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Biologically Inspired Pectoral Propulsors with Flapping and Rowing Control for a Specified Stroke Plane Angle 具有特定冲程平面角度的拍打和划船控制功能的生物启发胸桨
IF 3 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2024-09-17 DOI: 10.1007/s00332-024-10078-8
Bing Luo, Wei Li

Many flying and swimming creatures have morphing pectoral propulsors (wings or fins) for propulsion, typically with flapping, rowing, and pitching motions; flapping and rowing motions are responsible for the stroke plane angle that is important for a broader performance space of the propulsor, while the stroke plane angle has been less characterized and implemented by artificial propulsors of biomimetic vehicles and thus has lack of stroke plane angle control. In this paper, we consider robotic pectoral propulsors with combined flapping and rowing motions for a stroke plane angle that can be generally specified. We consider two possible rotation axes configurations (i.e., the dependence of the rotation axes for flapping and rowing). For each rotation axes configuration, we propose the kinematic relations between the flapping and rowing motions for a generally specified stroke plane angle and provide the general flapping (or rowing) kinematics as a function of the rowing (or flapping) kinematics, which have not been characterized previously. These results serve as the reference trajectories of the propulsor for specified stroke plane angles and have implications for stroke plane angle control and thus have implications to achieve a broader performance space for biomimetic propulsors.

许多飞行和游泳生物都有用于推进的变形胸推进器(翼或鳍),通常具有拍打、划船和俯仰运动;拍打和划船运动负责冲程平面角度,而冲程平面角度对于推进器更广阔的性能空间非常重要,而生物仿生飞行器的人工推进器对冲程平面角度的描述和实现较少,因此缺乏冲程平面角度控制。在本文中,我们考虑了具有拍打和划船组合运动的机器人胸肌推进器,其冲程平面角度可以大致确定。我们考虑了两种可能的旋转轴配置(即拍打和划船旋转轴的依赖关系)。对于每种旋转轴配置,我们都提出了在一般指定的划水平面角度下拍打运动和划船运动之间的运动学关系,并提供了作为划船(或拍打)运动学函数的一般拍打(或划船)运动学,而这些运动学特征之前还没有被描述过。这些结果可作为特定冲程平面角度下推进器的参考轨迹,对冲程平面角度控制有影响,因此对生物仿生推进器实现更广阔的性能空间有影响。
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引用次数: 0
Critical Transitions for Asymptotically Concave or d-Concave Nonautonomous Differential Equations with Applications in Ecology 渐近凹或 d-Concave 非自治微分方程的临界转换及其在生态学中的应用
IF 3 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2024-09-16 DOI: 10.1007/s00332-024-10088-6
Jesús Dueñas, Carmen Núñez, Rafael Obaya

The occurrence of tracking or tipping situations for a transition equation (x'=f(t,x,Gamma (t,x))) with asymptotic limits (x'=f(t,x,Gamma _pm (t,x))) is analyzed. The approaching condition is just (lim _{trightarrow pm infty }(Gamma (t,x)-Gamma _pm (t,x))=0) uniformly on compact real sets, and so there is no restriction to the dependence on time of the asymptotic equations. The hypotheses assume concavity in x either of the maps (xmapsto f(t,x,Gamma _pm (t,x))) or of their derivatives with respect to the state variable (d-concavity), but not of (xmapsto f(t,x,Gamma (t,x))) nor of its derivative. The analysis provides a powerful tool to analyze the occurrence of critical transitions for one-parametric families (x'=f(t,x,Gamma ^c(t,x))). The new approach significatively widens the field of application of the results, since the evolution law of the transition equation can be essentially different from those of the limit equations. Among these applications, some scalar population dynamics models subject to nontrivial predation and migration patterns are analyzed, both theoretically and numerically. Some key points in the proofs are: to understand the transition equation as part of an orbit in its hull which approaches the -limit and -limit sets; to observe that these sets concentrate all the ergodic measures; and to prove that in order to describe the dynamical possibilities of the equation it is sufficient that the concavity or d-concavity conditions hold for a complete measure subset of the equations of the hull.

