Finite-time Lyapunov Exponents Research Articles

Strange Nonchaotic Attractors in Memristor-Based Shimizu–Morioka Oscillator

The existence of strange nonchaotic attractors (SNAs) is examined using a memristor-based Shimizu–Morioka oscillator. In particular, we analyze the birth of SNAs by varying one of the frequency of external quasi-periodic forces. The occurrence and its mechanism are further analyzed through rational approximation theory. Following that, we also perform different nonlinear dynamical characterizations, including Poincaré surface sections, singular continuous spectrum, separation of nearby trajectories, 0–1 test, as well as finite-time Lyapunov exponents for more precise validation. Finally, the experimental evidence for the occurrence of SNA is inspected through a real-time analog circuit in the laboratory.

Applying finite-time lyapunov exponent to study the tidal dispersion on oil spill trajectory in Burrard Inlet

This study used a combination of Finite-Time Lyapunov Exponent (FTLE) values, residual currents, and tidal excursion lengths to systematically investigate the effects of tidal dispersion on oil spill trajectories in Burrard Inlet, BC, Canada, which is a tidally dominated estuary. The FTLE analysis results showed that tidal type and tidal phase significantly influenced the FTLE fields because the flow structure and the location of saddle points varied as a function of the tidal type and tidal phase. Some transport barriers formed in the Inner Harbour, which blocked the water exchange between the western and eastern parts of the inlet. Moreover, tidal mixing in the wider regions of Burrard Inlet (i.e., the western Outer Harbour) was relatively weak than in the narrower areas (i.e., First Narrows and Second Narrows). The observations from the FTLE analysis agreed well with the residual currents and tidal excursion results. The drifter trajectories were consistent with the Lagrangian coherent structure lines extracted from the FTLE analysis. To verify the tidal dispersion impact on an oil spill trajectory in the inlet, the FTLE fields were compared with a real oil spill that occurred in Burrard Inlet in 2015 (the M/V Marathassa oil spill). The FTLE fields reasonably explained the spilled oil’s trajectories from the real event. In addition, a set of stochastic oil spill models were run in this study and found that the FTLE analysis was a reliable tool for oil spill tracking. Overall, the FTLE method would be a valuable addition to practical oil spill response planning.

Planktos: An Agent-Based Modeling Framework for Small Organism Movement and Dispersal in a Fluid Environment with Immersed Structures.

Multiscale modeling of marine and aerial plankton has traditionally been difficult to address holistically due to the challenge of resolving individual locomotion dynamics while being carried with larger-scale flows. However, such problems are of paramount importance, e.g., dispersal of marine larval plankton is critical for the health of coral reefs, and aerial plankton (tiny arthropods) can be used as effective agricultural biocontrol agents. Here we introduce the open-source, agent-based modeling software Planktos targeted at 2D and 3D fluid environments in Python. Agents in this modeling framework are relatively tiny organisms in sufficiently low densities that their effect on the surrounding fluid motion can be considered negligible. This library can be used for scientific exploration and quantification of collective and emergent behavior, including interaction with immersed structures. In this paper, we detail the implementation and functionality of the library along with some illustrative examples. Functionality includes arbitrary agent behavior obeying either ordinary differential equations, stochastic differential equations, or coded movement algorithms, all under the influence of time-dependent fluid velocity fields generated by computational fluid dynamics, experiments, or analytical models in domains with static immersed mesh structures with sliding or sticky collisions. In addition, data visualization tools provide images or animations with kernel density estimation and velocity field analysis with respect to deterministic agent behavior via the finite-time Lyapunov exponent.

Computation of Lagrangian coherent structures from experimental fluid trajectory measurements in a mechanically agitated vessel

• Lagrangian coherent structures are computed from 3D particle tracking measurements. • LCSs characterise the hidden complex topology of turbulent flow in a stirred vessel. • Results show that LCSs are related to mixer configuration and macroscale mixing. In mechanically agitated vessels, bulk flow circulation which plays a leading role in macroscale mixing is controlled by hidden Lagrangian coherent structures (LCSs). We use a numerical finite-time Lyapunov exponent (FTLE) approach, for the first time, to resolve such LCSs. Experimental 3D Lagrangian trajectories obtained from a unique positron emission particle tracking (PEPT) technique are used to drive the FTLE model. By computing forward and backward FTLE fields and extracting repelling and attracting FTLE ridges in various azimuthal planes of the flow, a highly complex flow topology is unravelled which varies significantly with azimuthal position. We demonstrate how LCSs organise and quantify the chaotic behaviour of fluid particle paths that underpin mixing through the exchange of fluid between zones of different kinematics. This new Lagrangian approach driven by unique PEPT data is able to unfold some of the complexities of turbulent flow that are beyond the capability of traditional methods.

