Searcharxiv⌕ Search

arXiv subjects

Yiming Long

Publications and source records attributed to Yiming Long.

At least 19 recordsLinked to original sources
arXiv2025

On the common index jump theorem and further developments

In [LZ02] published in Annals of Mathematics, Long and Zhu established originally the common index jump theorem (CIJT) for symplectic paths in 2002, which has played an important role in later studies on periodic solution orbits for Hamiltonian systems, Reeb flows, and geodesic problems. This (CIJT) was generalized to its enhanced version (ECIJT) by Duan, Long and Wang in [DLW16] in 2016. Started from [GG20] of 2020, and finally in [CGG24] of 2024, a similar index theorem was obtained, i.e., Theorem 3.3 of [CGG24], which was given the name "index recurrence theorem" there. In this short note, we give detailed proofs to show that the major assertions, i.e., the first 4 assertions in the total of 5 assertions, in Theorem 3.3 of [CGG24] as well as all the assertions in [GG20] actually coincide completely with results in (ECIJT) of [DLW16].

math.SG↗
arXiv2022

Generalized common index jump theorem with applications to closed characteristics on star-shaped hypersurfaces and beyond

In this paper, we first generalize the common index jump theorem of Long-Zhu in 2002 and Duan-Long-Wang in 2016 to the case where the mean indices of symplectic paths are not required to be all positive. As applications, we study closed characteristics on compact star-shaped hypersurfaces in ${\bf R}^{2n}$, when both positive and negative mean indices may appear simultaneously. Specially we establish the existence of at least $n$ geometrically distinct closed characteristics on every compact non-degenerate perfect star-shaped hypersurface $\Sigma$ in ${\bf R}^{2n}$ provided that every prime closed characteristic possesses nonzero mean index. Furthermore, in the case of ${\bf R}^6$ we remove the nonzero mean index condition by showing that the existence of only finitely many geometrically distinct closed characteristics implies that each of them must possess nonzero mean index. We also generalize the above results about closed characteristics on non-degenerate star-shaped hypersurfaces to closed Reeb orbits of non-degenerate contact forms on a broad class of prequantization bundles.

math.DS↗
arXiv2019

Linear stability of the elliptic relative equilibrium with $(1 +n)$-gon central configurations in planar $n$-body problem

We study the linear stability of $(1+n)$-gon elliptic relative equilibrium (ERE for short), that is the Kepler homographic solution with the $(1+n)$-gon central configurations. We show that for $n\geq 8$ and any eccentricity $e\in[0,1)$, the $(1+n)$-gon ERE is stable when the central mass $m$ is large enough. Some linear instability results are given when $m$ is small.

math.DS↗
arXiv2019

Concentrated steady vorticities of the Euler equation on 2-d domains and their linear stability

We consider concentrated vorticities for the Euler equation on a smooth domain $Ω\subset \mathbf{R}^2$ in the form of \[ ω= \sum_{j=1}^N ω_j χ_{Ω_j}, \quad |Ω_j| = πr_j^2, \quad \int_{Ω_j} ω_j dμ=μ_j \ne 0, \] supported on well-separated vortical domains $Ω_j$, $j=1, \ldots, N$, of small diameters $O(r_j)$. A conformal mapping framework is set up to study this free boundary problem with $Ω_j$ being part of unknowns. For any given vorticities $μ_1, \ldots, μ_N$ and small $r_1, \ldots, r_N\in \mathbf{R}^+$, through a perturbation approach, we obtain such piecewise constant steady vortex patches as well as piecewise smooth Lipschitz steady vorticities, both concentrated near non-degenerate critical configurations of the Kirchhoff-Routh Hamiltonian function. When vortex patch evolution is considered as the boundary dynamics of $\partial Ω_j$, through an invariant subspace decomposition, it is also proved that the spectral/linear stability of such steady vortex patches is largely determined by that of the $2N$-dimensional linearized point vortex dynamics, while the motion is highly oscillatory in the $2N$-codim directions corresponding to the vortical domain shapes.

