Time-dependent Polynomial Research Articles

Lagrange stability for a class of impulsive Duffing equation with low regularity

We discuss the Lagrange stability for a class of impulsive Duffing equation with time-dependent polynomial potentials. More precisely, we prove that under suitable impulses, all the solutions of the impulsive Duffing equation (with low regularity in time) are bounded for all time and that there are many (positive Lebesgue measure) quasi-periodic solutions clustering at infinity.

Emergence of polynomial external potentials in solitonic hierarchies: Applications to the nonisospectral LPDE model

Inspired by the work of Akira Hasegawa, recently published in Optik 279 (2023) 170769, we set ourselves the task of explaining in detail how external nonstationary polynomial potentials appear in the nonisospectral NLSE hierarchy. We consider possible applications for observing solitonic regimes of propagation of ultrashort laser pulses in optical fibers. Our main goal is pedagogical, and our central point is the detailed derivation of new equations resulting from the application of the concept of nonautonomous solitons. It is revealed that complex potentials arise as polynomial series of the fourth degree of the spatial coordinate Uext(x,t)=α0(t)+α1(t)x+α2(t)x2+α3(t)x3+α4(t)x4, if the derivative of the spectral parameter is also expanded into a polynomial series of the spectral parameter up to the second order Λt(t)=λ0t+λ1(t)Λ+λ2(t)Λ2. It is predicted that new fourth-order nonlinear models arise if and only if the NLSE operator, the mKdV complex operator, and the fourth-order LPDE operators are connected by the extended Hirota constraint. We focus on exact analytical solutions and basic dynamic regimes of formation and interaction of new phase-modulated nonautonomous solitons of the nonisospectral LPDE model with fourth-order and time-dependent polynomial potentials.

A multi-agent approach to the impact of epidemic spreading on commercial activities

We introduce a class of one-dimensional linear kinetic equations of Boltzmann and Fokker–Planck type, describing the dynamics of the amount of consumer spending transactions of a multi-agent society, that, in the presence of social distancing laws consequent to an epidemic spreading, can reduce the volume of a consistent part of economic commercial activities requiring the presence. At the Boltzmann level, the microscopic variation of the number of transactions around a desired commercial target, is built up by joining a random risk with a deterministic change. Both the risk and target value parameters are here dependent of the number of social contacts of the multi-agent system, which can vary in time in reason of the governments decisions to control the epidemic spreading by introducing restrictions on the social activities. In the asymptotics of grazing interactions, the solution density of the Boltzmann-type kinetic equation is shown to converge towards the solution of a Fokker–Planck-type equation with variable coefficients of diffusion and drift, characterized by the details of the elementary interaction. The economic description is then coupled with the evolution equations of a new SIR-type compartmental epidemic system suitable to describe both the classical epidemic spreading and the social contacts evolution in dependence of the multi-agent social heterogeneity. The (local in time) steady states of the statistical distribution of the amount of economic transactions predicted by the Fokker–Planck equation are shown to be inverse Gamma densities, with time-dependent polynomial tails at infinity which characterize the consequences of the control policies on the time evolution of the Pareto index.

Private Trajectory Intersection Testing: Is Garbled Circuit Better than Custom Protocols?

In this paper, two protocols are presented for private intersection detection of two moving objects’ trajectories. Assuming that the movement trajectory of an object can be described by a time-dependent polynomial function, the problem of finding the intersection points is simplified to the problem of finding the common roots of the corresponding polynomials. Thereafter, GrObner Basis is used to design a novel secure protocol of finding the common roots of the polynomials. Another protocol is also designed based on the distance computation of two trajectories’ curves. Moreover, we present the complexity analysis of the protocol for private trajectory intersection testing of two moving objects, which is based on GrObner Basis. Then, we compare its complexity by the garbled circuit-based protocol for Euclidean Distance Computation of l points. We also prove the security of our proposed protocol, which is based on the distance computation of two curves. Keywords: Private Trajectory Intersection Testing, GrObner Basis, Distance Computation, Complexity , Garbled Circuit, Euclidean Distance

