Nonlinearity最新文献

英文中文

Parametric approximations of fast close encounters of the planar three-body problem as arcs of a focus-focus dynamics 平面三体问题快速近距离相遇的参数近似，作为焦点-焦点动力学的弧线

IF 1.7 2区数学 Q2 MATHEMATICS, APPLIED

Nonlinearity

Pub Date : 2024-09-09 DOI: 10.1088/1361-6544/ad72c6

Massimiliano Guzzo

A gravitational close encounter of a small body with a planet may produce a substantial change of its orbital parameters which can be studied using the circular restricted three-body problem. In this paper we provide parametric representations of the fast close encounters with the secondary body of the planar CRTBP as arcs of non-linear focus-focus dynamics. The result is the consequence of a remarkable factorisation of the Birkhoff normal forms of the Hamiltonian of the problem represented with the Levi–Civita regularisation. The parameterisations are computed using two different sequences of Birkhoff normalisations of given order N. For each value of N, the Birkhoff normalisations and the parameters of the focus-focus dynamics are represented by polynomials whose coefficients can be computed iteratively with a computer algebra system; no quadratures, such as those needed to compute action-angle variables of resonant normal forms, are needed. We also provide some numerical demonstrations of the method for values of the mass parameter representative of the Sun–Earth and the Sun–Jupiter cases.

小天体与行星的引力近距离相遇可能会使其轨道参数发生重大变化，这可以利用环形受限三体问题进行研究。在本文中，我们以非线性聚焦-聚焦动力学弧线的形式提供了与平面 CRTBP 次级天体快速近距离相遇的参数表示。这一结果是对用 Levi-Civita 正则化表示的问题的哈密顿的 Birkhoff 正则形式进行显著因式分解的结果。对于每个 N 值，伯克霍夫正则表达式和焦点-焦点动力学参数都用多项式表示，其系数可以用计算机代数系统迭代计算；不需要二次方程，如计算共振正则表达式的作用角变量所需的二次方程。我们还提供了该方法在太阳-地球和太阳-木星情况下质量参数值的一些数值演示。

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引用次数: 0

Simultaneous local normal forms of dynamical systems with singular underlying geometric structures 具有奇异基础几何结构的动力系统的同步局部正态形式

IF 1.7 2区数学 Q2 MATHEMATICS, APPLIED

Nonlinearity

Pub Date : 2024-09-09 DOI: 10.1088/1361-6544/ad700d

Kai Jiang, Tudor S Ratiu and Nguyen Tien Zung

The aim of this paper is to develop, for the first time, a general theory of simultaneous local normalisation of couples , where X is a dynamical system (vector field) and is an underlying geometric structure preserved by X, even if both have singularities. Such couples appear naturally in many problems, e.g. Hamiltonian dynamics, where is a symplectic structure and one has the theory of Birkhoff normal forms, or constrained dynamics, where is a smooth, in general singular, distribution of tangent subspaces, etc. In this paper, the geometric structure is of the following types: volume form, symplectic form, contact form, Poisson tensor, as well as their singular versions. The paper addresses mainly the more difficult situations when both X and are singular at a point and its results prove the existence of natural simultaneous normal forms in these cases. In general, the normalisation is only formal, but when and X are (real or complex) analytic and X is analytically or Darboux integrable, then the simultaneous normalisation is also analytic. Our theory is based on a new approach, called the Toric Conservation Principle, as well as the classical step-by-step normalisation technique, and the equivariant path method.

