Journal of Nonlinear Science最新文献_第9页

Geometry-Preserving Numerical Methods for Physical Systems with Finite-Dimensional Lie Algebras 有限维李代数物理系统的几何保全数值方法

IF 3 2区数学

Journal of Nonlinear Science Pub Date : 2023-12-23 DOI: 10.1007/s00332-023-10000-8

L. Blanco, F. Jiménez, J. de Lucas, C. Sardón

{"title":"Geometry-Preserving Numerical Methods for Physical Systems with Finite-Dimensional Lie Algebras","authors":"L. Blanco, F. Jiménez, J. de Lucas, C. Sardón","doi":"10.1007/s00332-023-10000-8","DOIUrl":"https://doi.org/10.1007/s00332-023-10000-8","url":null,"abstract":"<h3>Abstract</h3> We propose a geometric integrator to numerically approximate the flow of Lie systems. The key is a novel procedure that integrates the Lie system on a Lie group intrinsically associated with a Lie system on a general manifold via a Lie group action and then generates the discrete solution of the Lie system on the manifold via a solution of the Lie system on the Lie group. One major result from the integration of a Lie system on a Lie group is that one is able to solve all associated Lie systems on manifolds at the same time, and that Lie systems on Lie groups can be described through first-order systems of linear homogeneous ordinary differential equations (ODEs) in normal form. This brings a lot of advantages, since solving a linear system of ODEs involves less numerical cost. Specifically, we use two families of numerical schemes on the Lie group, which are designed to preserve its geometrical structure: the first one is based on the Magnus expansion, whereas the second is based on Runge–Kutta–Munthe–Kaas (RKMK) methods. Moreover, since the aforementioned action relates the Lie group and the manifold where the Lie system evolves, the resulting integrator preserves any geometric structure of the latter. We compare both methods for Lie systems with geometric invariants, particularly a class on Lie systems on curved spaces. We also illustrate the superiority of our method for describing long-term behavior and for differential equations admitting solutions whose geometric features depends heavily on initial conditions. As already mentioned, our milestone is to show that the method we propose preserves all the geometric invariants very faithfully, in comparison with non-geometric numerical methods.","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139029492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Geometric Methods for Adjoint Systems 用于邻接系统的几何方法

IF 3 2区数学

Journal of Nonlinear Science Pub Date : 2023-12-19 DOI: 10.1007/s00332-023-09999-7

Brian Kha Tran, Melvin Leok

{"title":"Geometric Methods for Adjoint Systems","authors":"Brian Kha Tran, Melvin Leok","doi":"10.1007/s00332-023-09999-7","DOIUrl":"https://doi.org/10.1007/s00332-023-09999-7","url":null,"abstract":"Adjoint systems are widely used to inform control, optimization, and design in systems described by ordinary differential equations or differential-algebraic equations. In this paper, we explore the geometric properties and develop methods for such adjoint systems. In particular, we utilize symplectic and presymplectic geometry to investigate the properties of adjoint systems associated with ordinary differential equations and differential-algebraic equations, respectively. We show that the adjoint variational quadratic conservation laws, which are key to adjoint sensitivity analysis, arise from (pre)symplecticity of such adjoint systems. We discuss various additional geometric properties of adjoint systems, such as symmetries and variational characterizations. For adjoint systems associated with a differential-algebraic equation, we relate the index of the differential-algebraic equation to the presymplectic constraint algorithm of Gotay et al. (J Math Phys 19(11):2388–2399, 1978). As an application of this geometric framework, we discuss how the adjoint variational quadratic conservation laws can be used to compute sensitivities of terminal or running cost functions. Furthermore, we develop structure-preserving numerical methods for such systems using Galerkin Hamiltonian variational integrators (Leok and Zhang in IMA J. Numer. Anal. 31(4):1497–1532, 2011) which admit discrete analogues of these quadratic conservation laws. We additionally show that such methods are natural, in the sense that reduction, forming the adjoint system, and discretization all commute, for suitable choices of these processes. We utilize this naturality to derive a variational error analysis result for the presymplectic variational integrator that we use to discretize the adjoint DAE system. Finally, we discuss the application of adjoint systems in the context of optimal control problems, where we prove a similar naturality result.\u0000","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138744862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Numerical Computation of Dark Solitons of a Nonlocal Nonlinear Schrödinger Equation 非局部非线性薛定谔方程暗孤子的数值计算

