomous system with only two symmetric stable equilibria, and further introduce a sawtooth wave modulation function to configure more equilibria. The highlight is that multiwing chaotic attractors can be generated from system with two or multiple stable node-foci. This characteristic indicates that the attractors can be hidden oscillating and are difficult to be located in the small neighborhoods of equilibria, which makes the system has good concealment performance for security applications. The design mechanism and system properties are theoretically analyzed and numerically investigated using phase portraits, finite-time Lyapunov exponents, and basins of attraction. Hardware experiments based on field-programmable gate array verify the feasibility of the system. Finally, based on the generated multiwing hidden chaotic attractors, an image encryption scheme with high security performance is designed to enhance its application.

Connecting finite-time Lyapunov exponents with supersaturation and droplet dynamics in a turbulent bulk flow

Connecting finite-time Lyapunov exponents with supersaturation and droplet dynamics in a turbulent bulk flow

Quantifying the role of submesoscale Lagrangian transport features in the concentration of phytoplankton in a coastal system

Abstract Food resources in the ocean are often found in low densities, and need to be concentrated for efficient consumption. This is done in part by oceanographic features transporting and locally concentrating plankton, creating a highly patchy resource. Lagrangian approaches applied to ocean dynamics can identify these transport features, linking Lagrangian transport and spatial ecology. However, little is known about how Lagrangian approaches perform in ageostrophic coastal flows. This study evaluates two Lagrangian Coherent Structure metrics against the distribution of phytoplankton; Finite Time Lyapunov Exponents (FTLE) and Relative Particle Density (RPD). FTLE and RPD are applied to High Frequency Radar (HFR) observed surface currents within a biological hotspot, Palmer Deep Canyon Antarctica. FTLE and RPD identify different transport patterns, with RPD mapping single particle trajectories and FTLE tracking relative motion of paired particles. Simultaneous measurements of circulation and phytoplankton were gathered through the integration of vessel and autonomous glider surveys within the HFR footprint. Results show FTLE better defined phytoplankton patches compared to RPD, with the strongest associations occurring in stratified conditions, suggesting that phytoplankton congregate along FTLE ridges in coastal flows. This quantified relationship between circulation and phytoplankton patches emphasizes the role of transport in the maintenance of coastal food webs.

Flow stirred by open turbine: A modal analysis of multiscale flow characteristics

AbstractThe turbulence observed in the stirred tank is a result of the superposition of flows at different scales, each exerting various influences on the operational processes. To understand the flow characteristics associated with stirring, this paper undertakes particle image velocimetry (PIV) experimental research on an unbaffled tank stirred by an open turbine featuring four distinct blade deflection angles. The flow field in the unbaffled tank manifests as a double‐circulation flow pattern; however, it is worth noting that the blade deflection angle exerts a significant impact on the stirring process. Specifically, when the blade deflection angle reached 90°, the flow discharge aligned closely with the radial direction, accompanied by pronounced fluctuations in the high‐velocity region. The Lagrangian coherent structures (LCS) formed were subsequently extracted and subjected to finite‐time Lyapunov exponent (FTLE) analysis. Notably, a more regular wake pattern emerged in proximity to the blade tip, while a larger scale flow structure persisted along the flow direction. To systematically analyze the flow field stirred by the open turbine, modal analysis was employed. By reconstructing the flow field based on the principle of 50% energy content, the decomposition of flows at different scales was achieved. The turbulent kinetic energy (TKE) generated by the various scale flows was subsequently calculated, revealing that the high TKE values primarily originate from the large‐scale flow structure. Conversely, the TKE generated by the small‐scale flow exhibits a uniform distribution across different regions. Notably, in regions characterized by higher velocities, the peak TKE associated with the small‐scale flow also manifests; however, it is significantly smaller in magnitude compared to that generated by the original flow or the large‐scale flow.

Magnetohydrodynamic Poynting Flux Vortices in the Solar Atmosphere and Their Role in Concentrating Energy

The nature of energy generation, transport, and effective dissipation responsible for maintaining a hot solar upper atmosphere is still elusive. The Poynting flux is a vital parameter for describing the direction and magnitude of the energy flow, which is mainly used in solar physics for estimating the upward energy generated by photospheric plasma motion. This study presents a pioneering 3D mapping of the magnetic energy transport within a numerically simulated solar atmosphere. By calculating the Finite Time Lyapunov Exponent of the energy velocity, defined as the ratio of the Poynting flux to the magnetic energy density, we precisely identify the sources and destinations of the magnetic energy flow throughout the solar atmosphere. This energy mapping reveals the presence of transport barriers in the lower atmosphere, restricting the amount of magnetic energy from the photosphere reaching the chromosphere and corona. Interacting kinematic and magnetic vortices create energy channels, breaking through these barriers and allowing three times more energy input from photospheric motions to reach the upper atmosphere than before the vortices formed. The vortex system also substantially alters the energy mapping, acting as a source and deposition of energy, leading to localized energy concentration. Furthermore, our results show that the energy is transported following a vortical motion: the Poynting flux vortex. In regions where these vortices coexist, they favor conditions for energy dissipation through ohmic and viscous heating, since they naturally create large gradients in the magnetic and velocity fields over small spatial scales. Hence, the vortex system promotes local plasma heating, leading to temperatures around a million Kelvins.