本文的目的是首次提出一种对偶同时局部归一化的一般理论，其中 X 是一个动力学系统（向量场），X 是一个由 X 保留的底层几何结构，即使两者都有奇点。这种耦合自然出现在许多问题中，例如汉密尔顿动力学，其中 X 是交映结构，我们有伯克霍夫正形式理论；或约束动力学，其中 X 是切分子空间的平滑分布，一般是奇异分布，等等。在本文中，几何结构有以下几种类型：体积形式、交映形式、接触形式、泊松张量以及它们的奇异版本。本文主要讨论了当 X 和都是奇异点时较为困难的情况，其结果证明了在这些情况下存在自然的同时正则表达式。一般来说，正化只是形式上的，但当和 X 是（实或复）解析的，并且 X 是解析或达布可积分的，那么同时正化也是解析的。我们的理论基于一种新方法，即 "环守恒原理"，以及经典的分步归一化技术和等变路径法。

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引用次数: 0

Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains 外域超临界椭圆问题的多重性和对称性突破

IF 1.7 2区数学 Q2 MATHEMATICS, APPLIED

Nonlinearity

Pub Date : 2024-09-09 DOI: 10.1088/1361-6544/ad74d0

Alberto Boscaggin, Francesca Colasuonno, Benedetta Noris, Tobias Weth

We deal with the following semilinear equation in exterior domains

−Δu+u=a(x)|u|p−2u,u∈H01(AR),

where

AR:={x∈RN:|x|>R}

N⩾3

, R > 0. Assuming that the weight a is positive and satisfies some symmetry and monotonicity properties, we exhibit a positive solution having the same features as a, for values of p > 2 in a suitable range that includes exponents greater than the standard Sobolev critical one. In the special case of radial weight a, our existence result ensures multiplicity of nonradial solutions. We also provide an existence result for supercritical p in nonradial exterior domains.

我们处理的是外部域中的半线性方程 -Δu+u=a(x)|u|p-2u,u∈H01(AR)，其中 AR:={x∈RN:|x|>R}, N⩾3, R > 0。假定权重 a 为正值，并满足一些对称性和单调性特性，对于 p > 2 的取值范围（包括大于标准索波列夫临界值的指数），我们展示了与 a 具有相同特征的正解。在径向权 a 的特殊情况下，我们的存在性结果确保了非径向解的多重性。我们还提供了非径向外部域中超临界 p 的存在性结果。

引用次数: 0

Spectral triples and Dixmier trace representations of Gibbs measures: theory and examples 吉布斯度量的谱三元和迪克斯米尔痕量表示：理论与实例

IF 1.7 2区数学 Q2 MATHEMATICS, APPLIED

Nonlinearity

Pub Date : 2024-09-05 DOI: 10.1088/1361-6544/ad7009

L Cioletti, L Y Hataishi, A O Lopes, M Stadlbauer

In this paper we study spectral triples and non-commutative expectations associated to expanding and weakly expanding maps. In order to do so, we generalise the Perron–Frobenius–Ruelle theorem and obtain a polynomial decay of the operator, which allows to prove differentiability of a dynamically defined ζ-function at its critical parameter. We then generalise Sharp’s construction of spectral triples to this setting and provide criteria when the associated spectral metric is non-degenerate and when the non-commutative expectation of the spectral triple is colinear to the integration with respect to the associated equilibrium state from thermodynamic formalism. Due to our general setting, we are able to simultaneously analyse expanding maps on manifolds or connected fractals, subshifts of finite type as well as the Dyson model from statistical physics, which underlines the unifying character of noncommutative geometry. Furthermore, we derive an explicit representation of the ζ-function associated to a particular class of pathological continuous potentials, giving rise to examples where the representation as a non-commutative expectation via the associated zeta function holds, and others where it does not hold.

在本文中，我们研究了与膨胀和弱膨胀映射相关的谱三元组和非交换期望。为此，我们对 Perron-Frobenius-Ruelle 定理进行了推广，得到了算子的多项式衰减，从而证明了动态定义的 ζ 函数在其临界参数处的可微分性。然后，我们将夏普的谱三元构造推广到这一环境中，并在相关谱度量是非退化的、谱三元的非交换期望与热力学形式主义的相关平衡态的积分是共线的情况下提供了标准。由于我们的一般设置，我们能够同时分析流形或连通分形上的膨胀映射、有限类型的子移动以及统计物理学中的戴森模型，这凸显了非交换几何的统一性。此外，我们还推导出了与某类病态连续势相关的ζ函数的明确表示，并举例说明了通过相关zeta函数作为非交换期望的表示在哪些情况下成立，以及在哪些情况下不成立。