Using Lagrangian coherent structures to investigate upwelling and physical process in the Gladstone coastal region

Coastal upwelling plays an important role in the exchange of nutrients and pollutants. Understanding of upwelling and its predictors allows for important management of nutrient-enriched coastal waters. This study presents an analysis of coastal upwelling with a focus on Gladstone, Queensland (Australia). Particularly, Lagrangian coherent structures (LCSs) are used to quantify the spatio-temporal variability of upwellings, and to reveal the relationships between the surface mixing and physical properties of water masses and oceanographic activity in the Gladstone region. Four different seasonal selections of water surface velocity data (for summer, autumn, winter and spring) are compared with wind, sea surface temperature (SST), sea surface salinity and density data to investigate the coastal upwelling area. The surface mixing is calculated based on the backward Finite-Time Lyapunov Exponent (FTLE) diagnostic approach of LCSs. The strong seasonality of upwelling largely follows seasonal patterns of winds. Results highlight how upwelling behaviour varied with the intensity of cyclones. The core conclusion is that upwelling dynamics, estimated using classic means such as SST, have a strong correlation with the FTLE metric of LCSs. The potential upwelling hotspot shows that on average, it has 17.3% of its area above the FTLE threshold, compared to a non-upwelling area which has a corresponding percentage of 0.64%. This relationship makes it possible to define different upwelling locations, and properties of upwellings (e.g., seasonal variability, cyclone effect) as a function of FTLE, a derivative of the surface velocity field. Therefore, LCSs can be used to predict surface mixing and upwelling behaviour in coastal water systems.

Lagrangian Study of Several Long-Lived Agulhas Rings

AbstractBy using a Lagrangian-averaged vorticity deviation (LAVD)-based vortex detection scheme, rotationally coherent Lagrangian vortices in the South Atlantic Ocean are detected. These vortices act as agents for water transport and can stay coherent in a limited time scale. Our study starts from the life cycle of several long-lived Agulhas rings detected with the LAVD-based vortex detection method. The life cycle of those long-lived Agulhas rings can be separated into two distinct stages: the growing stage and the decaying stage. It is found that at the growing stage, the ambient water spins in and provides effective shielding for the coherent core. The rate of change of material belt width with respect to the detection time scale at the end of the growing stage can represent the decay rate of coherence. We further find a linear relationship between the mean strain rate and the mean square root of kinetic energy (KE1/2). Mean finite-time Lyapunov exponents (FTLE) increase monotonically with the mean strain rate or mean KE1/2. The long existence of the Agulhas rings can be partly attributed to the energetic boundaries around the rings. The ratio of the boundary kinetic energy to the spatial mean kinetic energy (KE/MKE) is also found to be a contributing factor that can influence the lifetime of Agulhas rings. In the retroflection area, the short-lived Agulhas rings might be attributed to the low KE/MKE in this area.

Lagrangian characteristics in the western North Pacific help to explain variability in Pacific saury fishery

A new model for estimation of daily probability for the Pacific saury (Cololabis saira) encounter was proposed. The model performance was tested for the period of 2004–2018 (August–November) using the data from the Russian vessel monitoring system. The following physical oceanographic variables were used for encounter probability prediction: the absolute values and gradients (∇) of speed (V) of passive particles, imitating water parcels, and Lagrangian indicators. The positive effects on the encounter probability of saury were found for V, ∇V, and for the gradient of the finite-time Lyapunov exponent (∇Λ), while the effect of particle path length was negative. That means that saury preferred places close to the boundaries of the oceanographic features, where Lagrangian fronts are situated, but not inside the features themselves, because Λ is small in regular flows and large at Lagrangiam fronts. The model did not include information about years and volume of saury catches, but its monthly mean of catch probability in September had the highest correlation with Russian annual catches outside the national waters between Russia and Japan (r = 0.76, p = 0.001) and total annual catches there (r = 0.73, p = 0.002). Timeseries analysis of principle components (PC) from daily predictions of saury catch probabilities has also shown that the third PC correlated highly with the annual biomass of saury (r ≥ 0.8, p < 0.05). The model seems to be useful to manage Russian fishery and may help to explain the reasons for the saury biomass decline. The latter is very important to take into account for development of the stock assessment models.