Solving Time-dependent reliability-based design optimization by adaptive differential evolution algorithm and time-dependent polynomial chaos expansions (ADE-T-PCE)

It remains a great many of challenge to explore time-dependent reliability-based design and optimization problems, owing to the high computational cost of time-dependent reliability assessment and analysis, the difficulties to life-cycle reliability modelling. To address these issues, ADE-T-PCE, a methodology incorporating an adaptive differential evolution algorithm and an enhanced instantaneous time-dependent polynomial chaos expansion is proposed. The polynomial chaos expression (PCE) method was improved based on time-domain extension and principal component selection partial least squares, the life-cycle reliability model is formed. Consequently, dynamic niche Pareto optimal solution selection strategy, particle variations and optimal solution screening strategy, number and strategy of particle variation for the differential evolution algorithm were proposed to meet demands for multi-object time-dependent reliability design optimization. The effectiveness of proposed method in this study was verified through an electromagnetic relay in hybrid vehicle.

1-d Quantum harmonic oscillator with time quasi-periodic quadratic perturbation: Reducibility and growth of Sobolev norms

For a family of 1-d quantum harmonic oscillators with a perturbation which is C2 parametrized by E∈I⊂R and quadratic on x and −i∂x with coefficients quasi-periodically depending on time t, we show the reducibility (i.e., conjugation to time-independent) for a.e. E. As an application of reducibility, we describe the behaviors of solutions in Sobolev space:•Boundedness w.r.t. t is always true for “most” E∈I.•For “generic” time-dependent perturbation, polynomial growth and exponential growth to infinity w.r.t. t occur for E in a “small” part of I. Concrete examples are given for which the growths of Sobolev norm do occur.

Solving Burnup Equations by Numerical Inversion of the Laplace Transform Using Padé Rational Approximation

Burnup calculations play a very important role in nuclear reactor design and analysis, and solving burnup equations is an essential topic in burnup calculations. In the last decade, several high-accuracy methods, mainly including the Chebyshev rational approximation method (CRAM), the quadrature-based rational approximation method, the Laguerre polynomial approximation method, and the mini-max polynomial approximation method, have been proposed to solve the burnup equations. Although these methods have been demonstrated to be quite successful in the burnup calculations, limitations still exist in some cases, one of which is that the accuracy becomes compromised when treating the time-dependent polynomial external feed rate. In this work, a new method called the Padé rational approximation method (PRAM) is proposed. Without directly approximating the matrix exponential, this new method is derived by using the Padé rational function to approximate the scalar exponential function in the formula of the inverse Laplace transform of burnup equations. Several test cases are carried out to verify the proposed new method. The high accuracy of the PRAM is validated by comparing the numerical results with the high-precision reference solutions. Against CRAM, PRAM is significantly superior in handling the burnup equations with time-dependent polynomial external feed rates and is much more efficient in improving the accuracy by using substeps, which demonstrates that PRAM is the attractive method for burnup calculations.

Reachability analysis using dissipation inequalities for uncertain nonlinear systems

We propose a method to outer bound forward reachable sets on finite horizons for uncertain nonlinear systems with polynomial dynamics. This method makes use of time-dependent polynomial storage functions that satisfy appropriate dissipation inequalities that account for time-varying uncertain parameters, L2 disturbances, and perturbations Δ characterized by integral quadratic constraints (IQCs) with both hard and soft factorizations. The use of IQCs in forward reachability analysis allows for various types of uncertainty, including unmodeled dynamics. The generalized S-procedure and Sum-of-Squares techniques are used to derive algorithms with the goal of finding the tightest outer bound with a desired shape. Both pedagogical and practically motivated examples are presented, including a 7-state F-18 aircraft model.