IF 3 2区数学

Journal of Nonlinear Science Pub Date : 2023-12-18 DOI: 10.1007/s00332-023-10001-7

André de Laire, Guillaume Dujardin, Salvador López-Martínez

引用次数: 0

Spike Solutions to the Supercritical Fractional Gierer–Meinhardt System 超临界分式吉勒-梅因哈特系统的尖峰解决方案

IF 3 2区数学

Journal of Nonlinear Science Pub Date : 2023-12-18 DOI: 10.1007/s00332-023-10002-6

Daniel Gomez, Markus De Medeiros, Jun-cheng Wei, Wen Yang

{"title":"Spike Solutions to the Supercritical Fractional Gierer–Meinhardt System","authors":"Daniel Gomez, Markus De Medeiros, Jun-cheng Wei, Wen Yang","doi":"10.1007/s00332-023-10002-6","DOIUrl":"https://doi.org/10.1007/s00332-023-10002-6","url":null,"abstract":"Localized solutions are known to arise in a variety of singularly perturbed reaction–diffusion systems. The Gierer–Meinhardt (GM) system is one such example and has been the focus of numerous rigorous and formal studies. A more recent focus has been the study of localized solutions in systems exhibiting anomalous diffusion, particularly with Lévy flights. In this paper, we investigate localized solutions to a one-dimensional fractional GM system for which the inhibitor’s fractional order is supercritical. Specifically, we assume the fractional orders of the activator and inhibitor are, respectively, in the ranges (s_1in (1/4,1)) and (s_2in (0,1/2)). Using the method of matched asymptotic expansions, we reduce the construction of multi-spike solutions to solving a nonlinear algebraic system. The linear stability of the resulting multi-spike solutions is then addressed by studying a globally coupled eigenvalue problem. In addition to these formal results, we also rigorously establish the existence and stability of ground state solutions when the inhibitor’s fractional order is nearly critical. The fractional Green’s function, for which we present a rapidly converging series expansion, is prominently featured throughout both the formal and rigorous analysis in this paper. Moreover, we emphasize that the striking similarities between the one-dimensional supercritical GM system and the classical three-dimensional GM system can be attributed to the leading-order singular behaviour of the fractional Green’s function.","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138745963","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Nonlinear Model Reduction for Slow–Fast Stochastic Systems Near Unknown Invariant Manifolds 未知不变曲面附近慢-快随机系统的非线性模型还原

IF 3 2区数学

Journal of Nonlinear Science Pub Date : 2023-12-08 DOI: 10.1007/s00332-023-09998-8

Felix X.-F. Ye, Sichen Yang, Mauro Maggioni

{"title":"Nonlinear Model Reduction for Slow–Fast Stochastic Systems Near Unknown Invariant Manifolds","authors":"Felix X.-F. Ye, Sichen Yang, Mauro Maggioni","doi":"10.1007/s00332-023-09998-8","DOIUrl":"https://doi.org/10.1007/s00332-023-09998-8","url":null,"abstract":"We introduce a nonlinear stochastic model reduction technique for high-dimensional stochastic dynamical systems that have a low-dimensional invariant effective manifold with slow dynamics and high-dimensional, large fast modes. Given only access to a black-box simulator from which short bursts of simulation can be obtained, we design an algorithm that outputs an estimate of the invariant manifold, a process of the effective stochastic dynamics on it, which has averaged out the fast modes, and a simulator thereof. This simulator is efficient in that it exploits of the low dimension of the invariant manifold, and takes time-steps of size dependent on the regularity of the effective process, and therefore typically much larger than that of the original simulator, which had to resolve the fast modes. The algorithm and the estimation can be performed on the fly, leading to efficient exploration of the effective state space, without losing consistency with the underlying dynamics. This construction enables fast and efficient simulation of paths of the effective dynamics, together with estimation of crucial features and observables of such dynamics, including the stationary distribution, identification of metastable states, and residence times and transition rates between them.","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138562677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Error Estimates of hp Spectral Element Methods in Nonlinear Optimal Control Problem 非线性最优控制问题中hp谱元方法的误差估计

IF 3 2区数学

Journal of Nonlinear Science Pub Date : 2023-12-01 DOI: 10.1007/s00332-023-09991-1

Xiuxiu Lin, Yanping Chen, Yunqing Huang

引用次数: 0

Well-Posedness and Stability Analysis of a Landscape Evolution Model 景观演化模型的拟合性与稳定性分析

