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Construction of quasi-periodic solutions of state-dependent delay differential equations by the parameterization method II: Analytic case

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JOURNAL OF DIFFERENTIAL EQUATIONS 2016年第3期261卷 2068-2108页

作者： He, Xiaolong de la Llave, Rafael Hunan Univ Coll Math & Econometr Changsha 410082 Hunan Peoples R China Georgia Inst Technol Sch Math 686 Cherry St Atlanta GA 30332 USA

We construct analytic quasi-periodic solutions of a state-dependent delay differential equation with quasi periodically forcing. We show that if we consider a family of problems that depends on one dimensional parameters (with some non-degeneracy conditions), there is a positive measure set Pi of parameters for which the system admits analytic quasi-periodic solutions. The main difficulty to be overcome is the appearance of small divisors and this is the reason why we need to exclude parameters. Our main result is proved by a Nash-Moser fast convergent method and is formulated in the a-posteriori format of numerical analysis. That is, given an approximate solution of a functional equation which satisfies some non-degeneracy conditions, we can find a true solution close to it. This is in sharp contrast with the finite regularity theory developed in [18]. We conjecture that the exclusion of parameters is a real phenomenon and not a technical difficulty. More precisely, for generic families of perturbations, the quasi-periodic solutions are only finitely differentiable in open sets in the complement of parameters set Pi. (C) 2016 Elsevier Inc. All rights reserved.

关键词： State-dependent delay parameterization method Analytic solution Foliation-preserving torus map KAM theory A-posteriori

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An Airfoil parameterization method for the Representation and Optimization of Wind Turbine Special Airfoil

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Journal of Thermal Science 2015年第2期24卷 99-108页

作者： LIU Yixiong YANG Ce SONG Xiancheng School of Mechanical Engineering Beijing Institute of Technology School of Energy and Power Engineering Beihang University

A new airfoil shape parameterization method is developed, which extended the Bezier curve to the generalized form with adjustable shape parameters. The local control parameters at airfoil leading and trailing edge regions are enhanced, where have significant effect on the aerodynamic performance of wind turbine. The results show this improved parameterization method has advantages in the fitting characteristics of geometry shape and aero- dynamic performance comparing with other three common airfoil parameterization methods. The new paramete- rization method is then applied to airfoil shape optimization for wind turbine using Genetic Algorithm （GA）, and the wind turbine special airfoil, DU93-W-210, is optimized to achieve the favorable C1/Cd at specified flow con- ditions. The aerodynamic characteristic of the optimum airfoil is obtained by solving the RANS equations in computational fluid dynamics （CFD） method, and the optimization convergence curves show that the new para- meterization method has good convergence rate in less number of generations comparing with other methods. It is concluded that the new method not only has well controllability and completeness in airfoil shape representation and provides more flexibility in expressing the airfoil geometry shape, but also is capable to find efficient and op- timal wind turbine airfoil. Additionally, it is shown that a suitable parameterization method is helpful for improv- ing the convergence rate of the optimization algorithm.

关键词： wind turbine airfoil parameterization method optimization

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Flow Map parameterization methods for Invariant Tori in Quasi-Periodic Hamiltonian Systems

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SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 2024年第1期23卷 127-166页

作者： Fernandez-Mora, Alvaro Haro, Alex Mondelo, J. M. Univ Barcelona Dept Matemat & Informat Gran Via 585 Barcelona 08007 Spain Univ Autonoma Barcelona CERES IEEC Bellaterra 08193 Barcelona Spain Ctr Recerca Matemat Edif CCampus Bellaterra Bellaterra 08193 Barcelona Spain Univ Autonoma Barcelona Dept Matemat Av Eix CentEdif C Bellaterra 08193 Barcelona Spain

The aim of this paper is to present a method to compute parameterizations of partially hyperbolic invariant tori and their invariant bundles in nonautonomous quasi -periodic Hamiltonian systems. We generalize flow map parameterization methods to the quasi -periodic setting. To this end, we introduce the notion of fiberwise isotropic tori and sketch definitions and results on fiberwise symplectic deformations and their moment maps. These constructs are vital to work in a suitable setting and lead to the proofs of ``magic cancellations"" that guarantee the existence of solutions of cohomological equations. We apply our algorithms in the elliptic restricted three body problem and compute nonresonant 3 -dimensional invariant tori and their invariant bundles around the L1 point.