DualSmoke: Sketch-based smoke illustration design with two-stage generative model

AbstractThe dynamic effects of smoke are impressive in illustration design, but it is a troublesome and challenging issue for inexpert users to design smoke effects without domain knowledge of fluid simulations. In this work, we propose DualSmoke, a two-stage global-to-local generation framework for interactive smoke illustration design. In the global stage, the proposed approach utilizes fluid patterns to generate Lagrangian coherent structures from the user’s hand-drawn sketches. In the local stage, detailed flow patterns are obtained from the generated coherent structure. Finally, we apply a guiding force field to the smoke simulator to produce the desired smoke illustration. To construct the training dataset, DualSmoke generates flow patterns using finite-time Lyapunov exponents of the velocity fields. The synthetic sketch data are generated from the flow patterns by skeleton extraction. Our user study verifies that the proposed design interface can provide various smoke illustration designs with good user usability. Our code is available at https://githubcom/shasph/DualSmoke.

Spatial super-resolution of finite-time Lyapunov exponent fields with generative adversarial networks

In this article, two generative adversarial networks that enhance the spatial resolution of two- and three-dimensional finite-time Lyapunov fields by a factor of 4 in each dimension are presented. Using these models, high-resolution distributions can be approximated based on low-resolution input, the latter of which is associated with a computational cost and storage reduced by factors of about 42 and 43, respectively. Evaluating various test cases, the performance and generalizability of this approach are assessed. Shortcomings are only observed in the case of high-frequency spatial fluctuations where no sufficient statistical information is available in the low-resolution space. The major flow structures, however, are adequately rendered, giving rise to Lagrangian analyses of complex flow configurations that may otherwise remain elusive due to an excessive computational cost.

Finite-Time Lyapunov Exponents of Deep Neural Networks.

We compute how small input perturbations affect the output of deep neural networks, exploring an analogy between deep feed-forward networks and dynamical systems, where the growth or decay of local perturbations is characterized by finite-time Lyapunov exponents. We show that the maximal exponent forms geometrical structures in input space, akin to coherent structures in dynamical systems. Ridges of large positive exponents divide input space into different regions that the network associates with different classes. These ridges visualize the geometry that deep networks construct in input space, shedding light on the fundamental mechanisms underlying their learning capabilities.

Superconductivity coupling of harmonic resonant oscillators: Homogeneous and heterogeneous extreme multistability with multi-scrolls.

Understanding and characterizing multistabilities, whether homogeneous or heterogeneous, is crucial in various fields as it helps to unveil complex system behaviors and provides insights into the resilience and adaptability of these systems when faced with perturbations or changes. Homogeneous and heterogeneous multistabilities refer, respectively, to situation in which various multiple stable states within a system are qualitatively similar or distinct. Generating such complex phenomena with multi-scrolls from inherent circuits is less reported. This paper aims to investigate extreme multistability dynamics with homogeneous and heterogeneous multi-scrolls in two coupled resonant oscillators through a shunted Josephson junction. Analysis of equilibrium points revealed that the system supports both hidden and self-excited attractors. Various dynamical tools, including bifurcation diagrams, spectrum of Lyapunov exponents, and phase portraits, are exploited to establish the connection between the system parameters and various complicated dynamical features of the system. By tuning both system parameters and initial conditions, some striking phenomena, such as homogeneous and heterogeneous extreme multistability, along with the emergence of multi-scrolls, are illustrated. Furthermore, it is observed that one can readily control the number of scrolls purely by varying the initial conditions of the investigated system. A multi-metastable phenomenon is also captured in the system and confirmed using the finite-time Lyapunov exponents. Finally, the microcontroller implementation of the system demonstrates strong alignment with the numerical investigations.

