Integrodifference equations are versatile models in theoretical ecology for the spatial dispersal of species evolving in non-overlapping generations. The dynamics of these infinite-dimensional discrete dynamical systems is often illustrated using computational simulations. This paper studies the effect of Nyström discretization to the local dynamics of periodic integrodifference equations with Hölder continuous functions over a compact domain as state space. We prove persistence and convergence for hyperbolic periodic solutions and their associated stable and unstable manifolds respecting the convergence order of the quadrature/cubature method.
We study how the "full hierarchy" of invariant manifolds for nonautonomous integrodifference equations on the Banach spaces of $p$-integrable functions behaves under spatial discretization of Galerkin type. These manifolds include the classical stable, center-stable, center, center-unstable and unstable ones, and can be represented as graphs of $C^m$-functions. For kernels with a smoothing property, our main result establishes closeness of these graphs in the $C^{m-1}$-topology under numerical discretizations preserving the convergence order of the method. It is formulated in a quantitative fa…
In this paper, we study the propagation dynamics for a class of integrodifference competition models in a periodic habitat. An interesting feature of such a system is that multiple spreading speeds can be observed, which biologically means different species may have different spreading speeds. We show that the model system admits a single spreading speed, and it coincides with the minimal wave speed of the spatially periodic traveling waves. A set of sufficient conditions for linear determinacy of the spreading speed is also given.
In this paper, we study a class of time-periodic stochastic Tonelli Lagrangians on compact manifolds. Precisely, we discuss the stochastic Mane Critical Value, prove the existence of stochastic Weak KAM solutions of the related Hamilton-Jacobi equation. Furthermore, we survey the global minimizer.
We describe the topological structure of closed manifolds of dimension no less than four which admit Morse-Smale diffeomorphisms such that its non-wandering set contains any number of sink periodic points, and any number of source periodic points, and few saddle periodic points.
On any closed Riemannian 3-manifold which is not a torus bundle, every nonvanishing analytic solution of the stationary Euler equations has a periodic trajectory. This result is originally due to A. Rechtman (arXiv:0904.2719) and K. Cieliebak and E. Volkov (arXiv:1402.6484); here we present an alternative proof of it.
The goal of this paper is to study periodic geodesics for sub-Riemannian metrics on a contact 3D-manifold.We develop two rather independent subjects:1) The existence of closed geodesics spiraling around periodic Reeb orbits for a generic metric.2) The precise study of the periodic geodesics for a right invariant metric on a quotient of SL2(R)
When the planar circular restricted 3-body problem (RTBP) is periodically perturbed, families of unstable periodic orbits break up into whiskered tori, with most tori persisting into the perturbed system. In this study, we 1) develop a quasi-Newton method which simultaneously solves for the tori and their center, stable, and unstable directions; 2) implement continuation by both perturbation as well as rotation numbers; 3) compute Fourier-Taylor parameterizations of the stable and unstable manifolds; 4) regularize the equations of motion; and 5) globalize these manifolds. Our methodology impro…
In this paper we study the existence of periodic orbits with prescribed energy levels of convex Lagrangian systems on complete Riemannian manifolds. We extend the existence results of Contreras by developing a modified minimax principal to a class of Lagrangian systems on noncompact Riemannian manifolds, namely the so called $\lsh$ Lagrangian systems. In particular, we prove that for almost every $k\in(0,c_u(L))$ the exact magnetic flow associated to a $\lsh$ Lagrangian has a contractible periodic orbit with energy $k$. We also discuss the existence and non-existence of closed geodesics on the…
This paper is concerned with the derivative nonlinear Schrodinger equation with periodic boundary conditions $$\mathbf{i}u_t+u_{xx}+\mathbf{i}\Big(f(x,u,\bar{u})\Big)_x=0,\quad x\in\mathbb{T}:=\mathbb{R}/2π\mathbb{Z},$$ where $f$ is an analytic function of the form $$f(x,u,\bar{u})=μ|u|^2u+f_{\geq4}(x,u,\bar{u}),\quad 0\neqμ\in\mathbb{R},$$ and $f_{\geq4}(x,u,\bar{u})$ denotes terms of order at least four in $u,\bar{u}$. We show the above equation possesses Cantor families of smooth quasi-periodic solutions of small amplitude. The proof is based on an infinite dimensional KAM theorem for unbou…
