Simplified Dynamical System Research Articles

Hybrid Method of Uncertainty Propagation for Near-Earth Conjunction Analysis

In this research, several existing semi-analytical dynamics and uncertainty propagation techniques are combined with conjunction models to achieve fast and accurate probability of collision results. Initial Gaussian uncertainty associated with two objects is split into smaller Gaussian distributions using Gaussian mixture models to achieve mixture components that will maintain linearity over longer propagation times. These mixture components are propagated forward using second-order state transition tensors that can capture the nonlinearity in the propagation accurately by taking into account the desired dynamics. The dynamic solution and these tensors are computed using the Deprit–Lie averaging approach, including transformations between mean and osculating states, which accounts for perturbations due to solar radiation pressure and J2. This simplified dynamic system allows fast and accurate propagation by combining the speed of propagation with averaged dynamics and the accuracy of short-period variation addition. Combined, these mathematical tools are used to propagate the object’s uncertainties forward. The final distributions are compared using analytical conjunction methods to compute the probability of collision, which is then compared to the Monte Carlo result to confirm the method’s validity.

Characteristics and Controlling Factors of Rapid Weakening of Tropical Cyclones After Reaching Their Intensity Peaks Over the Western North Pacific

AbstractThe sudden transition to rapid weakening immediately after a tropical cyclone (TC) reaches its intensity peak (hereafter TP‐RW) is a challenge to operational TC intensity forecasts, but the characteristics and the controlling factors of the TP‐RW have not been investigated so far. In this study, the main characteristics of the TP‐RW events over the western North Pacific (WNP) and the corresponding key controlling factors and their relative importance are analyzed and quantified. Results show that about 2.5 TP‐RW events occurred each year, which corresponds to about 10% of total TCs over the WNP during 1982–2018. The TP‐RW events occurred in all months but peaked in fall and were the most likely to occur for TCs with intensities between 90 and 145 kt. The sea surface temperature, the TC maximum potential intensity and translational speed, the environmental vertical wind shear and relative humidity are key to the TP‐RW. Four distinct clusters of the TP‐RW events are identified based on their tracks using the K‐means clustering algorithm. The relative contribution of each of the identified key factors to each cluster is quantified using the growth rate from the simplified dynamical system for TC intensity prediction developed by DeMaria. The reversal of the growth rate from positive to negative values is shown to be a good indicator of the TP‐RW processes. Different clusters are predominantly controlled by different key factors. The findings from this study can help better understand TC intensity changes and improve TC intensity forecasts for such kind of RW.

Long-term degradation estimation of wind turbine drive-train under a gain-scheduling control strategy according to the weather conditions

This paper proposes a methodology to analyze the long-term degradation of a wind turbine drive-train using a gain-scheduling control strategy considering the variation in the wind conditions. The degradation in the transmission is modeled through a simplified dynamical system based on contact mechanics, employing the dissipated energy as an indicator of the degradation in the shaft. In addition, a gain-scheduling control strategy is considered to assign a control gain according to the wind flow conditions. The long-term simulation is based on the accelerated degradation due to turbulent wind compared to the laminar flow. Numerical experiments were considered to illustrate the proposed approach, assuming a 2 MW variable speed-fixed pitch turbine with a horizontal axis and fixed gearbox.

Comparison of digestibility and potential allergenicity of raw shrimp (Litopenaeus vannamei) extracts in static and dynamic digestion systems

In this work, a simplified dynamic digestion system was developed, and used for comparing the digestibility and potential allergenicity of raw shrimp extracts (RSE) in static and dynamic digestion systems. Protein hydrolysis was analyzed by electrophoresis, and the potential allergenicity was reflected in IgG/IgE binding ability and activation of basophils. In comparison with static digestion, protein hydrolysis indicated different kinetic behaviors, especially tropomyosin (TM) showed better digestion stability during dynamic digestion. The potential allergenicity of RSE exhibited different changing trends with digestion in the two systems. However, the apparent molecular weight (Mw) of immune fragments (>11 kDa) showed good approximation, and the IgE-binding fragment near 70 kDa revealed outstanding digestion stability than primordial protein in both systems. In conclusion, the dynamic conditions had a significant impact on the assessment of the persistence and potential allergenicity of digestion-resistant allergens, while the apparent Mw of IgG/IgE binding hydrolysate was not affected.

Brain as a Dynamical System

This article aims at understanding the essential dynamical characteristics of the brain by modeling the brain as a simplified dynamical control system. Although the brain is a highly complex system, such a simplified model provides us insights into its essential features. Using a control system model, the first half of this article describes the dynamics of a system comprising the dorsolateral prefrontal cortex and striatum with dopaminergic modulation and argue its relevance to the functions of the brain with schizophrenia. The second half of this article introduces a model-based analysis of molecular imaging data.