Long duration response evaluation of linear structural system with random system properties using time dependent polynomial chaos

Polynomial Chaos (PC) is one of the popular methods for approximate evaluation of responses of stochastic structural mechanics problems. In the case of a structural system under dynamic loading, the PC method is combined with a time integration scheme. The PC method with Galerkin projection converts the stochastic PDE to a set of simultaneous deterministic PDE, which can be solved using any standard time integration scheme. However, the convergence of responses is known to fail for long duration time integration problem due to the development of unacceptable error in calculated response. The probability density functions of the responses are not of constant type as the time progresses. However, in the case of generalized PC (gPC), these are approximated using initially considered gPC, thus loses its efficiency. Time-dependent general polynomial chaos (TDgPC) can be considered to overcome these difficulties, where polynomials are generated on the fly to match with the pdf of the responses. The present study considered an investigation of structural mechanics responses with both Gaussian and non-Gaussian random system properties under dynamic loading using TDgPC. An adaptive updation scheme based on the RMS value of the coefficients of PC expansion is investigated for a proper updation of PC expansion. The curse of dimensionality of PC expansion is addressed by considering only the dominant components of the responses evaluated using KL expansion. Further, starting with lower-order PC, the order of PC expansion is increased only after the first updation, thus reducing computational complexities.

Polynomial Chaos-Based Flight Control Optimization with Guaranteed Probabilistic Performance

A probabilistic performance-oriented controller design approach based on polynomial chaos expansion and optimization is proposed for flight dynamic systems. Unlike robust control techniques where uncertainties are conservatively handled, the proposed method aims at propagating uncertainties effectively and optimizing control parameters to satisfy the probabilistic requirements directly. To achieve this, the sensitivities of violation probabilities are evaluated by the expansion coefficients and the fourth moment method for reliability analysis, after which an optimization that minimizes failure probability under chance constraints is conducted. Afterward, a time-dependent polynomial chaos expansion is performed to validate the results. With this approach, the failure probability is reduced while guaranteeing the closed-loop performance, thus increasing the safety margin. Simulations are carried out on a longitudinal model subject to uncertain parameters to demonstrate the effectiveness of this approach.

Bundle Adjustment of a Time-Sequential Spectral Camera Using Polynomial Models

Lightweight hyperspectral cameras based on frame geometry have been used for several applications in unmanned aerial vehicles (UAVs). The camera used in this investigation is based on a tunable Fabry-Pérot interferometer (FPI) and works on the time-sequential principle for band acquisition. Due to this feature, when collecting images in movement, hypercubes are generated with unregistered bands, and consequently, the individual bands in each hypercube have different exterior orientation parameters (EOPs), which must be estimated by an image orientation procedure. The objective of this paper was to develop an approach for bundle block adjustment (BBA) using time-dependent polynomial models for simultaneous image orientation of all bands. The procedure consists of using a minimum number of bands to estimate the polynomial parameters. From the estimated polynomial parameters, the EOPs (position and attitude) of all bands can be determined. In tests with backprojecting ground points to interpolated bands, the average error was smaller than 1 pixel, which indicates excellent potential for orthomosaic generation. The polynomial technique was also compared to the conventional BBA. The discrepancies assessed at checkpoints indicated a similar error for both techniques, which were approximately less than the pixel size in planimetry and less than 2.8 times the pixel size in height. Therefore, the results show that the spectral band orientation can be performed with the proposed technique, assuming that the trajectory during the cube can be modeled with the polynomial model, which reduces the workload while achieving the same accuracy as conventional BBA for all bands.

Time-dependent polynomials with one multiple root and new solvable dynamical systems

A time-dependent monic polynomial in the z variable with N distinct roots such that exactly one root has multiplicity m¯≥2 is considered. For k = 1, 2, the kth derivatives of the N roots are expressed in terms of the derivatives of order j ≤ k of the first N coefficients of the polynomial and of the derivatives of order j ≤ k − 1 of the roots themselves. These relations are utilized to construct new classes of algebraically solvable first order systems of ordinary differential equations as well as N-body problems. Multiple examples of solvable isochronous (all solutions are periodic with the same period) 2- and 3-body problems are provided.