分析了具有渐近极限 (x'=f(t,x,Gamma _pm (t,x)) 的过渡方程 (x'=f(t,x,Gamma _pm (t,x)) 的跟踪或临界情况的发生。逼近条件只是 (lim _{trightarrow pm infty }(Gamma (t,x)-Gamma _pm (t,x))=0/)均匀地在紧凑实集上,因此对渐近方程对时间的依赖没有限制。假设假定映射(x/mapsto f(t,x,Gamma _pm (t,x))或它们关于状态变量的导数(d-凹性)在x上是凹性的,但不是映射(x/mapsto f(t,x,Gamma (t,x))或它的导数的凹性。该分析为分析一参数族 (x'=f(t,x,Gamma ^c(t,x))) 临界转换的发生提供了强有力的工具。新方法大大拓宽了结果的应用领域,因为过渡方程的演化规律可能与极限方程的演化规律有本质区别。在这些应用中,我们从理论和数值两方面分析了一些受非琐碎捕食和迁移模式影响的标量种群动力学模型。证明中的一些关键点是:将过渡方程理解为其全壳中接近-极限集和-极限集的轨道的一部分;观察到这些集集中了所有的遍历度量;证明为了描述方程的动力学可能性,全壳方程的完整度量子集的凹性或d-凹性条件成立就足够了。
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引用次数: 0
A Resonant Lyapunov Centre Theorem with an Application to Doubly Periodic Travelling Hydroelastic Waves 共振李亚普诺夫中心定理在双周期水弹性游波中的应用
IF 3 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2024-09-14 DOI: 10.1007/s00332-024-10073-z
R. Ahmad, M. D. Groves, D. Nilsson

We present a Lyapunov centre theorem for an antisymplectically reversible Hamiltonian system exhibiting a nondegenerate 1 : 1 or (1:-1) semisimple resonance as a detuning parameter is varied. The system can be finite- or infinite-dimensional (and quasilinear) and have a non-constant symplectic structure. We allow the origin to be a ‘trivial’ eigenvalue arising from a translational symmetry or, in an infinite-dimensional setting, to lie in the continuous spectrum of the linearised Hamiltonian vector field provided a compatibility condition on its range is satisfied. As an application, we show how Kirchgässner’s spatial dynamics approach can be used to construct doubly periodic travelling waves on the surface of a three-dimensional body of water (of finite or infinite depth) beneath a thin ice sheet (‘hydroelastic waves’). The hydrodynamic problem is formulated as a reversible Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable, and the infinite-dimensional phase space consists of wave profiles which are periodic (with fixed period) in a second, different horizontal direction. Applying our Lyapunov centre theorem at a point in parameter space associated with a 1 : 1 or (1:-1) semisimple resonance yields a periodic solution of the spatial Hamiltonian system corresponding to a doubly periodic hydroelastic wave.

我们提出了一个反交可逆哈密顿系统的李雅普诺夫中心定理,该系统随着解谐参数的变化而呈现出非enerate 1 : 1 或 (1:-1) 半简单共振。该系统可以是有限维或无限维(和准线性)的,并具有非恒定交映结构。我们允许原点是一个由平移对称性产生的 "琐碎 "特征值,或者在无限维环境中,只要满足其范围的相容性条件,原点就可以位于线性化哈密顿矢量场的连续谱中。作为应用,我们展示了如何利用基尔希格斯纳的空间动力学方法在薄冰下的三维水体(有限或无限深度)表面上构建双周期行波("水弹性波")。水动力问题被表述为一个可逆哈密顿系统,其中任意水平空间方向是类时间变量,而无限维相空间由在第二个不同水平方向上周期性(具有固定周期)的波浪剖面组成。在参数空间中与 1 : 1 或 (1:-1)半简单共振相关的点上应用我们的 Lyapunov 中心定理,可以得到空间哈密顿系统的周期解,该解对应于双周期水弹性波。
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引用次数: 0
Solitary-Wave Solutions of the Fractional Nonlinear Schrödinger Equation: I—Existence and Numerical Generation 分数非线性薛定谔方程的孤波解:I-存在性与数值生成
IF 3 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2024-09-11 DOI: 10.1007/s00332-024-10086-8
Angel Durán, Nuria Reguera

The present paper is the first part of a project devoted to the fractional nonlinear Schrödinger (fNLS) equation. It is concerned with the existence and numerical generation of the solitary-wave solutions. For the first point, some conserved quantities of the problem are used to search for solitary-wave solutions from a constrained critical point problem and the application of the concentration-compactness theory. Several properties of the waves, such as the regularity and the asymptotic decay in some cases, are derived from the existence result. Some other properties, such as the monotone behavior and the speed-amplitude relation, will be explored computationally. To this end, a numerical procedure for the generation of the profiles is proposed. The method is based on a Fourier pseudospectral approximation of the differential system for the profiles and the use of Petviashvili’s iteration with extrapolation.