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引用次数: 0

New notion of nonuniform exponential dichotomy with applications to the theory of pullback and forward attractors 非均匀指数二分法的新概念及其在回拉和前向吸引子理论中的应用

IF 1.7 2区数学 Q2 MATHEMATICS, APPLIED

Nonlinearity

Pub Date : 2024-09-04 DOI: 10.1088/1361-6544/ad700a

José A Langa, Rafael Obaya, Alexandre N Oliveira-Sousa

In this work we study nonuniform exponential dichotomies and existence of pullback and forward attractors for evolution processes associated to nonautonomous differential equations. We define a new concept of nonuniform exponential dichotomy, for which we provide several examples, study the relation with the standard notion, and establish a robustness under perturbations. We provide a dynamical interpretation of admissibility pairs related with exponential dichotomies to obtain existence of pullback and forward attractors. We apply these abstract results for ordinary and parabolic differential equations.

在这项研究中，我们研究了与非自治微分方程相关的演化过程的非均匀指数二分法以及回拉和前向吸引子的存在性。我们定义了非均匀指数二分法的新概念，并提供了几个例子，研究了它与标准概念的关系，并建立了扰动下的稳健性。我们提供了与指数二分法相关的可接纳性对的动力学解释，以获得回拉和前向吸引子的存在性。我们将这些抽象结果应用于常微分方程和抛物线微分方程。

引用次数: 0

Continuity of the Lyapunov exponents of non-invertible random cocycles with constant rank 具有恒定秩的非可逆随机循环的李亚普诺夫指数的连续性

IF 1.7 2区数学 Q2 MATHEMATICS, APPLIED

Nonlinearity

Pub Date : 2024-09-04 DOI: 10.1088/1361-6544/ad72c7

Pedro Duarte, Catalina Freijo

In this paper we establish uniform large deviations estimates of exponential type and Hölder continuity of the Lyapunov exponents for random non-invertible cocycles with constant rank.

在本文中，我们建立了指数型的均匀大偏差估计，以及具有恒定秩的随机非可逆循环的 Lyapunov 指数的荷尔德连续性。

引用次数: 0

Singular limit of a chemotaxis model with indirect signal production and phenotype switching 具有间接信号产生和表型转换功能的趋化模型的奇异极限

IF 1.7 2区数学 Q2 MATHEMATICS, APPLIED

Nonlinearity

Pub Date : 2024-09-03 DOI: 10.1088/1361-6544/ad6bdf

Philippe Laurençot, Christian Stinner

Convergence of solutions to a partially diffusive chemotaxis system with indirect signal production and phenotype switching is shown in a two-dimensional setting when the switching rate increases to infinity, thereby providing a rigorous justification of formal computations performed in the literature. The expected limit system being the classical parabolic–parabolic Keller–Segel system, the obtained convergence is restricted to a finite time interval for general initial conditions but valid for arbitrary bounded time intervals when the mass of the initial condition is appropriately small. Furthermore, if the solution to the limit system blows up in finite time, then neither of the two phenotypes in the partially diffusive system can be uniformly bounded with respect to the L₂-norm beyond that time.

在二维环境中，当切换率增加到无穷大时，显示了具有间接信号产生和表型切换的部分扩散趋化系统解的收敛性，从而为文献中的正式计算提供了严格的证明。预期的极限系统是经典的抛物线-抛物线凯勒-西格尔系统，对于一般初始条件，所获得的收敛性被限制在有限的时间间隔内，但当初始条件的质量适当小时，收敛性对任意有界时间间隔有效。此外，如果极限系统的解在有限时间内炸毁，那么部分扩散系统中的两个表型在该时间之后都不能均匀地受 L2 规范约束。