On the probability of positive finite-time Lyapunov exponents on strange nonchaotic attractors

On the probability of positive finite-time Lyapunov exponents on strange nonchaotic attractors

Lyapunov exponents and Lagrangian chaos suppression in compressible homogeneous isotropic turbulence

We study Lyapunov exponents of tracers in compressible homogeneous isotropic turbulence at different turbulent Mach numbers Mt and Taylor-scale Reynolds numbers Reλ. We demonstrate that statistics of finite-time Lyapunov exponents have the same form as that in incompressible flow due to density-velocity coupling. The modulus of the smallest Lyapunov exponent λ3 provides the principal Lyapunov exponent of the time-reversed flow, which is usually wrong in a compressible flow. This exponent, along with the principal Lyapunov exponent λ1, determines all the exponents due to vanishing of the sum of all Lyapunov exponents. Numerical results by high-order schemes for solving the Navier–Stokes equations and tracking particles verify these theoretical predictions. We found that (1) the largest normalized Lyapunov exponent λ1τη, where τη is the Kolmogorov timescale, is a decreasing function of Mt. Its dependence on Reλ is weak when the driving force is solenoidal, while it is an increasing function of Reλ when the solenoidal and compressible forces are comparable. Similar facts hold for |λ3|, in contrast to well-studied short-correlated model; (2) the ratio of the first two Lyapunov exponents λ1/λ2 decreases with Reλ and is virtually independent of Mt for Mt≤1 in the case of solenoidal force but decreases as Mt increases when solenoidal and compressible forces are comparable; (3) for purely solenoidal force, λ1:λ2:λ3≈4:1:−5 for Reλ>80, which is consistent with incompressible turbulence studies; and (4) the ratio of dilation-to-vorticity is a more suitable parameter to characterize Lyapunov exponents than Mt.

Rate-induced tracking for concave or d-concave transitions in a time-dependent environment with application in ecology

This paper investigates biological models that represent the transition equation from a system in the past to a system in the future. It is shown that finite-time Lyapunov exponents calculated along a locally pullback attractive solution are efficient indicators (early-warning signals) of the presence of a tipping point. Precise time-dependent transitions with concave or d-concave variation in the state variable giving rise to scenarios of rate-induced tracking are shown. They are classified depending on the internal dynamics of the set of bounded solutions. Based on this classification, some representative features of these models are investigated by means of a careful numerical analysis.

Non-Boussinesq effect on the internal solitary wave propagation under a Lagrangian-like description for various pycnocline thickness conditions

The Dubreil–Jacotin–Long (DJL) equation is an exact model for the steady propagation of internal solitary waves (ISWs) under Boussinesq approximation. The aim of this study is to discuss the non-Boussinesq effect on ISWs initialized by DJL equation for various pycnocline thickness conditions. The simulated results show that the non-Boussinesq effect can indeed lead to instability of DJL equation, and the pycnocline thickness will affect the amplitude of re-stable ISWs and the separation process of tail waves. Moreover, in order to extract features of the non-Boussinesq instability and the location of the tail wave separation, we propose an effective analytical approach for the quasi-traveling wave problem inspired by the definition of classical Lagrangian coherent structure (LCS). The improved finite-time Lyapunov exponent with a moving window (FTLEmw) is a Lagrangian-like description with considering the prior information of physical phenomenon. For a steady ISW propagation in the Boussinesq system, the repelling and attracting FTLEmv fields are symmetrical and the field ridge behaves as a continuous spiral topology, which suggests that particles on different streamlines in the flow structure have different rotation periods, while the streamlines are not actually completely closed due to the forward propagation of the wave. During the main unstable stage of non-Boussinesq effect, the topology of separated tail wave which is disconnected from the ISW main part can be clearly distinguished. The tail wave separation locations identified by the improved FTLEmv field and the classical FTLE field are not the same, which illustrates the difference between the two perspectives of flow structure change and fluid particle motion. In addition, the leading edge, trailing edge and main part of ISWs will gradually be divided into three-part FTLEmv topologies in the thicker pycnocline conditions.

Mixing time of heterogeneities in a buoyancy-dominated magma ocean

SUMMARY During the accretion stage, large impacts provided sufficient energy to melt the entire mantle into a terrestrial magma ocean. Processes occurring in the magma ocean may have led to the formation of heterogeneities still found in modern ocean island basalts. So far, no definitive mechanism exists to explain the survival of early heterogeneities for approximately 4.5 Ga. Addressing this question requires understanding the efficiency of convective mixing during both the early molten and the solid-state stages experienced by the Earth’s mantle. While mixing in the solid mantle and in an essentially crystallized magma ocean has been relatively well documented, the efficiency of convective mixing in a liquid magma ocean has received less attention. In this paper we characterized the mixing efficiency of a convecting fluid in a rotating spherical shell, accounting for inertial effects, by computing finite-time Lyapunov exponents (i.e. the Lagrangian strain rate). We conducted a series of numerical experiments for a regime where the influence of the buoyancy force dominates that of rotation and we derived scaling laws to predict the mixing efficiency. We found that for a terrestrial magma ocean, in its fully liquid state, mixing time is of the order of a few minutes or less, even for initially large (∼1000 km) heterogeneities. Therefore, passive early mantle heterogeneities cannot survive a fully molten magma ocean stage. This suggests that short-lived heterogeneities (e.g. 182Hf−182W) were either created at the end of the accretional stage, or were stored in deeper regions of the Earth.