本文是分数非线性薛定谔方程(fNLS)项目的第一部分。它关注孤波解的存在和数值生成。对于第一部分,问题的一些守恒量被用来从受约束临界点问题和集中-紧密性理论的应用中寻找孤波解。根据存在性结果推导出了孤波的一些性质,如规律性和某些情况下的渐近衰减。其他一些特性,如单调行为和速度-振幅关系,将通过计算来探索。为此,我们提出了一种生成剖面的数值程序。该方法基于轮廓微分系统的傅立叶伪谱近似和 Petviashvili 外推法迭代。
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引用次数: 0
Chaotic Dynamics at the Boundary of a Basin of Attraction via Non-transversal Intersections for a Non-global Smooth Diffeomorphism 通过非全局光滑衍射的非横向交叉在吸引盆地边界的混沌动力学
IF 3 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2024-09-11 DOI: 10.1007/s00332-024-10079-7
Ernest Fontich, Antonio Garijo, Xavier Jarque

In this paper, we give analytic proofs of the existence of transversal homoclinic points for a family of non-globally smooth diffeomorphisms having the origin as a fixed point which come out as a truncated map governing the local dynamics near a critical period three-cycle associated with the Secant map. Using Moser’s version of Birkhoff–Smale’s theorem, we prove that the boundary of the basin of attraction of the origin contains a Cantor-like invariant subset such that the restricted dynamics to it is conjugate to the full shift of N-symbols for any integer (Nge 2) or infinity.

在本文中,我们给出了以原点为定点的非全局平滑差分变换系的横向同轴点存在性的解析证明,该系以截断映射的形式出现,支配着与塞康特映射相关的临界周期三周期附近的局部动力学。利用莫泽尔(Moser)版本的伯克霍夫-斯梅尔(Birkhoff-Smale)定理,我们证明了原点吸引盆的边界包含一个类似康托尔(Cantor)的不变量子集,对于任意整数(Nge 2)或无穷大,该子集的受限动力学与N符号的全移共轭。
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引用次数: 0
Stability and Bifurcation Analysis of a Reaction–Diffusion SIRS Epidemic Model with the General Saturated Incidence Rate 具有一般饱和发病率的反应-扩散 SIRS 流行模型的稳定性和分岔分析
IF 3 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2024-09-04 DOI: 10.1007/s00332-024-10081-z
Gaoyang She, Fengqi Yi

In this paper, we are concerned with the dynamics of a reaction–diffusion SIRS epidemic model with the general saturated nonlinear incidence rates. Firstly, we show the global existence and boundedness of the in-time solutions for the parabolic system. Secondly, for the ODEs system, we analyze the existence and stability of the disease-free equilibrium solution, the endemic equilibrium solutions as well as the bifurcating periodic solution. In particular, in the language of the basic reproduction number, we are able to address the existence of the saddle-node-like bifurcation and the secondary bifurcation (Hopf bifurcation). Our results also suggest that the ODEs system has a Allee effect, i.e., one can expect either the coexistence of a stable disease-free equilibrium and a stable endemic equilibrium solution, or the coexistence of a stable disease-free equilibrium solution and a stable periodic solution. Finally, for the PDEs system, we are capable of deriving the Turing instability criteria in terms of the diffusion rates for both the endemic equilibrium solutions and the Hopf bifurcating periodic solution. The onset of Turing instability can bring out multi-level bifurcations and manifest itself as the appearance of new spatiotemporal patterns. It seems also interesting to note that p and k, appearing in the saturated incidence rate (kSI^p/(1+alpha I^p)), tend to play far reaching roles in the spatiotemporal pattern formations.