We address the problem of counting periodic orbits of vector fields on smooth closed manifolds. The space of non-constant periodic orbits is enlarged to a complete space by adding the ghost orbits, which are decorations of the zeros of vector fields. Associated with any compact and open subset $Γ$ of the moduli space of periodic and ghost orbits, we define an integer weight. When the vector field moves along a path, and $Γ$ deforms in a compact and open family, we show that the weight function stays constant. We also give a number of examples and computations, which illustrate the applications…
M. Kruskal showed that each nearly-periodic dynamical system admits a formal $U(1)$ symmetry, generated by the so-called roto-rate. We prove that such systems also admit nearly-invariant manifolds of each order, near which rapid oscillations are suppressed. We study the nonlinear normal stability of these slow manifolds for nearly-periodic Hamiltonian systems on barely symplectic manifolds -- manifolds equipped with closed, non-degenerate $2$-forms that may be degenerate to leading order. In particular, we establish a sufficient condition for long-term normal stability based on second derivati…
In this manuscript we consider the stability of periodic solutions to Lambda-Omega lattice dynamical systems. In particular, we show that an appropriate ansatz casts the lattice dynamical system as an infinite-dimensional fast-slow differential equation. In a neighbourhood of the periodic solution an invariant slow manifold is proven to exist, and that this slow manifold is uniformly exponentially attracting. The dynamics of solutions on the slow manifold become significantly more complicated and require a more delicate treatment. We present sufficient conditions to guarantee convergence on th…
We study the dynamics of magnetic flows on Heisenberg groups. Let $H$ denote the three-dimensional simply connected Heisenberg Lie group endowed with a left-invariant Riemannian metric and an exact, left-invariant magnetic field. Let $Γ$ be a lattice subgroup of $H,$ so that $Γ\backslash H$ is a closed nilmanifold. We first find an explicit description of magnetic geodesics on $H$, then determine all closed magnetic geodesics and their lengths for $Γ\backslash H$. We then consider two applications of these results: the density of periodic magnetic geodesics and marked magnetic length spectrum …
We prove the equidistribution of (weighted) periodic orbits of the geodesic ow on noncompact negatively curved manifolds toward equilibrium states in the narrow topology, i.e. in the dual of bounded continuous functions. We deduce an exact asymptotic counting for periodic orbits (weighted or not), which was previously known only for geometrically nite manifolds.
Multi-shooting parameterization methods for invariant manifolds and heteroclinics of 2 DOF Hamiltonian Poincaré maps, with applications to celestial resonant dynamics
A Lagrangian system with singularities is considered. The configuration space is a non-compact manifold that depends on time. A set of periodic solutions has been found.
We show that the existence of noncontractible periodic orbits for compactly supported time-dependent Hamiltonian on the disk cotangent bundle of a Finsler manifold provided that the Hamiltonian is sufficiently large over the zero section. We generalize the BPS capacities and earlier constructions of Weber (2006 Duke Math. J. 133, 527-568) and other authors Biran et al (2003 Duke Math. J. 119, 65-118) to the Finsler setting. We then obtain a number of applications including: (1) generalizing the main theorem of Xue (2017 J. Symplectic Geom. 15, 905-936) to the Lie group setting, (2) preservatio…
Symmetric Lyapunov Subcenter Manifolds for Periodic Regulation of Mechanical Systems
University of Twente (UT)（特文特大学）；Technical University of Munich (TUM)（慕尼黑技术大学）；German Aerospace Center (DLR)（德国航空航天中心）；Delft University of Technology (TU Delft)（代尔夫特理工大学）
Consider a transitive Anosov flow on a closed $3$-manifold. After removing a finite set of null-homologous periodic orbits, we study the distribution of the remaining periodic orbits in the homology of the knot complement.
This paper concerns connections between dynamical systems, knots and helicity of vector fields. For a divergence-free vector field on a closed $3$-manifold that generates an Anosov flow, we show that the helicity of the vector field may be recovered as the limit of appropriately weighted averages of linking numbers of periodic orbits, regarded as knots. This complements a classical result of Arnold and Vogel that, when the manifold is a real homology $3$-sphere, the helicity may be obtained as the limit of the normalised linking numbers of typical pairs of long trajectories. We also obtain res…