引用次数: 0

Extending discrete geometric singular perturbation theory to non-hyperbolic points 将离散几何奇异扰动理论扩展到非双曲点

IF 1.7 2区数学 Q2 MATHEMATICS, APPLIED

Nonlinearity

Pub Date : 2024-09-02 DOI: 10.1088/1361-6544/ad72c5

S Jelbart, C Kuehn

We extend the recently developed discrete geometric singular perturbation theory to the non-normally hyperbolic regime. Our primary tool is the Takens embedding theorem, which provides a means of approximating the dynamics of particular maps with the time-1 map of a formal vector field. First, we show that the so-called reduced map, which governs the slow dynamics near slow manifolds in the normally hyperbolic regime, can be locally approximated by the time-1 map of the reduced vector field which appears in continuous-time geometric singular perturbation theory. In the non-normally hyperbolic regime, we show that the dynamics of fast-slow maps with a unipotent linear part can be locally approximated by the time-1 map induced by a fast-slow vector field in the same dimension, which has a nilpotent singularity of the corresponding type. The latter result is used to describe (i) the local dynamics of two-dimensional fast-slow maps with non-normally singularities of regular fold, transcritical and pitchfork type, and (ii) dynamics on a (potentially high-dimensional) local center manifold in n-dimensional fast-slow maps with regular contact or fold submanifolds of the critical manifold. In general, our results show that the dynamics near a large and important class of singularities in fast-slow maps can be described via the use of formal embedding theorems which allow for their approximation by the time-1 map of a fast-slow vector field featuring a loss of normal hyperbolicity.

我们将最近开发的离散几何奇异扰动理论扩展到非正态双曲系统。我们的主要工具是塔肯斯嵌入定理，它提供了一种用形式向量场的时间-1映射来近似特定映射动态的方法。首先，我们证明了所谓的还原映射（它支配着常态双曲系统中慢流形附近的慢动力学）可以用连续时间几何奇异扰动理论中出现的还原矢量场的时间-1映射局部逼近。在非正态双曲系统中，我们证明了具有单能线性部分的快慢映射的动力学可以由同一维度的快慢矢量场诱导的时间-1映射局部逼近，该矢量场具有相应类型的无能奇点。后一结果用于描述：(i) 具有规则折叠、跨临界和干草叉类型非正则奇点的二维快慢图的局部动力学；(ii) 具有临界流形的规则接触或折叠子流形的 n 维快慢图中（可能是高维）局部中心流形上的动力学。一般来说，我们的结果表明，可以通过使用形式嵌入定理来描述快慢图中一大类重要奇点附近的动力学，这些定理允许用具有法向双曲性损失的快慢向量场的时间-1映射来逼近它们。

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引用次数: 0

Complex-plane singularity dynamics for blow up in a nonlinear heat equation: analysis and computation 非线性热方程炸裂的复平面奇点动力学：分析与计算

IF 1.7 2区数学 Q2 MATHEMATICS, APPLIED

Nonlinearity

Pub Date : 2024-08-28 DOI: 10.1088/1361-6544/ad700b

M Fasondini, J R King, J A C Weideman

Blow-up solutions to a heat equation with spatial periodicity and a quadratic nonlinearity are studied through asymptotic analyses and a variety of numerical methods. The focus is on tracking the dynamics of the singularities in the complexified space domain all the way from the initial time until the blow-up time, which occurs when the singularities reach the real axis. This widely applicable approach gives forewarning of the possibility of blow up and an understanding of the influence of singularities on the solution behaviour on the real axis, aiding the (perhaps surprisingly involved) asymptotic analysis of the real-line behaviour. The analysis provides a distinction between small and large nonlinear effects, as well as insight into the various time scales over which blow up is approached. The solution to the nonlinear heat equation in the complex spatial plane is shown to be related asymptotically to a nonlinear ordinary differential equation. This latter equation is studied in detail, including its computation on multiple Riemann sheets, providing further insight into the singularities of blow-up solutions of the nonlinear heat equation when viewed as multivalued functions in the complex space domain and illustrating the potential intricacy of singularity dynamics in such (non-integrable) nonlinear contexts.