本文关注的是具有一般饱和非线性发病率的反应扩散 SIRS 流行病模型的动力学。首先,我们证明了抛物线系统的全局存在性和实时解的有界性。其次,对于 ODEs 系统,我们分析了无病平衡解、流行平衡解以及分岔周期解的存在性和稳定性。特别是,在基本繁殖数的语言中,我们能够解决鞍结状分岔和二次分岔(霍普夫分岔)的存在问题。我们的结果还表明,ODEs 系统具有阿利效应,即可以预期稳定的无病平衡解和稳定的地方病平衡解共存,或者稳定的无病平衡解和稳定的周期解共存。最后,对于 PDEs 系统,我们能够根据地方性平衡解和霍普夫分岔周期解的扩散率推导出图灵不稳定性标准。图灵不稳定性的出现会带来多级分岔,并表现为新的时空模式的出现。值得注意的是,出现在饱和发生率 (kSI^p/(1+alpha I^p)) 中的 p 和 k 往往在时空模式形成中发挥深远的作用。
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引用次数: 0
Measure-Valued Structured Deformations 量值结构变形
IF 3 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2024-09-03 DOI: 10.1007/s00332-024-10076-w
Stefan Krömer, Martin Kružík, Marco Morandotti, Elvira Zappale

Measure-valued structured deformations are introduced to present a unified theory of deformations of continua. The energy associated with a measure-valued structured deformation is defined via relaxation departing either from energies associated with classical deformations or from energies associated with structured deformations. A concise integral representation of the energy functional is provided both in the unconstrained case and under Dirichlet conditions on a part of the boundary.

引入量值结构变形是为了提出连续体变形的统一理论。与量值结构变形相关的能量是通过从与经典变形相关的能量或与结构变形相关的能量出发的松弛来定义的。无论是在无约束情况下,还是在部分边界的迪里希特条件下,都提供了能量函数的简明积分表示。
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引用次数: 0
The Dynamics of Periodic Traveling Interfacial Electrohydrodynamic Waves: Bifurcation and Secondary Bifurcation 周期性行进的界面电流体动力波的动力学:分岔和二次分岔
IF 3 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2024-09-02 DOI: 10.1007/s00332-024-10085-9
Guowei Dai, Fei Xu, Yong Zhang

In this paper, we consider two-dimensional periodic capillary-gravity waves traveling under the influence of a vertical electric field. The full system is a nonlinear, two-layered, free boundary problem. The interface dynamics are derived by coupling Euler equations for the velocity field of the fluid with voltage potential equations governing the electric field. We first introduce the naive flattening technique to transform the free boundary problem into a fixed boundary problem. We then prove the existence of small-amplitude electrohydrodynamic waves with constant vorticity using local bifurcation theory. Moreover, we show that these electrohydrodynamic waves are formally stable in the linearized sense. Furthermore, we obtain a secondary bifurcation curve that emerges from the primary branch, consisting of ripple solutions on the interface. As far as we know, such solutions in electrohydrodynamics are established for the first time. It is worth noting that the electric field (E_0) plays a key role in controlling the shapes and types of waves on the interface.

在本文中,我们考虑了在垂直电场影响下行进的二维周期性毛细重力波。整个系统是一个非线性、双层、自由边界问题。通过将流体速度场的欧拉方程与控制电场的电压电势方程耦合,得出了界面动力学。我们首先引入了天真平坦化技术,将自由边界问题转化为固定边界问题。然后,我们利用局部分岔理论证明了具有恒定涡度的小振幅电流体动力波的存在。此外,我们还证明了这些电流体动力波在线性化意义上是形式稳定的。此外,我们还得到了一条从主分支中产生的次级分岔曲线,由界面上的波纹解组成。据我们所知,这是首次在电流体力学中建立这种解。值得注意的是,电场 (E_0) 在控制界面上波的形状和类型方面起着关键作用。
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引用次数: 0
Integrable Variants of the Toda Lattice 户田格子的可积分变体
IF 3 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2024-08-29 DOI: 10.1007/s00332-024-10072-0
Ya-Jie Liu, Hui Alan Wang, Xiang-Ke Chang, Xing-Biao Hu, Ying-Nan Zhang