Polynomials with multiple zeros and solvable dynamical systems including models in the plane with polynomial interactions

The interplay among the time-evolution of the coefficients ymt and the zeros xnt of a generic time-dependent (monic) polynomial provides a convenient tool to identify certain classes of solvable dynamical systems. Recently, this tool has been extended to the case of nongeneric polynomials characterized by the presence, for all time, of a single double zero; subsequently, significant progress has been made to extend this finding to the case of polynomials featuring a single zero of arbitrary multiplicity. In this paper, we introduce an approach suitable to deal with the most general case, i.e., that of a nongeneric time-dependent polynomial with an arbitrary number of zeros each of which features, for all time, an arbitrary (time-independent) multiplicity. We then focus on the special case of a polynomial of degree 4 featuring only 2 different zeros, and by using a recently introduced additional twist of this approach, we thereby identify many new classes of solvable dynamical systems of the following type: ẋn=Pnx1,x2, n=1,2 , with Pnx1,x2 being two polynomials in the two variables x1t and x2t.

PERFORMANCE EVALUATION OF SEQUENTIAL BAND ORIENTATION BY POLYNOMIAL MODELS IN HYPERSPECTRAL CUBES COLLECTED WITH UAV

Abstract. A study on orientation of hyperspectral band cubes acquired with frame camera is presented in this paper. The camera technology is based on a tuneable Fabry-Perot Interferometer (FPI) and captures cubes of images sequentially using two sensors. However, the bands are not recorded at the same instant, which results different exterior orientation parameters (EOPs) for each image band. A technique based on polynomial model is assessed, which determines the EOPs within the hypercube from few sample bands, since a large number of bands are generated. Experiments were performed to assess the feasibility of using the polynomial technique. An analysis of the UAV trajectory was performed and the results of the polynomial technique were compared with those obtained by a conventional bundle adjustment. The trials showed that the results of both techniques were comparable, indicating that the time-dependent polynomial model can be used to estimate the EOPs of all spectral bands, without requiring a bundle adjustment including all bands. The accuracy of the block adjustment was analysed based on the discrepancies obtained from independent checkpoints. The root mean square error (RMSE) was calculated and showed an accuracy of approximately 1 GSD in planimetry and 1.5 GSD in altimetry. This accurate result is important because the proposed technique can significantly reduce the processing workload.

A time-variant extreme-value event evolution method for time-variant reliability analysis

Abstract In this paper, we propose a time-variant extreme-value event evolution method (TEEM). The time-evolution process of extreme-value event is firstly proposed in this paper. And by solving it, we can obtain the time-variant reliability of arbitrary time interval and arbitrary failure threshold. In this method, the random process in limit-state function is firstly expanded by an improved orthogonal series expansion method (iOSE). Second, we introduce the idea of extreme-value event to describe the time-variant reliability problem. And by discretizing the time domain, we can obtain a series of extreme-value events. The moments of extreme-value event in every discrete time interval will be solved by the integration of Broyden–Fletcher–Goldfarb–Shanno (BFGS) method and univariate dimension reduction method (UDRM). Third, a time-dependent polynomial chaos expansion method (t-PCE) is proposed to simulate the extreme-value event's time-evolution process, and it will be simulated as a function in terms of a standard normal variable and time. Finally, Monte Carlo simulation (MCS) is adopted to sample the standard normal variable to obtain the time-variant reliability of arbitrary failure threshold and time interval. Three numerical examples are investigated to demonstrate the effectiveness of the proposed methods.