Wave-packet spreading in disordered soft architected structures.

We study the dynamical and chaotic behavior of a disordered one-dimensional elastic mechanical lattice, which supports translational and rotational waves. The model used in this work is motivated by the recent experimental results of Deng et al. [Nat. Commun. 9, 1 (2018)]. This lattice is characterized by strong geometrical nonlinearities and the coupling of two degrees-of-freedom (DoFs) per site. Although the linear limit of the structure consists of a linear Fermi-Pasta-Ulam-Tsingou lattice and a linear Klein-Gordon (KG) lattice whose DoFs are uncoupled, by using single site initial excitations on the rotational DoF, we evoke the nonlinear coupling between the system's translational and rotational DoFs. Our results reveal that such coupling induces rich wave-packet spreading behavior in the presence of strong disorder. In the weakly nonlinear regime, we observe energy spreading only due to the coupling of the two DoFs (per site), which is in contrast to what is known for KG lattices with a single DoF per lattice site, where the spreading occurs due to chaoticity. Additionally, for strong nonlinearities, we show that initially localized wave-packets attain near ballistic behavior in contrast to other known models. We also reveal persistent chaos during energy spreading, although its strength decreases in time as quantified by the evolution of the system's finite-time maximum Lyapunov exponent. Our results show that flexible, disordered, and strongly nonlinear lattices are a viable platform to study energy transport in combination with multiple DoFs (per site), also present an alternative way to control energy spreading in heterogeneous media.

Flow Patterns of Self-Sustained Oscillations in Fluidic Diverters

A fluidic oscillator is a device capable of transforming a steady input flow into an oscillatory output using solely fluid dynamic principles and without requiring any moving components. Although the basic operation principles of fluidic oscillator are known to some extent, the unsteady and turbulent nature of its internal flow remains difficult to understand. The current study aims to further clarify the complicated flow physics involved in fluidic oscillators by investigating a device based on the fluid diverter principle. Spatiotemporally resolved velocity and pressure data obtained from a large-eddy simulation of the internal flow are decomposed into the proper orthogonal modes. The finite-time Lyapunov exponents are computed on the flowfield reconstructed using the proper orthogonal modes to perform a Lagrangian analysis of the switching mechanism. It is found that four dominant modes are sufficient to describe the jet switching. Strong coherent structures are found to separate the flow going through the oscillator from secondary flows through the feedback tube. The deflection of the jet inside the device is found to depend only on the pressure difference across the control ports, whereas the change of this pressure difference is mediated by the flows in the feedback tube.

The statistical geometry of material loops in turbulence

Material elements – which are lines, surfaces, or volumes behaving as passive, non-diffusive markers – provide an inherently geometric window into the intricate dynamics of chaotic flows. Their stretching and folding dynamics has immediate implications for mixing in the oceans or the atmosphere, as well as the emergence of self-sustained dynamos in astrophysical settings. Here, we uncover robust statistical properties of an ensemble of material loops in a turbulent environment. Our approach combines high-resolution direct numerical simulations of Navier-Stokes turbulence, stochastic models, and dynamical systems techniques to reveal predictable, universal features of these complex objects. We show that the loop curvature statistics become stationary through a dynamical formation process of high-curvature folds, leading to distributions with power-law tails whose exponents are determined by the large-deviations statistics of finite-time Lyapunov exponents of the flow. This prediction applies to advected material lines in a broad range of chaotic flows. To complement this dynamical picture, we confirm our theory in the analytically tractable Kraichnan model with an exact Fokker-Planck approach.