Optimization of Microalgae Selection: Highlighting turnpike features

We consider a simplified dynamical system to represent the competition between two species of microalgae in a chemostat. The model is derived from the classical Droop model, assuming the substrate level can be directly controlled. An optimal control problem (OCP) is formulated, with the objective of finding the substrate concentration which would maximize the fraction of the species of interest over a fixed finite time-window. We characterize the substrate-based control strategy that steers the model trajectories and achieves species separation. Our objective is to highlight through a numerical optimal-synthesis - based on direct optimal control tools - the existence of a turnpike feature that appears in the optimal control law as well as in the optimal model trajectories and their co-states.

Robust Guidance Strategy for Target Circulation by Controlled UAV

A controlled fixed-wing unmanned aerial vehicle is modeled as a simplified low-dimensional nonlinear dynamical system with additive disturbances. The guidance strategy presented, considering simultaneously inputs and states constraints, is shown to be robust to additive perturbations representing norm-bounded uncertainties associated with modeling errors resulting from the lower order model approximation, together with unknown but limited atmospheric disturbances. Mathematical proofs are provided ensuring robustness in achieving target circulation with a minimum prescribed accuracy.

Can a Simple Dynamical System Describe the Interplay between Drag and Buoyancy in Terrain-Induced Canopy Flows?

Abstract Under nonneutral stratification and in the presence of topography, the dynamics of turbulent flow within a canopy is not yet completely understood. This has, among other consequences, serious implications for the measurement of surface–atmosphere exchange by means of eddy covariance: for example, the measurement of carbon dioxide fluxes is strongly influenced if drainage flows occur during night, when the flow within the canopy decouples from the flow aloft. An improved physical understanding of the behavior of scalars under canopy turbulence in complex terrain is urgently needed. In the present work, the authors investigate the dynamics of turbulent flow within sloped canopies, focusing on the slope wind and potential temperature. The authors concentrate on the presence of oscillatory behavior in the flow variables in terms of switching of flow regimes by conducting linear stability analysis. The authors revisit and correct the simplified theory that exists in the literature, which is based on the interplay between the drag force and the buoyancy. The authors find that the simplified description of this dynamical system cannot exhibit the observed richness of the dynamics. To augment the simplified dynamical system’s analysis, the authors make use of large-eddy simulation of a three-dimensional hill covered by a homogeneous forest and analyze the phase synchronization behavior of the buoyancy and drag forces in the momentum budget to explore the turbulent dynamics in more detail.

Hybrid Method for Uncertainty Propagation of Orbital Motion

The efficient and accurate representation of uncertainty for orbiting objects under nonlinear dynamics is a topic of great interest for space situational awareness. This paper presents a hybrid method that enables an efficient propagation of uncertainty while maintaining accuracy. The method proposed in this research is defined by combining two ideas. First, the second-order semianalytic solutions are derived, based on a Deprit–Lie transformation, in order to describe a perturbed orbiting motion due to central-body oblateness and solar radiation pressure. From this theory, a simplified dynamical system is defined by eliminating the short-period variation. Next, this simplified dynamical system is expanded by using the state transition tensors, which can directly map uncertainty while capturing the nonlinearity of higher-order dynamics. In this paper, it is verified that the hybrid method propagates uncertainty accurately by comparing uncertainty from Monte Carlo methods through statistical approaches. It is also shown that the method reduces the computational burden of the Monte Carlo methods to less than 0.002%.

The Relationship between Sea Surface Temperature and Maximum Intensification Rate of Tropical Cyclones in the North Atlantic

Abstract An empirical relationship between sea surface temperature (SST) and the maximum potential intensification rate (MPIR) of tropical cyclones (TCs) over the North Atlantic has been developed based on the best-track TC data and the observed SST during 1988–2014. Similar to the empirical relationship between SST and the maximum potential intensity of TCs previously documented, results from this study show a nonlinear increasing trend of the MPIR with increasing SST, with a more rapid increasing trend when SST is higher than 27°C. Further analyses indicate that about 28% of intensifying TCs over the North Atlantic reached 50% of their MPIR and only 7% reached 80% of their MPIR at the time when they were at their lifetime maximum intensification rates. Moreover, a TC tended to have a larger intensification rate when it was located in regions with higher SST and lower vertical wind shear (VWS). This indicates that although the MPIR–SST relationship is much stronger than that for the IR rate versus SST for most TCs, the actual intensification rate of a TC is determined by not only the SST but also other environmental effects, such as VWS. Additional results from a simplified dynamical system previously developed for TC intensity prediction suggest an SST-dependent TC MPIR, similar to that fitted from observations. However, the MPIR obtained from the observational fitting seems to underestimate the MPIR in regions with low SST at higher latitudes where VWS is often large. Nevertheless, this study provides the observational evidence for the existence of the MPIR for TCs.