Exact linearisation of the Van der Pol oscillator with fundamental linear differential equations for the tracking error

Abstract This paper deals with the computation of the exact linearisation of the Van der Pol oscillator with fundamental linear differential equations for the tracking error. The Van der Pol oscillator is proven as a highly nonlinear system and it is used as an application example for the analysis of the nonlinear tracking controller with and without disturbances. Tracking control achieves precise tracking performance of a system output for a desired trajectory and incorporates additionally the disturbance rejection. The simulation results show the precise tracking performance of the system output for a desired reference trajectory. The reference trajectory in the simulation study is chosen as a time dependent polynomial with the dependence on the relative degree. Two different disturbance signals show the well adjusted disturbance rejection and the advantage of using the disturbances as measurement signals in the tracking control law.

QoE-Driven Resource Allocation Optimized for Uplink Delivery of Delay-Sensitive VR Video Over Cellular Network

Uploading virtual reality (VR) video over cellular networks is expected to boom in the near future, as general consumers could generate the high-quality VR videos with portable 360-degree cameras and are willing to share with others. Consequently, the concerns of uplink bandwidth and delay arose for current popular technology of tile-based VR video streaming, which requires high-quality video to transcode into multiple representations for further adaptive streaming. Motivated by this, we proposed a novel scheme for uplink delivery of tile-based VR video over cellular networks, in which encoding bit rate of each tile is determined by the uplink resource allocation (RA), and the quality of content (QoC) contribution of each tile and channel quality of user equipments (UEs) are jointly considered during RA. Moreover, the RA problem is formulated as a frequency and time dependent non-deterministic polynomial (NP)-hard problem. Furthermore, we propose three algorithms to explore solving the RA problem. The simulation results show that the proposed approximate convex algorithm with low-complexity can achieve higher utility, i.e., higher total quality of experience (QoE) for viewers.

Lagrange stability for impulsive Duffing equations

This work discusses the boundedness of solutions for impulsive Duffing equation with time-dependent polynomial potentials. By KAM theorem, we prove that all solutions of the Duffing equation with low regularity in time undergoing suitable impulses are bounded for all time and that there are many (positive Lebesgue measure) quasi-periodic solutions clustering at infinity. This result extends some well-known results on Duffing equations to impulsive Duffing equations.

Time-Dependent Polynomials with One Double Root, and Related New Solvable Systems of Nonlinear Evolution Equations

Recently new solvable systems of nonlinear evolution equations—including ODEs, PDEs and systems with discrete time—have been introduced. These findings are based on certain convenient formulas expressing the kth time-derivative of a root of a time-dependent monic polynomial in terms of the kth time-derivative of the coefficients of the same polynomial and of the roots of the same polynomial as well as their time-derivatives of order less than k. These findings were restricted to the case of generic polynomials without any multiple root. In this paper some of these findings—those for $$k=1$$ and $$k=2$$ —are extended to polynomials featuring one double root; and a few representative examples are reported of new solvable systems of nonlinear evolution equations.

Zeros of rational functions and solvable nonlinear evolution equations

Recently a convenient technique to relate the time evolution of the N zeros of a time-dependent polynomial pNz;t of degree N in the (complex) variable z to the time evolution of its N coefficients has been exploited to identify large classes of dynamical systems solvable by algebraic operations. These models also include N-body problems that evolve in the complex plane (or, equivalently, in the real Cartesian plane) according to systems of nonlinearly-coupled equations of motion of Newtonian type (“accelerations equal forces”). Many of these models feature remarkable properties: for instance, they are Hamiltonian and integrable and/or multiply periodic or even isochronous (featuring completely periodic solutions with a fixed period largely independent of the initial data), or asymptotically isochronous (featuring isochrony only up to corrections vanishing in the remote future). In this paper, an analogous technique is introduced that focuses instead on the time evolution of the N zeros of an appropriate class of time-dependent rational functions RNz;t, thereby opening large vistas of new dynamical systems solvable by algebraic operations and featuring remarkable properties. A few examples are reported.