Effect of pressure evolution on the formation enhancement in dual interacting vortex rings

In the biological locomotion, the ambit pressure is of particular importance to use as a means of propulsion. The multiple vortex rings have been proved to generate additional thrust by interaction, but the mechanism of this thrust enhancement is still unknown. This study examines the effect of ambit pressure on formation enhancement in interacting dual vortex rings. The vortex rings, which have the same formation time, are successively generated in a piston-cylinder apparatus. The finite-time Lyapunov exponent (FTLE) visualizes the flow fields as an indication of Lagrangian coherent structures (LCSs), and the pressure field is calculated based on the digital particle image velocity (DPIV). We extract the back pressure of the rear vortex in dual vortices and the back pressure circulation Γ b, which is defined as a form of overpressure circulation Γ p. The Γ b has a positive linear relationship with Γ p. A critical interval distance in a range of 0.32–0.42 is found where Γ b and Γ p reach the maximum synchronously, leading to a full-interaction mode. Moreover, an over-interaction mode and an under-interaction mode are proposed when the dimensionless interval distance d* is smaller or larger than . To conclude, the high back pressure caused by vortex interaction can enhance the formation of vortex rings and lead to high thrust.

On the Lagrangian dynamics of saliva particles during normal mouth breathing

The flow and saliva particulate transport dynamics during normal human breathing through the mouth are simulated numerically using an Eulerian large-eddy simulation (LES) approach for the flow coupled with a Lagrangian approach for the transport of saliva particles. The coupled Eulerian–Lagrangian (EL) simulation results reveal new striking insights into the rich dynamics of the Lagrangian coherent structures (LCS) that arise from saliva particles during normal breathing. Specifically, they uncover a new time-periodic mechanism via which particles are introduced into the flow as individual breathing pulses and accumulate to form a slowly propagating vortex front that persists long distances away from the source. The simulated LCS reveal a wealth of recurrent material motion through which the biosols propagate forward while their cloud expands laterally with a slowly evolving vortex front. Also, the finite-time Lyapunov exponent (FTLE) field of human breathing was calculated using the flow map from the LES velocity field. The ridges of the calculated FTLE field revealed distinct hyperbolic LCS, which closely resemble trajectories of saliva particles obtained from the coupled EL simulation. Finally, simulation results for normal breathing with a non-medical face mask show that the mask can effectively disrupt the formation of coherent particle surfaces and, thus, effectively limit saliva particle propagation.

Turbulent coherent structures in channel flow with a wall-mounted hemisphere

Turbulent channel flows around a wall-mounted hemisphere numerically are investigated by large eddy simulation, and the Reynolds number based on the hemisphere’s diameter is 3 × 104. The statistical characteristics and turbulent structure evolution are revealed in the Eulerian frameworks and Lagrangian frameworks. The vortex identification and Dynamic Mode Decomposition (DMD) are used to study the evolution of turbulent structure in the Eulerian frameworks, and the finite-time Lyapunov exponents are applied to identify Lagrangian coherent structures (LCS) in the Lagrangian framework. It is found that the developing angle of the hairpin vortex is ∼7° at two frameworks. What is more, there are some hairpin vortices formed behind the hemisphere and some turbulent structures formed near the wall by DMD method. The correlation analysis is applied to investigate the angle variation and scale variation of turbulent structures, and it is observed that the angle of turbulent structures is negative at Y/d ≥ 1.2 and the spanwise length scales of turbulent structures increase as it moves downstream. By studying the LCS behind a wall-mounted hemisphere, there is formation of “kink” caused by viscous interaction between some hairpin vortex legs, which is the characteristic of hairpin vortex deformation. The comparisons of statistical characteristics between Eulerian frameworks and Lagrangian frameworks are conducted by the correlation analysis, the spectrum analysis, and the structure functions.

A Novel Compound-Coupled Hyperchaotic Map for Image Encryption

Considering a nonlinear dynamic oscillator, a high Lyapunov exponent indicates a high degree of randomness useful in many applications, including cryptography. Most existing oscillators yield very low Lyapunov exponents. The proposed work presents a general strategy to derive an n-D hyperchaotic map with a high Lyapunov exponent. A 2D case study was analyzed using some well-known nonlinear dynamic metrics including phase portraits, bifurcation diagrams, finite time Lyapunov exponents, and dimension. These metrics indicated that the state of the novel map was more scattered in the phase plane than in the case of some traditional maps. Consequently, the novel map could produce output sequences with a high degree of randomness. Another important observation was that the first and second Lyapunov exponents of the proposed 2D map were both positive for the whole parameter space. Consequently, the attractors of the map could be classified as hyperchaotic attractors. Finally, these hyperchaotic sequences were exploited for image encryption/decryption. Various validation metrics were exploited to illustrate the security of the presented methodology against cryptanalysts. Comparative analysis indicated the superiority of the proposed encryption/decryption protocol over some recent state-of-the-art methods.