通过渐近分析和各种数值方法，研究了具有空间周期性和二次非线性的热方程的炸裂解。重点是跟踪复数化空间域中奇点的动态，从初始时间一直到爆破时间（奇点到达实轴时）。这种广泛应用的方法可以提前预知炸裂的可能性，并了解奇点对实轴上求解行为的影响，有助于对实线行为进行渐近分析（可能涉及到令人惊讶的内容）。通过分析，我们可以区分小非线性效应和大非线性效应，并深入了解接近炸裂的各种时间尺度。复空间平面上的非线性热方程的解被证明与一个非线性常微分方程渐近相关。对后一个方程进行了详细研究，包括其在多个黎曼片上的计算，进一步深入了解了非线性热方程的炸裂解在复数空间域中被视为多值函数时的奇异性，并说明了在这种（不可积分的）非线性背景下奇异性动态的潜在复杂性。

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引用次数: 0

Nijenhuis geometry IV: conservation laws, symmetries and integration of certain non-diagonalisable systems of hydrodynamic type in quadratures 奈亨伊斯几何 IV：某些非对角流体力学类型系统的守恒定律、对称性和二次积分

IF 1.7 2区数学 Q2 MATHEMATICS, APPLIED

Nonlinearity

Pub Date : 2024-08-27 DOI: 10.1088/1361-6544/ad6acc

Alexey V Bolsinov, Andrey Yu Konyaev, Vladimir S Matveev

The paper contains two lines of results: the first one is a study of symmetries and conservation laws of

-regular Nijenhuis operators. We prove the splitting theorem for symmetries and conservation laws of Nijenhuis operators, show that the space of symmetries of a

-regular Nijenhuis operator forms a commutative algebra with respect to (pointwise) matrix multiplication. Moreover, all the elements of this algebra are strong symmetries of each other. We establish a natural relationship between symmetries and conservation laws of a

-regular Nijenhuis operator and systems of the first and second companion coordinates. Moreover, we show that the space of conservation laws is naturally related to the space of symmetries in the sense that any conservation laws can be obtained from a single conservation law by multiplication with an appropriate symmetry. In particular, we provide an explicit description of all symmetries and conservation laws for

-regular operators at algebraically generic points. The second line of results contains an application of the theoretical part to a certain system of partial differential equations of hydrodynamic type, which was previously studied by different authors, but mainly in the diagonalisable case. We show that this system is integrable in quadratures, i.e. its solutions can be found for almost all initial curves by integrating closed 1-forms and solving some systems of functional equations. The system is not diagonalisable in general, and construction and integration of such systems is an actively studied and explicitly stated problem in the literature.

本文包含两方面的成果：第一方面是对g-正规尼延胡斯算子的对称性和守恒定律的研究。我们证明了尼延胡伊算子对称性和守恒定律的分裂定理，证明了g-正则尼延胡伊算子的对称性空间形成了一个关于（点式）矩阵乘法的交换代数。此外，这个代数的所有元素都是彼此的强对称性。我们在gl-非正规尼延胡斯算子的对称性和守恒定律与第一和第二伴坐标系之间建立了一种自然的关系。此外，我们还证明了守恒定律空间与对称性空间的自然关系，即任何守恒定律都可以通过与适当的对称性相乘而从单一守恒定律中得到。特别是，我们提供了对代数通项点上 gl-regular 算子的所有对称性和守恒律的明确描述。第二部分结果包含理论部分在某个流体力学类型偏微分方程系统中的应用，该系统以前由不同作者研究过，但主要是在可对角情况下。我们证明了该系统在四元数中是可积分的，即通过积分封闭的 1-forms 和求解某些函数方程组，可以找到几乎所有初始曲线的解。一般来说，该系统不可对角，而此类系统的构造和积分是文献中一个积极研究并明确阐述的问题。

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