The all-source Green’s function (ASGF) and its applications to storm surge modeling, part II: from the ASGF convolution to forcing data compression and a regression model

This study first validates the ASGS algorithm developed in part I with an analytical solution in a simplified dynamical system and with a real storm surge event. It then assesses the computational efficiency by the ASGF method compared to the traditional method. By analyzing a realistic case, the ASGF method is shown to be three orders of magnitude more computationally efficient than the traditional method. Using the singular value decomposition (SVD) and the fast Fourier transform and its inverse (FFT/IFFT), this study further demonstrates how to compress atmospheric forcing data and how to cast the ASGF convolution as a simple and efficient regression model for data assimilation. When tested with the real storm surge event, the output from the regression model can account for 98 % of the observed variance.

Long-term evolution of space debris under the $$J_2$$ J 2 effect, the solar radiation pressure and the solar and lunar perturbations

The aim of this paper is the development of a model to propagate space debris in the geostationary ring considering the \(J_2\) effect due to the Earth oblateness, the Sun and Moon perturbations, and the solar radiation pressure. We justify the importance of considering the \(J_2\) effect when propagating space debris independently of the ratio A / m for short and long-term propagation. We study the role of the Sun and the Moon in the period and amplitude of the inclination for different values of A / m. Thanks to the Hamiltonian formulation of the problem and the use of Poincare’s variables it is possible to express the evolution of the space debris through a simplified dynamical system. We test and validate our obtained analytical solutions with the numerical ones, computed with a powerful integrator named NIMASTEP. We analyse the improvements obtained when we include the \(J_2\) effect and the third body perturbations by a rigorous comparison with a previous model, which only considers the solar radiation pressure. Finally, we study the effect of the area-to-mass ratio on short and long-term propagation.

Effect of Dynamical Accuracy for Uncertainty Propagation of Perturbed Keplerian Motion

This paper poses the following question: How precisely must an Earth orbiter’s dynamics be mapped to accurately capture the statistical evolution of its uncertainty? This question is addressed by using a simplified dynamical system that replaces the short-period variations with constants. The systems are defined and applied to the propagation of uncertainty for a non-Keplerian orbit. The simplified dynamical system is defined with two different types of approximate solutions: one is based on the Brouwer–Lyddane theory, and another is based on Lagrange planetary equations. The simplified dynamical system allows exploration of the relationship between dynamical model precision and uncertainty propagation accuracy, as well gained insight into effects due to individual variations: that is, secular, short, and long periods. Accuracy of propagation of uncertainty with the simplified dynamical system is verified with statistical methods for dynamical models including perturbations of and third-body gravity. In conclusion, the simplified dynamical system is capable of propagating uncertainty accurately and efficiently, as well as discovering that the secular variations are dominant in uncertainty mapping.

A Statistical Analysis on the Dependence of Tropical Cyclone Intensification Rate on the Storm Intensity and Size in the North Atlantic

AbstractThe dependence of tropical cyclone (TC) intensification rate IR on storm intensity and size was statistically analyzed for North Atlantic TCs during 1988–2012. The results show that IR is positively (negatively) correlated with storm intensity (the maximum sustained near-surface wind speed Vmax) when Vmax is below (above) 70–80 knots (kt; 1 kt = 0.51 m s−1), and negatively correlated with storm size in terms of the radius of maximum wind (RMW), the average radius of gale-force wind (AR34), and the outer-core wind skirt parameter DR34 (=AR34 − RMW). The turning point for Vmax of 70–80 kt is explained as a balance between the potential intensification and the maximum potential intensity (MPI). The highest IR occurs for Vmax = 80 kt, RMW ≤ 40 km, and AR34 = DR34 = 150 km. The high frequency of occurrence of intensifying TCs occurs for Vmax ≤ 80 kt and RMW between 20 and 60 km, AR34 ≤ 200 km, and DR34 ≤ 150 km. Rapid intensification (RI) often occurs in a relatively narrow parameter space in storm intensity and both inner- and outer-core sizes. In addition, a theoretical basis for the intensity dependency has also been provided based on a previously constructed simplified dynamical system for TC intensity prediction.

A Simplified Gravitational Model for Texture Analysis

Textures are among the most important features in the field of image analysis. This paper presents an innovative methodology to extract information from them, converting an image into a simplified dynamical system in gravitational collapse process whose collapsing states are described by using the lacunarity method. The paper compares the proposed approach to other classical methods using Brodatz textures and a second texture database as benchmark. The best classification results using the standard parameters of the method were 97.00 % and 54.10 % of success rate (percentage of samples correctly classified) for both databases, respectively. These results prove that the presented approach is an efficient tool for texture analysis.