关键词： invariant tori quasi-periodic Hamiltonian systems parameterization method ERTBP KAM

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Transfers to Earth-Moon triangular libration points by Sun-perturbed dynamics

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ADVANCES IN SPACE RESEARCH 2025年第3期75卷 2837-2855页

作者： Peng, Lei Liang, Yuying He, Xingji Beihang Univ Sch Astronaut Beijing 100191 Peoples R China Yunnan Police Coll Kunming 650223 Peoples R China

This paper is devoted to the construction of transfers to Earth- Moon triangular libration points based on their Sun-perturbed dynamics in the Sun-Earth-Moon system. In this bi-circular problem (B Moon triangular libration points with three remaining periodic substitutions. As the stable region, the invariant tori full of quasi the destination of triangular libration point transfers for a low st ation-keeping cost. However, the instability of the unstable periodic orbit is so mild that its stable and unstable manifolds are folded ar CP), the solar gravity perturbation breaks the stability of the Earth- orbits (two stable and one unstable) around them as their dynamical-periodic orbits around the two stable periodic orbits can be used as the authors approximate the stable and unstable manifolds of th from the Sun-perturbed Earth-Moon triangular libration points vi around triangular libration points. First, the transfers connecting t anew connection mechanism near L 4 and L 5 based on the N : M m skill to time-dependent situations. The quickest one is achieved by a 1:1 L 4-L5 heteroclinic-based connection with a TOF of less than 26.9 days at a fuel cost of 117.88 m/s. Note that such Sun-perturbed transfers cannot be produced by continuing any trajectory in the Earth-Moon circular restricted three-body problem but the continuation. ound the invariant tori surrounding the other two stable orbits. Thus, e unstable orbit through the parameterization method, which emerge cinity. They act as a bridge between outer space and the stable region he Earth's vicinity are constructed using the manifolds. Subsequently, ulti-loop transfers is developed by extending the conventional patching y can generate two-impulsive transfers in the ephemeris model by (c) 2024 COSPAR. Published by Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Triangular libration points Sun-Earth-Moon system Bi-circular problem parameterization method

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parameterization of surface roller evolution in wave-induced current modeling

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OCEAN MODELLING 2025年 196卷

作者： Ji, Chao Jiang, Qi Ma, Dianguang Wu, Yuefeng Ran, Guoquan Kong, Xianwei Zhang, Qinghe Minist Transport Tianjin Res Inst Water Transport Engn Natl Engn Res Ctr Port Hydraul Construct Technol Tianjin 300456 Peoples R China CCCC First Harbor Consultants Co Ltd Tianjin 300222 Peoples R China Tianjin Univ State Key Lab Hydraul Engn Intelligent Constructio Minist Educ Tianjin 300350 Peoples R China Chongqing Jiaotong Univ Sch River & Ocean Engn Chongqing 400074 Peoples R China

Surface rollers, which are onshore-traveling bores of broken waves, store dissipated wave energy and delay the transfer of energy to the mean flow. Traditional roller evolution models typically rely on two key parameters: the roller slope and the energy transfer fraction, both of which are often treated as empirical constants. However, this approach can result in unrealistic or inaccurate simulations of wave-induced currents. In this study, we propose parameterizations for the roller slope and energy transfer fraction in the roller evolution equation for wave-induced current modeling. Three roller slope parameterization methods were compared, and on the basis of their performance in simulating wave-induced currents under various conditions, one method was selected and modified to ensure both physical consistency and computational flexibility. Building on this framework, we further refined the energy transfer fraction by identifying optimal values for different locations. These values and local wave and bathymetric parameters were subsequently used to perform least-squares fitting, yielding an effective parameterization of the energy transfer fraction. Model evaluations demonstrate that our roller slope and energy transfer fraction parameterizations provide a robust theoretical foundation for wave-induced current modeling, significantly enhancing its accuracy and applicability, compared with previous methods.

关键词： Surface rollers parameterization method Wave-induced currents Coupled model Radiation stress

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Aerodynamic adjoint optimization of turbomachinery with direct control on blade design parameters

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Chinese Journal of Aeronautics 2023年第11期36卷 119-134页

作者： Xin LI Tongtong MENG Weiwei LI Ling ZHOU Lucheng JI Institute for Aero Engine Tsinghua UniversityBeijing 100084China School of Aerospace Engineering Beijing Institute of TechnologyBeijing 100081China

Nowadays,the adjoint method has become a popular approach in the optimization of turbomachinery to further improve its aerodynamic ***,design variables in these adjoint optimization applications are generally not direct design parameters of blade(such as wedge angles or maximum thickness),making the geometric variation by adjoint optimization can hardly be re-extracted as the change of each design *** giving considerations to the G1 continuity constraint of adjoint method on its parameterization method,this manuscript shows how to apply a parameterization method in 3D blade design process into adjoint *** all design parameters can therefore be treated as design variables in the adjoint method and then participate in the sensitivity-based ***,a fitted Rotor 67 blade is optimized and the adiabatic efficiency is significantly improved by nearly 0.91%.

