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Model reduction of high-dimensional self-excited nonlinear systems using floquet theory based parameterization method

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NONLINEAR DYNAMICS 2025年第2期113卷 1137-1161页

作者： Fan, Shan Hong, Ling Jiang, Jun Xi An Jiao Tong Univ State Key Lab Strength & Vibrat Xian 710049 Peoples R China

The parameterization method has demonstrated remarkable efficacy in the construction of reduced order models for nonlinear dynamical systems, which are mostly applied to depict the dynamics near a fixed point or a forced periodic solution perturbed from it. In this paper, we develop a method based on direct linear algebra to construct reduced order models around the limit cycles of high dimensional autonomous self-excited nonlinear systems. In particular, Taylor-Fourier series is adopted to parameterize the invariant manifold around a periodic solution of the self-excited systems. The basic theory and derivation of equations are presented in terms of matrices and Kronecker product, which benefits computer implementation and enhances computational efficiency. Two approaches to solve co-homological equations based on Floquet normal form and direct linear algebra are introduced and compared. The method based on Floquet normal form proves to be impossible to be applied in the high dimensional cases due to the extensive computing consumption. To facilitating the implementation, an iterative formula for the series expansion of generic nonlinear functions by composition of elementary function is also proposed. The reduced order models of a self-excited wing model, a rotor-stator rubbing model and a FEM rotor model are constructed. It is shown that the proposed approaches are valuable in predicting responses of high-dimensional self-excited nonlinear systems and can greatly reduce the computational costs.

关键词： Reduced order model Nonlinear normal mode Self-excited system parameterization method Floquet theory

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Weighted Birkhoff Averages and the parameterization method

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SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 2024年第3期23卷 1766-1804页

作者： Blessing, David James, J. D. Mireles Florida Atlantic Univ Dept Math Sci Boca Raton FL 33431 USA

This work provides a systematic recipe for computing accurate high order Fourier expansions of quasiperiodic invariant circles (and systems of such circles) in area preserving maps. The recipe requires only a finite data set sampled from the quasiperiodic circle. Our approach, being based pp. 1261--1300;A. Haro and R. de la Llave, J. Differential Equations, 228 (2006), pp. 530--579], uses a Newton scheme to iteratively solve a conjugacy equation describing the invariant circle (or systems of circles). A critical step in properly formulating the conjugacy equation is to determine the rotation number of the quasiperiodic subsystem. For this we exploit the weighted Birkhoff averaging method Revisited, Springer, Cham, 2016, pp. 103--118;S. Das and J. A. Yorke, Nonlinearity, 31 (2018), pp. 491--501]. This approach facilities accurate computation of the rotation number given nothing but the already mentioned orbit data. The weighted Birkhoff averages also facilitate the computation of other integral observables like Fourier coefficients of the parameterization of the invariant circle. Since, the parameterization method is based on a Newton scheme, we only need to approximate a small number of Fourier coefficients with low accuracy (say, a few correct digits) to find a good enough initial approximation so that Newton converges. Moreover, the Fourier coefficients may be computed independently, so we can sample the higher modes to guess the decay rate of the Fourier coefficients. This allows us to choose, a priori, an appropriate number of modes in the truncation. We illustrate the utility of the approach for explicit example systems including the area preserving He'\non map and the standard map (polynomial and trigonometric nonlinearity respectively). We present example computations for invariant circles and for systems of invariant circles with as many as 120 components. We also employ a numerical continuation scheme (where the rotation number is the continuation parameter

关键词： Key words. invariant circles parameterization method weighted Birkhoff averages numerical computation numerical continuation

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Semianalytical Computation of Heteroclinic Connections Between Center Manifolds with the parameterization method

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SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 2024年第1期23卷 98-126页

作者： Barcelona, Miquel Haro, Alex Mondelo, Josep-Maria Univ Autonoma Barcelona Dept Matemat Ave Eix CentEdifici C Bellaterra 08193 Spain Univ Autonoma Barcelona CERES IEEC Ave Eix CentEdifici C Bellaterra 08193 Spain Univ Barcelona Dept Matemat & Informat Gran Via 585 Barcelona 08007 Spain

This paper presents a methodology for the computation of whole sets of heteroclinic connections between isoenergetic slices of center manifolds of center \times center \times saddle fixed points of autonomous Hamiltonian systems. It involves (a) computing Taylor expansions of the center-unstable and center stable manifolds of the departing and arriving fixed points through the parameterization method, using a new style that uncouples the center part from the hyperbolic one, thus making the fibered structure of the manifolds explicit;(b) uniformly meshing isoenergetic slices of the center manifolds, using a novel strategy that avoids numerical integration of the reduced differential equations and makes an explicit three-dimensional representation of these slices as deformed solid ellipsoids;(c) matching the center-stable and center-unstable manifolds of the departing and arriving points in a Poincare'\ section. The methodology is applied to obtain the whole set of isoenergetic heteroclinic connections from the center manifold of L2 to the center manifold of L1 in the Earth-Moon circular, spatial restricted three-body problem, for nine increasing energy levels that reach the appearance of halo orbits in both L1 and L2. Some comments are made on possible applications to space mission design.