Comparing Three Lane Merging Schemes for Short-Term Work Zones: A Simulation Study

Traffic safety and mobility of roadway work zones have been considered to be one of the major concerns in highway traffic safety and operations in Florida. Dynamic lane merging (DLM) systems—ITS-based lane management technology—were introduced by several states in an attempt to enhance both safety and mobility of roadway work zones. Two forms of lane merging, namely, the early merge and the late merge were designed to advise drivers on definite merging locations. Up to date, there are no studies that contrast both merging schemes under matching work zone settings. This study simulates a two-to-one work zone lane closure configuration under three different Maintenance of Traffic (MOT) plans in VISSIM. The first MOT is the conventional plans used in Florida’s work zones, the second MOT is a simplified dynamic early merging system (early SDLMS), and the third MOT is a simplified dynamic late merging systems (late SDLMSs). Field data was collected to calibrate and validate the simulation models. Simulation results indicated that overall, under different levels of drivers’ compliance rate and different percentages of trucks in the traffic composition, the early SLDMS outperformed the conventional MOT and the late SDLMS in terms of travel times and throughputs.

A simplified linguistic information feedback-based dynamical fuzzy system

Inspired by the linguistic information feedback-based dynamical fuzzy system (LIFDFS) recently proposed by the authors, we present a simplified LIFDFS (S-LIFDFS) model in this paper, which has a simpler linguistic information feedback structure. Compared with the LIFDFS, the S-LIFDFS can offer us with a considerably reduced computational complexity. We first give a detailed description of its underlying principle. Based on the gradient descent method, an adaptive learning algorithm for the feedback parameters is next derived. We also discuss the application of this S-LIFDFS in time series prediction. Three evaluation examples including prediction of two artificial time sequences and the well-known Box–Jenkins gas furnace data are demonstrated here. Simulation results illustrate that with a compact structure, our S-LIFDFS can still retain the advantage of inherent dynamics of linguistic information feedback and is, therefore, well suited for handling temporal problems like prediction, modeling, and control.

Existence of oscillatory solutions in longitudinal flight dynamics

Abstract By means of coincidence degree theory and Mawhin's continuation theorem, a theoretical proof is given for the existence of oscillatory solutions of the simplified dynamical system which governs the motion around the center of mass in a longitudinal flight with constant forward velocity of a rigid aircraft, when the automatic flight control system is decoupled.

A Simplified Dynamical System for Tropical Cyclone Intensity Prediction

Abstract A simplified dynamical system for tropical cyclone intensity prediction based on a logistic growth equation (LGE) is developed. The time tendency of the maximum sustained surface winds is proportional to the sum of two terms: a growth term and a term that limits the maximum wind to an upper bound. The maximum wind evolution over land is determined by an empirical inland wind decay formula. The LGE contains four free parameters, which are the time-dependent growth rate and maximum potential intensity (MPI), and two constants that determine how quickly the intensity relaxes toward the MPI. The MPI is estimated from an empirical formula as a function of sea surface temperature and storm translational speed. The adjoint of the LGE provides a method for finding the other three free parameters to make the predictions as close as possible to the National Hurricane Center best-track intensities. The growth rate is assumed to be a linear function of the vertical shear (S), a convective instability parameter (C) determined from an entraining plume, and their product, where both S and C use global model fields as input. This assumption reduces the parameter estimation problem to the selection of six constants. Results show that the LGE optimized for the full life cycle of individual storms can very accurately simulate the intensity variations out to as long as 15 days. For intensity prediction, single values of the six constants are found by fitting the model to more than 2400 Atlantic forecasts from 2001 to 2006. Results show that the observed intensity variations can be fit more accurately with the LGE than with the linear Statistical Hurricane Intensity Prediction Scheme (SHIPS) formulation, and with a much smaller number of constants. Results also show that LGE model solution (and some properties of real storms) can be explained by the evolution in the two-dimensional S–C phase space. Forecast and other applications of the LGE model are discussed.

Nonlinear dynamics of a simplified engine-propeller system

This paper presents a procedure for studying dynamical behaviors of a simplified engine-propeller dynamical system consisting of a number of bodies of plane motions. The equation of motion of the complex system is obtained using the Lagrange equation and solved numerically using the 4th order Runge–Kutta method. Various simulations were performed to investigate the transient and steady state behaviors of the multiple body system while taking into consideration the engine pressure pulsations, nonlinear inertia of moving bodies, and nonlinear aerodynamic load. Sub-harmonics and super harmonics in the steady state responses for different power and propeller pitch settings are obtained using the fast Fourier transform. Numerical simulations indicate that the 1.5 order is the dominant order of harmonics in the steady state oscillatory motion of the crankshaft. The findings and procedure presented in the paper are useful to the aerospace industry in certifying reciprocating engines and propellers. The crankshaft oscillatory velocities obtained from the simplified rigid body model are in good agreement with the experimental data for a SAITO-450 engine and a SOLO propeller at a 6″ pitch setting.