Studying finite-time (non)-domination in dynamical systems using Oseledec’s splitting. Application to the standard map

Oseledec’s theorem provides the necessary conditions for the existence of the decomposition of the tangent bundle called Oseledec’s splitting and the formal definition of the Lyapunov exponents. Using the concept of τ-domination of expansion or contraction rates of invariant subspaces after a time τ, together with the Oseledec theorem, we propose three quantifiers of τ-domination and τ-non-domination, which can be used to study the complex dynamics of physical systems. The interpretation of the Oseledec τ-domination in terms of the finite-time Lyapunov exponents is of great advantage for this purpose. Numerical results for the quantifiers are presented using the conservative standard map in a regime with mixed phase-space dynamics. Distinct typical regions in the phase-space are chosen, two containing a completely chaotic motion and two containing regularity islands. Results show that all quantifiers recognize that regions close to hyperbolic periodic points have a larger Oseledec τ-domination than regions around islands of regularity. Furthermore, Oseledec τ-non-dominated regions are also identified and are mostly close to regularity islands. We found a relation between the mean of expansion rates between the Oseledec subspaces and the mean of the angles between them.

Lagrangian Coherent Structures and hypoxia in the Baltic Sea

The spatial and temporal variations of Lagrangian Coherent Structures (LCSs) in the Baltic Sea are extracted from finite-time Lyapunov exponent (FTLE) fields. A validated 3-D hydrodynamic model of the Baltic Sea coupled with a water quality model is applied for the years 2000–2009. The novelty of the work is on the variation of LCSs with the sea depth, the state of hypoxia and the possible relationship with the blooming patterns in the Baltic. The study reveals a variety of LCSs with a typical core diameter of 10–40 km and a duration of 2–7 days that are formed offshore in all the basins of the Baltic Sea. They occur throughout the year even during winter times when the sea at the northern basins is partially covered with ice. The LCSs are more abundant in the southern basins where extensive algae blooms occur. The dominant structures are large vortex dipoles and anti-rotating vortex pairs that are not limited to the surface water layer but spread to a depth of 143.5 m. Likely mechanisms for the formation and the spread of LCSs are Kelvin-Helmholtz type instabilities, Proudman-Taylor column and Ekman Spiral. In the vicinity of the shorelines, the LCSs are smaller in diameter scaling with the mean Rossby radius to around 5 km. In the southern basins of the Baltic Sea; the dissolved oxygen (DO) content is permanently below 2 mg/l at depths below 80 m. DO contents vary seasonally with high values during winter and early spring times as opposed to lower values during summer and autumn periods. During late summers, the decline in DO content appears related to the extensive algae blooming consuming oxygen through the decomposition process. The LCSs map the patterns of Algae blooms detected by satellites. The duration of Algae blooming agrees with the persisting time of LCSs as well as the spatial surface water extents. The major 2003 inflow (MBI) increased the seabed DO content only in the lower part of the Arkona Basin that lasted just for about 3 weeks. The inputs of DO from the rivers and the 2003 MBI were not sufficient to counteract the seabed hypoxia in the Baltic Proper for the years 2000–2009.

Finite-horizon, energy-efficient trajectories in unsteady flows.