IF 3 2区数学

Journal of Nonlinear Science Pub Date : 2023-11-24 DOI: 10.1007/s00332-023-09997-9

Julie Binard, Pierre Degond, Pascal Noble

引用次数: 0

Cognitive Consumer-Resource Spatiotemporal Dynamics with Nonlocal Perception 非局部感知的认知消费者-资源时空动态

IF 3 2区数学

Journal of Nonlinear Science Pub Date : 2023-11-18 DOI: 10.1007/s00332-023-09996-w

Yongli Song, Hao Wang, Jinfeng Wang

{"title":"Cognitive Consumer-Resource Spatiotemporal Dynamics with Nonlocal Perception","authors":"Yongli Song, Hao Wang, Jinfeng Wang","doi":"10.1007/s00332-023-09996-w","DOIUrl":"https://doi.org/10.1007/s00332-023-09996-w","url":null,"abstract":"Nonlocal perception is crucial to the mechanistic modeling of cognitive animal movement. We formulate a diffusive consumer-resource model with nonlocal perception on resource availability, where resource dynamics is explicitly modeled, to investigate the influence of nonlocal perception on stability and spatiotemporal patterns. For the finite domain, nonlocal perception described by two common types of resource detection function (spatial average or Green function) has no impact on the stability of the spatially homogeneous steady state. For the infinite domain, nonlocal perception described by the Laplacian or Gaussian detection function has no impact on stability either; however, the top-hat detection function can destabilize the spatially homogeneous steady state when the rate of perceptual movement is large and the detection scale belongs to an appropriate interval. Using the more realistic top-hat perception kernel, we investigate the influence of the detection scale, the perceptual movement rate and the resource’s carrying capacity on the spatiotemporal patterns and find the stripe spatial patterns, oscillatory patterns with different spatial profiles as well as spatiotemporal chaos.","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138518648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Oscillations in a Fully Connected Network of Leaky Integrate-and-Fire Neurons with a Poisson Spiking Mechanism 具有泊松尖峰机制的漏积分-火神经元全连接网络的振荡

IF 3 2区数学

Journal of Nonlinear Science Pub Date : 2023-11-17 DOI: 10.1007/s00332-023-09995-x

Grégory Dumont, Jacques Henry, Carmen Oana Tarniceriu

引用次数: 0

Geometric Blow-Up for Folded Limit Cycle Manifolds in Three Time-Scale Systems 三时间尺度系统中折叠极限环流形的几何爆破

IF 3 2区数学

Journal of Nonlinear Science Pub Date : 2023-11-16 DOI: 10.1007/s00332-023-09987-x

S. Jelbart, C. Kuehn, S.-V. Kuntz

{"title":"Geometric Blow-Up for Folded Limit Cycle Manifolds in Three Time-Scale Systems","authors":"S. Jelbart, C. Kuehn, S.-V. Kuntz","doi":"10.1007/s00332-023-09987-x","DOIUrl":"https://doi.org/10.1007/s00332-023-09987-x","url":null,"abstract":"Geometric singular perturbation theory provides a powerful mathematical framework for the analysis of ‘stationary’ multiple time-scale systems which possess a critical manifold, i.e. a smooth manifold of steady states for the limiting fast subsystem, particularly when combined with a method of desingularisation known as blow-up. The theory for ‘oscillatory’ multiple time-scale systems which possess a limit cycle manifold instead of (or in addition to) a critical manifold is less developed, particularly in the non-normally hyperbolic regime. We use the blow-up method to analyse the global oscillatory transition near a regular folded limit cycle manifold in a class of three time-scale ‘semi-oscillatory’ systems with two small parameters. The systems considered behave like oscillatory systems as the smallest perturbation parameter tends to zero, and stationary systems as both perturbation parameters tend to zero. The additional time-scale structure is crucial for the applicability of the blow-up method, which cannot be applied directly to the two time-scale oscillatory counterpart of the problem. Our methods allow us to describe the asymptotics and strong contractivity of all solutions which traverse a neighbourhood of the global singularity. Our main results cover a range of different cases with respect to the relative time-scale of the angular dynamics and the parameter drift. We demonstrate the applicability of our results for systems with periodic forcing in the slow equation, in particular for a class of Liénard equations. Finally, we consider a toy model used to study tipping phenomena in climate systems with periodic forcing in the fast equation, which violates the conditions of our main results, in order to demonstrate the applicability of classical (two time-scale) theory for problems of this kind.","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138518656","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 2