关键词： parameterization method heteroclinic connections invariant tori libration point orbits RTBP center manifolds

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parameterization method for Unstable Manifolds of Standing Waves on the Line

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SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 2020年第3期19卷 1758-1797页

作者： Barker, Blake James, Jason Mireles Morgan, Jalen Brigham Young Univ Dept Math Provo UT 84602 USA Florida Atlantic Univ Dept Math Sci Boca Raton FL 33431 USA

We consider a linearly unstable standing wave solution of a parabolic partial differential equation (PDE) on the real line and develop a high order method for polynomial approximation of the local unstable manifold. The unstable manifold describes the breakdown of the nonlinear wave after the loss of stability. Our method is based on the parameterization method for invariant manifolds and studies an invariance equation describing a local chart map. This invariance equation is a PDE posed on the product of a disk and the line. The dimension of the disk is equal to the Morse index of the wave. We develop a formal series solution for the invariance equation, and show that the coefficients of the series solve certain boundary value problems (BVPs) on the line. We solve these BVPs numerically to any desired order. The result is a polynomial describing the dynamics of the PDE in a macroscopic neighborhood of the unstable standing wave. The method is implemented for a number of example problems. Truncation/numerical errors are quantified via a posteriori indicators.

关键词： traveling waves unstable manifold PDE on the line parameterization method formal series expansion numerical methods

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Assembly pin factor parameterization method based on the proper orthogonal decomposition

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ANNALS OF NUCLEAR ENERGY 2020年 139卷 107262-107262页

作者： Li, Zhuo Ma, Yu Zhang, Qian Cao, Liangzhi Wu, Hongchun Li, Yunzhao Sun Yat Sen Univ Sino French Inst Nucl Engn & Technol Zhuhai 519082 Peoples R China Harbin Engn Univ Coll Nucl Sci & Technol Fundamental Sci Nucl Safety & Simulat Technol Lab Harbin 150001 Peoples R China Xi An Jiao Tong Univ Sch Nucl Sci & Technol Xian 710049 Peoples R China

The proper orthogonal decomposition (POD) is introduced into the assembly pin factor parameterization. The decomposition is performed first to the pin factors with specific discrete state parameters to capture the spatial characteristic of pin factors. Then the POD basis are stored, and the POD coefficients are functionalized according to the state parameters. While the few-group constants are needed in the core calculation, the functionalized POD coefficients are calculated. Then the required pin factors are obtained by multiplying the POD basis and the functionalized POD coefficients. This assembly pin factor parameterization procedure based on the POD is carried out and verified based on the assembly in the BEAVRS benchmark core. Moreover, the comparison is performed between the POD method and the traditional method for the assembly pin factor parameterization. The numerical results show that this procedure based on the POD is feasibility and effective with high accuracy and calculation speed. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Assembly pin factor Proper orthogonal decomposition parameterization method Few-group constants

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A Geometric Approach to Phase Response Curves and Its Numerical Computation Through the parameterization method

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JOURNAL OF NONLINEAR SCIENCE 2019年第6期29卷 2877-2910页

作者： Perez-Cervera, Alberto Seara, Tere M. Huguet, Gemma Univ Politecn Cataluna Dept Matemat Avda Diagonal 647 E-08028 Barcelona Spain BGSMATH Barcelona Spain

The phase response curve (PRC) is a tool used in neuroscience that measures the phase shift experienced by an oscillator due to a perturbation applied at different phases of the limit cycle. In this paper, we present a new approach to PRCs based on the parameterization method. The underlying idea relies on the construction of a periodic system whose corresponding stroboscopic map has an invariant curve. We demonstrate the relationship between the internal dynamics of this invariant curve and the PRC, which yields a method to numerically compute the PRCs. Moreover, we link the existence properties of this invariant curve as the amplitude of the perturbation is increased with changes in the PRC waveform and with the geometry of isochrons. The invariant curve and its dynamics will be computed by means of the parameterization method consisting of solving an invariance equation. We show that the method to compute the PRC can be extended beyond the breakdown of the curve by means of introducing a modified invariance equation. The method also computes the amplitude response functions (ARCs) which provide information on the displacement away from the oscillator due to the effects of the perturbation. Finally, we apply the method to several classical models in neuroscience to illustrate how the results herein extend the framework of computation and interpretation of the PRC and ARC for perturbations of large amplitude and not necessarily pulsatile.