Intelligent mobile sensors, such as uninhabited aerial or underwater vehicles, are becoming prevalent in environmental sensing and monitoring applications. These active sensing platforms operate in unsteady fluid flows, including windy urban environments, hurricanes and ocean currents. Often constrained in their actuation capabilities, the dynamics of these mobile sensors depend strongly on the background flow, making their deployment and control particularly challenging. Therefore, efficient trajectory planning with partial knowledge about the background flow is essential for teams of mobile sensors to adaptively sense and monitor their environments. In this work, we investigate the use of finite-horizon model predictive control (MPC) for the energy-efficient trajectory planning of an active mobile sensor in an unsteady fluid flow field. We uncover connections between trajectories optimized over a finite-time horizon and finite-time Lyapunov exponents of the background flow, confirming that energy-efficient trajectories exploit invariant coherent structures in the flow. We demonstrate our findings on the unsteady double gyre vector field, which is a canonical model for chaotic mixing in the ocean. We present an exhaustive search through critical MPC parameters including the prediction horizon, maximum sensor actuation, and relative penalty on the accumulated state error and actuation effort. We find that even relatively short prediction horizons can often yield energy-efficient trajectories. We also explore these connections on a three-dimensional flow and ocean flow data from the Gulf of Mexico. These results are promising for the adaptive planning of energy-efficient trajectories for swarms of mobile sensors in distributed sensing and monitoring.

Lagrangian Coherent Structure Analysis on the Vegetated Compound Channel with Numerical Simulation

Natural channels often consist of a mainstream near their thalwegs and shallow vegetated areas near shores. The compounded and partially vegetated cross-sections play a significantly role in determining the hydrodynamic characteristics of a channel. By employing the Lagrangian Coherent Structure (LCS) analysis, the present work unravels the effect of vegetation and geometry on the hydrodynamic interactions between mainstreams with the various depths and vegetated shallow areas. The LCS method is the concept of dynamical system analyses, which is determined by the finite-time Lyapunov exponents (FTLE) field of fluid particles. It enables to overcome the limitations of using the particle tracking method in cost and time for simulations. Since the LCSs represent material surfaces or asymptotic lines which particles approach, but do not pass through, they match well with the trajectories of particles or materials obtained by solving particle motion equations. Therefore, the temporal and spatial developments of the interfacial layers could be investigated by using the FTLE. As the difference of depth becomes appreciable, the values of FTLE are relatively larger farther from the vegetated area. It implies that the interfacial layer becomes wider with the larger size of vortex produced by the differences of velocities between the mainstreams and the vegetated areas. In other words, as depth differences become large, materials and momentum can be spread from the vegetated area to or collected from a wider area of the mainstream.

Near-wake dynamics of a vertical-axis turbine

Cross-flow, or vertical-axis, turbines are a promising technology for capturing kinetic energy in wind or flowing water and their inherently unsteady fluid mechanics present unique opportunities for control optimization of individual rotors or arrays. To explore the potential for beneficial interactions between turbines in an array, as well as to characterize important cycle-to-cycle variations, coherent structures in the wake of a single two-bladed cross-flow turbine are examined using planar stereo particle image velocimetry in a water channel experiment. There are three main objectives in the present work. First, the mean wake structure of this high chord-to-radius ratio rotor is described, compared with previous studies, and a simple explanation for observed wake deflection is presented. Second, the unsteady flow is then analysed via the triple decomposition, with the periodic component extracted using a combination of traditional techniques and a novel implementation of the optimized dynamic mode decomposition. The latter method is shown to outperform conditional averaging and Fourier methods, as well as uncover frequencies suggesting a transition to bluff-body shedding in the far wake. Third, vorticity and finite-time Lyapunov exponents are then employed to further analyse the oscillatory wake component. Vortex streets on both sides of the wake are identified, and their formation mechanisms and effects on the mean flow are discussed. Strong axial (vertical) flow is observed in vortical structures shed on the retreating side of the rotor where the blades travel downstream. Time-resolved tracking of these vortices is performed, which demonstrates that vortex trajectories have significant rotation-to-rotation variation within one diameter downstream. This variability suggests it would be challenging to harness or avoid such structures at greater downstream distances.

Existence of multiple noise-induced transitions in Lasota-Mackey maps.

We prove the existence of multiple noise-induced transitions in the Lasota-Mackey map, which is a class of one-dimensional random dynamical system with additive noise. The result is achieved with the help of rigorous computer assisted estimates. We first approximate the stationary distribution of the random dynamical system and then compute certified error intervals for the Lyapunov exponent. We find that the sign of the Lyapunov exponent changes at least three times when increasing the noise amplitude. We also show numerical evidence that the standard non-rigorous numerical approximation by finite-time Lyapunov exponent is valid with our model for a sufficiently large number of iterations. Our method is expected to work for a broad class of nonlinear stochastic phenomena.