Understanding surgical smoke in laparoscopy through Lagrangian Coherent Structures.

In laparoscopic surgery, one of the main byproducts is the gaseous particles, called surgical smoke, which is found hazardous for both the patient and the operating room staff due to their chemical composition, and this implies a need for its effective elimination. The dynamics of surgical smoke are monitored by the underlying flow inside the abdomen and the hidden Lagrangian Coherent Structures (LCSs) present therein. In this article, for an insufflated abdomen domain, we analyse the velocity field, obtained from a computational fluid dynamics model, first, by calculating the flow rates for the outlets and then by identifying the patterns which are responsible for the transportation, mixing and accumulation of the material particles in the flow. From the finite time Lyapunov exponent (FTLE) field calculated for different cross-sections of the domain, we show that these material curves are dependent on the angle, positions and number of the outlets, and the inlet. The ridges of the backward FTLE field reveal the regions of vortex formation, and the maximum accumulation, details which can inform the effective placement of the instruments for efficient removal of the surgical smoke.

A Three Dimensional Lagrangian Analysis of the Smoke Plume From the 2019/2020 Australian Wildfire Event

AbstractDuring the 2019/2020 Australian bushfire season, intense wildfires generated a rising plume with a record concentration of smoke in the lower stratosphere. Motivated by this event, we use the atmospheric wind reanalysis model ERA5 to characterize the three dimensional atmospheric transport in the general region of the plume following a dynamical system approach in the Lagrangian framework. Aided by the Finite Time Lyapunov Exponent tool (FTLE), we identify Lagrangian Coherent Structures (LCS) which simplify the three‐dimensional transport description. Different reduced FTLE formulations are compared to study the impact of the vertical velocity and the vertical shear on the movement of the plume. We then consider in detail some of the uncovered LCS that are directly relevant for the evolution of the plume, as well as other LCS that are less relevant for the plume but have interesting geometries, and we show the presence of 3D lobe dynamics at play. Also, we unveil the qualitatively different dynamical fates of the smoke parcels trajectories depending on the region in which they originated. One feature that had a pronounced influence on the evolution of the smoke plume is a synoptic‐scale anticyclone that was formed near the same time as, and close to the region of, intense wildfires. We analyze this anticyclone in detail, including its formation, the entrainment of the smoke plume, and how it maintained coherence for a long time. Transport paths obtained with the inclusion of the buoyancy effects are compared with those obtained considering only the reanalysis velocity.

Heterogeneity of the attractor of the Lorenz ’96 model: Lyapunov analysis, unstable periodic orbits, and shadowing properties

It is well known that the predictability of weather and climate is strongly state-dependent. Special, easily recognisable, and extremely relevant atmospheric states like blockings are associated with anomalous instability. This reflects the general property that the attractors of chaotic systems can feature considerable heterogeneity in terms of dynamical properties, and specifically, of their instability. The attractor of a chaotic system is densely populated by unstable periodic orbits (UPOs) that can be used to approximate any forward trajectory through the so-called shadowing. Dynamical heterogeneity can lead to the presence of UPOs with different number of unstable dimensions. This phenomenon – unstable dimensions variability – is a serious breakdown of hyperbolicity and has considerable implications in terms of the structural stability of the system and of the possibility to model accurately its behaviour through numerical models. As a step in the direction of better understanding the properties of high-dimensional chaotic systems, here we provide an extensive numerical investigation of the variability of the dynamical properties across the attractor of the much studied Lorenz ’96 model. By combining the Lyapunov analysis of the tangent space with the study of the shadowing of the chaotic trajectory performed by a very large set of UPOs, we show that the observed variability in the number of unstable dimensions is associated with the presence of a substantial number of finite-time Lyapunov exponents that fluctuate about zero also when very long averaging times are considered. The transition between regions of the attractor with different degrees of instability is associated with a significant drop of the quality of the shadowing. By performing a coarse graining based on the shadowing UPOs, we are able to characterise the slow fluctuations of the system between regions featuring, on the average, anomalously high and anomalously low instability. In turn, such regions are associated, respectively, with states of anomalously high and low energy, thus providing a clear link between the microscopic and thermodynamical properties of the system.