By introducing bilinear operators of trigonometric type, we propose several novel integrable variants of the famous Toda lattice, two of which can be regarded as integrable discretizations of the Kadomtsev–Petviashvili equation—a universal model describing weakly nonlinear waves in media with dispersion of velocity. We also demonstrate that two one-dimensional reductions of these variants can approximate the nonlinear Schrödinger equation and a generalized nonlinear Schrödinger equation well. It turns out that these equations admit meaningful solutions including solitons, breathers, lumps and rogue waves, which are expressed in terms of explicit and closed forms. In particular, it seems to be the first time that rogue wave solutions have been obtained for Toda-type equations. Furthermore, g-periodic wave solutions are also produced in terms of Riemann theta function. An approximation solution of the three-periodic wave is successfully carried out by using a deep neural network. The introduction of trigonometric-type bilinear operators is also efficient in generating new variants together with rich properties for some other integrable equations.

通过引入三角函数类型的双线性算子,我们提出了著名的户田晶格的几种新的可积分变体,其中两种可被视为卡多姆采夫-佩特维亚什维利方程的可积分离散化--该方程是描述具有速度色散的介质中弱非线性波的通用模型。我们还证明,这些变体的两个一维还原可以很好地逼近非线性薛定谔方程和广义非线性薛定谔方程。结果表明,这些方程允许有意义的解,包括孤子、呼吸器、肿块和流氓波,这些都可以用显式和闭式表达。尤其是,这似乎是第一次获得托达方程的流氓波解。此外,还用黎曼θ函数得到了g周期波解。利用深度神经网络成功地实现了三周期波的近似解。三角型双线性算子的引入也有效地为其他一些可积分方程生成了新的变体和丰富的特性。
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引用次数: 0
Finite-Time Analysis of Crises in a Chaotically Forced Ocean Model 混沌强迫海洋模型危机的有限时间分析
IF 3 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2024-08-28 DOI: 10.1007/s00332-024-10077-9
Andrew R. Axelsen, Courtney R. Quinn, Andrew P. Bassom

We consider a coupling of the Stommel box model and the Lorenz model, with the goal of investigating the so-called crises that are known to occur given sufficient forcing. In this context, a crisis is characterized as the destruction of a chaotic attractor under a critical forcing strength. We document the variety of chaotic attractors and crises possible in our model, focusing on the parameter region where the Lorenz model is always chaotic and where bistability exists in the Stommel box model. The chaotic saddle collisions that occur in a boundary crisis are visualized, with the chaotic saddle computed using the Saddle-Straddle Algorithm. We identify a novel sub-type of boundary crisis, namely a vanishing basin crisis. For forcing strength beyond the crisis, we demonstrate the possibility of a merging between the persisting chaotic attractor and either a chaotic transient or a ghost attractor depending on the type of boundary crisis. An investigation of the finite-time Lyapunov exponents around crisis levels of forcing reveals a convergence between two near-neutral exponents, particularly at points of a trajectory most sensitive to divergence. This points to loss of hyperbolicity associated with crisis occurrence. Finally, we generalize our findings by coupling the Stommel box model to other strange attractors and thereby show that the behaviors are quite generic and robust.

我们考虑了斯托梅尔箱体模型和洛伦兹模型的耦合,目的是研究已知在足够的作用力下会发生的所谓危机。在这种情况下,危机的特征是混沌吸引子在临界强迫强度下的破坏。我们记录了模型中可能出现的各种混沌吸引子和危机,重点是洛伦兹模型总是混沌的参数区域和斯托梅尔箱模型中存在双稳态的参数区域。边界危机中发生的混沌鞍碰撞是可视化的,混沌鞍是用鞍-鞍算法计算出来的。我们发现了一种新的边界危机子类型,即消失盆地危机。对于超越危机的强迫强度,我们证明了持续混沌吸引子与混沌瞬态或幽灵吸引子合并的可能性,这取决于边界危机的类型。对危机水平附近的有限时间李亚普诺夫指数的研究表明,两个接近中性的指数之间出现了趋同,特别是在对发散最敏感的轨迹点上。这表明危机发生时双曲线的丧失。最后,我们通过将斯托梅尔箱模型与其他奇异吸引子耦合,对我们的发现进行了归纳,从而表明这些行为是非常通用和稳健的。
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引用次数: 0
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Journal of Nonlinear Science
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