关键词： Phase response curves Isochrons Phase equation parameterization method NHIM Synchronization

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Invariant manifolds and the parameterization method in coupled energy harvesting piezoelectric oscillators

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PHYSICA D-NONLINEAR PHENOMENA 2017年 351卷 14-29页

作者： Granados, Albert Tech Univ Denmark Dept Appl Math & Comp Sci Bldg 303B DK-2800 Lyngby Denmark

Energy harvesting systems based on oscillators aim to capture energy from mechanical oscillations and convert it into electrical energy. Widely extended are those based on piezoelectric materials, whose dynamics are Hamiltonian submitted to different sources of dissipation: damping and coupling. These dissipations bring the system to low energy regimes, which is not desired in long term as it diminishes the absorbed energy. To avoid or to minimize such situations, we propose that the coupling of two oscillators could benefit from theory of Arnold diffusion. Such phenomenon studies 0(1) energy variations in Hamiltonian systems and hence could be very useful in energy harvesting applications. This article is a first step towards this goal. We consider two piezoelectric beams submitted to a small forcing, and coupled through an electric circuit. By considering the coupling, damping and forcing as perturbations, we prove that the unperturbed system possesses a 4-dimensional Normally Hyperbolic Invariant Manifold with 5 and 4-dimensional stable and unstable manifolds, respectively. These are locally unique after the perturbation. By means of the parameterization method, we numerically compute parameterizations of the perturbed manifold, its stable and unstable manifolds and study its inner dynamics. We show evidence of homoclinic connections when the perturbation is switched on. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Damped oscillators Energy harvesting systems parameterization method Normally hyperbolic invariant manifolds Homoclinic connections Arnold diffusion

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Construction of Quasi-periodic Solutions of State-Dependent Delay Differential Equations by the parameterization method I: Finitely Differentiable, Hyperbolic Case

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JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS 2017年第4期29卷 1503-1517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

In this paper, we use the parameterization method to construct quasi-periodic solutions of state-dependent delay differential equations. For example Under the assumption of exponential dichotomies for the case, we use a contraction mapping argument to prove the existence and smoothness of the quasi-periodic solution. Furthermore, the result is given in an a posteriori format. The method is very general and applies also to equations with several delays, distributed delays etc.

关键词： State-dependent delay Quasi-periodic solution Exponential dichotomies parameterization method Interpolation inequalities A posteriori

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Invariant manifolds of a non-autonomous quasi-bicircular problem computed via the parameterization method

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NONLINEARITY 2017年第8期30卷 3040-3075页

作者： Le Bihan, B. Masdemont, J. J. Gomez, G. Lizy-Destrez, S. ISAE SUPAERO 10 Ave Edouard BelinBP 54032 F-31055 Toulouse 4 France Univ Politecn Cataluna IEEC Diagonal 647 E-08028 Barcelona Spain Univ Politecn Cataluna Dept Matemat Diagonal 647 E-08028 Barcelona Spain Univ Barcelona IEEC Gran Via 585 E-08007 Barcelona Spain Univ Barcelona Dept Matemat & Informat Gran Via 585 E-08007 Barcelona Spain

The parameterization method (pm) has been used to compute high-order parameterizations of invariant manifolds of vector fields at fixed points. This paper extends such approach to invariant manifolds of periodically-perturbed vector fields about a periodic orbit with the same frequency, with a direct application on the libration points of the Sun-Earth-Moon system. The Sun-Earth-Moon environment is modeled by the so-called quasi-bicircular model (qbcp), which is a coherent restricted four-body model that describes the motion of a spacecraft under the simultaneous gravitational influences of the Earth, the Moon, and the Sun. The pm is adapted to account for the explicit time-dependency of the corresponding vector field. This new procedure yields high-order periodic semi-analytical approximations of the center manifolds about the libration points L-1,L-2 of the periodically-perturbed Sun(Earth + Moon) and Earth-Moon systems. These approximations are then used to initialize the computation of Poincare maps, which allow to get a qualitative description of the non-autonomous dynamics near the equilibrium points. It is shown that, with this new approach, the semi-analytical description of the center manifolds in a coherent four-body environment is valid in a neighborhood significant enough to be used in practice. In particular, the well-known Halo orbit bifurcation is recovered in all cases.

关键词： invariant manifolds parameterization method quasi-periodic systems Sun-Earth-Moon system

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Computation of maximal local (un)stable manifold patches by the parameterization method

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INDAGATIONES MATHEMATICAE-NEW SERIES 2016年第1期27卷 340-367页

作者： Breden, Maxime Lessard, Jean-Philippe James, Jason D. Mireles Univ Paris Saclay CMLA ENS Cachan CNRS 61 Ave President Wilson F-94230 Cachan France Univ Laval Dept Math & Stat 1045 Ave Med Quebec City PQ G1V 0A6 Canada Florida Atlantic Univ Sci BldgRoom 234777 Glades Rd Boca Raton FL 33431 USA

In this work we develop some automatic procedures for computing high order polynomial expansions of local (un)stable manifolds for equilibria of differential equations. Our method incorporates validated truncation error bounds, and maximizes the size of the image of the polynomial approximation relative to some specified constraints. More precisely we use that the manifold computations depend heavily on the scalings of the eigenvectors: indeed we study the precise effects of these scalings on the estimates which determine the validated error bounds. This relationship between the eigenvector scalings and the error estimates plays a central role in our automatic procedures. In order to illustrate the utility of these methods we present several applications, including visualization of invariant manifolds in the Lorenz and FitzHugh-Nagumo systems and an automatic continuation scheme for (un)stable manifolds in a suspension bridge problem. In the present work we treat explicitly the case where the eigenvalues satisfy a certain non-resonance condition. (C) 2015 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.

关键词： Invariant manifold parameterization method Radii polynomials Algorithms A posteriori analysis

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