关键词： Adjoint method parameterization method Aerodynamic optimization Customized blading method Compressor Turbomachinery

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Invariant manifolds of maps and vector fields with nilpotent parabolic tori

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JOURNAL OF DIFFERENTIAL EQUATIONS 2024年 396卷 314-362页

作者： Cufi-Cabre, Clara Fontich, Ernest Univ Autonoma Barcelona UAB Dept Matemat Barcelona 08193 Spain Univ Barcelona UB Ctr Recerca Matemat CRM Dept Matemat & Informat Gran Via 585 Barcelona 08007 Spain

We consider analytic maps and vector fields defined in R 2 x T d , having a d -dimensional invariant torus T . The map (resp. vector field) restricted to T defines a rotation of Diophantine frequency vector omega is an element of R d , and its derivative restricted to transversal directions to T does not diagonalize. In this context, we give conditions on the coefficients of the nonlinear terms of the map (resp. vector field) under which T possesses stable and unstable invariant manifolds, and we show that such invariant manifolds are analyitic away from the invariant torus. We also provide effective algorithms to compute approximations of parameterizations of the invariant manifolds, and a posteriori theorems that can be used to validate numerical computations. Moreover, we present some applications of the results. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY -NC -ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).

关键词： Parabolic torus Invariant manifold parameterization method

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Quasi-periodic Response Solution to Scalar State-Dependent Delay Differential Equation with Degenerate Equilibrium

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QUALITATIVE THEORY OF DYNAMICAL SYSTEMS 2024年第5期23卷 244-244页

作者： He, Xiaolong Jin, Feng Song, Yongli Hangzhou Normal Univ Dept Math Hangzhou 311121 Peoples R China

We consider the state-dependent delay differential equation (SDDE) obtained by adding delayed perturbation to a one-dimensional ODE with a degenerate equilibrium. We prove the existence of the response solution of the equation, i.e., the quasi-periodic solution with the same frequency as the forcing. The novelty of our paper is to provide a concrete example to discuss the smoothness issues of SDDE, especially showing the analyticity of quasi-periodic solutions in some probability sense.

关键词： State-dependent delay Quasi-periodic parameterization method KAM Degenerate equilibrium

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Bifurcations and limit cycle prediction of rotor systems with fluid-film bearings using center manifold reduction

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NONLINEAR DYNAMICS 2023年第19期111卷 17749-17767页

作者： Mereles, Arthur Alves, Diogo Stuani Cavalca, Katia Lucchesi Univ Estadual Campinas Fac Mech Engn Lab Rotating Machinery Rua Mendeleyev 200 BR-13083860 Campinas SP Brazil Univ Bath Dept Mech Engn Bath BA2 7AY England

Many rotating machines utilize a fluid-type bearing. Despite their reliability and high load capacity, these bearings often show instabilities due to the interaction between the fluid media and the rotating shaft. These instabilities, known as oil-whirl and oil-whip, occur due to a Hopf bifurcation;being the parameter the shaft speed. Identifying the type of bifurcation, either sub-critical or super-critical, is an important task to determine the safety of the machines near the instability speed, and it tells whether one experiences oil-whip or oil-whirl. This work presents an approach, based on the center manifold reduction (CMR) method, to obtain limit cycles near Hopf bifurcations of rotors supported on fluid-film bearings. The basis of the method is the obtention of the center manifold of the system, which allows one to assess the type of bifurcation at hand. To obtain the center manifold, the parameterization method for invariant manifolds is used, which is a powerful tool to obtain invariant manifolds of high-dimensional dynamical systems. The proposed method is evaluated by comparing its results with an open-source numerical continuation package (MATCONT) in two systems: a simple and a realistic rotor system. The results show that the CMR, together with the parameterization method, can be used reliably to learn if the rotor system presents sub- or super-critical bifurcations, to perform parametric studies with different bearing properties, and also to predict the amplitude of the limit cycles.

关键词： Rotordynamics Hopf bifurcation Fluid-film bearing parameterization method Center manifold

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Validated Numerical Approximation of Stable Manifolds for Parabolic Partial Differential Equations

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JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS 2023年第4期35卷 3589-3649页

作者： van den Berg, Jan Bouwe Jaquette, Jonathan James, J. D. Mireles Vrije Univ Amsterdam Dept Math de Boelelaan 1111 NL-1081 HV Amsterdam Netherlands Boston Univ Dept Math & Stat Boston MA 02215 USA Brandeis Univ Dept Math Waltham MA 02453 USA Florida Atlantic Univ Dept Math Sci Boca Raton FL 33431 USA

This paper develops validated computational methods for studying infinite dimensional stable manifolds at equilibrium solutions of parabolic PDEs, synthesizing disparate errors resulting from numerical approximation. To construct our approximation, we decompose the stable manifold into three components: a finite dimensional slow component, a fast-but-finite dimensional component, and a strongly contracting infinite dimensional "tail". We employ the parameterization method in a finite dimensional projection to approximate the slow-stable manifold, as well as the attached finite dimensional invariant vector bundles. This approximation provides a change of coordinates which largely removes the nonlinear terms in the slow stable directions. In this adapted coordinate system we apply the Lyapunov-Perron method, resulting in mathematically rigorous bounds on the approximation errors. As a result, we obtain significantly sharper bounds than would be obtained using only the linear approximation given by the eigendirections. As a concrete example we illustrate the technique for a 1D Swift-Hohenberg equation.

关键词： Parabolic partial differential equations stable manifold Lyapunov-Perron method parameterization method Rigorous numerics Computer assisted proof

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