math.AP↗
arXiv2017

On bifurcation of eigenvalues along convex symplectic paths

We consider a continuously differentiable curve $t\mapsto γ(t)$ in the space of $2n\times 2n$ real symplectic matrices, which is the solution of the following ODE: $\frac{\mathrm{d}γ}{\mathrm{d}t}(t)=J_{2n}A(t)γ(t), γ(0)\in\operatorname{Sp}(2n,\mathbb{R})$, where $J=J_{2n}\overset{\text{def}}{=}\begin{bmatrix}0 & \operatorname{Id}_n\\-\operatorname{Id}_n & 0\end{bmatrix}$ and $A:t\mapsto A(t)$ is a continuous in the space of $2n\times2n$ real matrices which are symmetric. Under certain convexity assumption (which includes the particular case that $A(t)$ is strictly positive definite for all $t\in\mathbb{R}$), we investigate the dynamics of the eigenvalues of $γ(t)$ when $t$ varies, which are closely related to the stability of such Hamiltonian dynamical systems. We rigorously prove the qualitative behavior of the branching of eigenvalues and explicitly give the first order asymptotics of the eigenvalues. This generalizes classical Krein-Lyubarskii theorem on the analytic bifurcation of the Floquet multipliers under a linear perturbation of the Hamiltonian. As a corollary, we give a rigorous proof of the following statement of Ekeland: $\{t\in\mathbb{R}:γ(t)\text{ has a Krein indefinite eigenvalue of modulus }1\}$ is a discrete set.

math.DS↗
arXiv2017

The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form

Let $M=S^n/ Γ$ and $h$ be a nontrivial element of finite order $p$ in $π_1(M)$, where the integer $n\geq2$, $Γ$ is a finite group which acts freely and isometrically on the $n$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we establish first the resonance identity for non-contractible homologically visible minimal closed geodesics of the class $[h]$ on every Finsler compact space form $(M, F)$ when there exist only finitely many distinct non-contractible closed geodesics of the class $[h]$ on $(M, F)$. Then as an application of this resonance identity, we prove the existence of at least two distinct non-contractible closed geodesics of the class $[h]$ on $(M, F)$ with a bumpy Finsler metric, which improves a result of Taimanov in [Taimanov 2016] by removing some additional conditions. Also our results extend the resonance identity and multiplicity results on $\mathcal{R}P^n$ in [arXiv:1607.02746] to general compact space forms.

math.DS↗
arXiv2016

The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds

In this paper, we first generalize the common index jump theorem for symplectic matrix paths proved in 2002 by Long and Zhu in [LoZ], and get an enhanced version of it. As its applications, we further prove that for a compact simply-connected manifold $(M,F)$ with a bumpy, irreversible Finsler metric $F$ and $H^*(M;{\bf Q})\cong T_{d,n+1}(x)$ for some even integer $d\ge 2$ and integer $n\ge 1$, there exist at least $\frac{dn(n+1)}{2}$ distinct non-hyperbolic closed geodesics with odd Morse indices, provided the number of distinct prime closed geodesics is finite and every prime closed geodesic satisfies $i(c)>0$. Note that the last non-zero index condition is satisfied if the flag curvature $K$ satisfies $K\ge 0$. For an odd-dimensional bumpy Finsler sphere $(S^d,F)$, there exist at least $(d+1)$ distinct prime closed geodesics with even Morse indices, and at least $(d-1)$ of which are non-hyperbolic, provided the number of distinct prime closed geodesics is finite and every prime closed geodesic $c$ satisfies $i(c)\ge 2$. Note that the last index condition $i(c)\ge 2$ is satisfied if the reversibility $λ$ and the flag curvature $K$ of $(M,F)$ satisfy $\frac{λ^2}{(1+λ)^2}<K\le 1$. Note that the first two in the above three lower bound estimates are sharp due to Katok's examples. In addition, we also prove that either there exists at least one non-hyperbolic closed geodesic, or there exist infinitely many distinct closed geodesics on a compact simply connected bumpy Finsler $(M,F)$ satisfying the above cohomological condition with some even integer $d\ge 2$ and integer $n\ge 1$.

math.SG↗
arXiv2015

Non-hyperbolic closed characteristics on non-degenerate star-shaped hypersurfaces in ${\bf R}^{2n}$

In this paper, we prove that for every index perfect non-degenerate compact star-shaped hypersurface $Σ\subset{\bf R}^{2n}$, there exist at least $n$ non-hyperbolic closed characteristics with even Maslov-type indices on $Σ$ when $n$ is even. When $n$ is odd, there exist at least $n$ closed characteristics with odd Maslov-type indices on $Σ$ and at least $(n-1)$ of them are non-hyperbolic. Here we call a compact star-shaped hypersurface $Σ\subset {\bf R}^{2n}$ {\rm index perfect} if it carries only finitely many geometrically distinct prime closed characteristics, and every prime closed characteristic $(τ,y)$ on $Σ$ possesses positive mean index and whose Maslov-type index $i(y, m)$ of its $m$-th iterate satisfies $i(y, m)\not= -1$ when $n$ is even, and $i(y, m)\not\in \{-2,-1,0\}$ when $n$ is odd for all $m\in {\bf N}$.

math.SG↗
arXiv2015

The reduction on the linear stability of elliptic Euler-Moulton solutions of the n-body problem to those of 3-body problems

In this paper, we consider the elliptic collinear solutions of the classical $n$-body problem, where the $n$ bodies always stay on a straight line, and each of them moves on its own elliptic orbit with the same eccentricity. Such a motion is called an elliptic Euler-Moulton collinear solution. Here we prove that the corresponding linearized Hamiltonian system at such an elliptic Euler-Moulton collinear solution of $n$-bodies splits into $(n-1)$ independent linear Hamiltonian systems, the first one is the linearized Hamiltonian system of the Kepler $2$-body problem at Kepler elliptic orbit, and each of the other $(n-2)$ systems is the essential part of the linearized Hamiltonian system at an elliptic Euler collinear solution of a $3$-body problem whose mass parameter is modified. Then the linear stability of such a solution in the $n$-body problem is reduced to those of the corresponding elliptic Euler collinear solutions of the $3$-body problems, which for example then can be further understood using numerical results of Martin\'ez, Sam\`a and Sim\'o in \cite{MSS1} and \cite{MSS2} on $3$-body Euler solutions in 2004-2006. As an example, we carry out the detailed derivation of the linear stability for an elliptic Euler-Moulton solution of the $4$-body problem with two small masses in the middle.

math.DS↗
arXiv2015

A New Class of Problems in the Calculus of Variations

This paper investigates an infinite-horizon problems in the one-dimensional calculus of variations, arising from the Ramsey model of endogeneous economic growth. Following Chichilnisky, we introduce an additional term, which models concern for the well-being of future generations. We show that there are no optimal solutions, but that there are equilibrium strateges, i.e. Nash equilibria of the leader-follower game between successive generations. To solve the problem, we approximate the Chichilnisky criterion by a biexponential criterion, we characterize its equilibria by a pair of coupled differential equations of HJB type, and we go to the limit. We find all the equilibrium strategies for the Chichilnisky criterion. The mathematical analysis is difficult because one has to solve an implicit differential equation in the sense of Thom. Our analysis extends earlier work by Ekeland and Lazrak. It is shown that optimal solutions a class of problems raising from time inconsistency problems in the framework of the neoclassical one-sector model of economic growth, and contains new results in environment economics. Without exogenous commitment mechanism, a notion of the equilibrium strategies instead of the optimal strategies is introduced. We characterized the equilibrium strategies by an integro-differential equation system. For two special criteria, the bi-exponential criteria and the Chichilnisky criteria, we established the existence of the equilibrium strategies.

econ.GN↗
arXiv2015

Maslov-type indices and linear stability of elliptic Euler solutions of the three-body problem

In this paper, we use the central configuration coordinate decomposition to study the linearized Hamiltonian system near the elliptic Euler solutions. Then using the Maslov-type \omega-index theory of symplectic paths and the theory of linear operators we compute the \omega-indices and obtain certain properties of linear stability of the Euler elliptic solutions of the classical three-body problem.

math.DS↗
arXiv2015

Equivalence of linear stabilities of elliptic triangle solutions of the planar charged and classical three-body problems

In this paper, we prove that the linearized system of elliptic triangle homographic solution of planar charged three-body problem can be transformed to that of the elliptic equilateral triangle solution of the planar classical three-body problem. Consequently, the results of Mart\'{\i}nez, Sam\`{a} and Sim\'{o} ([15] in J. Diff. Equa.) of 2006 and results of Hu, Long and Sun ([6] in Arch. Ration. Mech.Anal.) of 2014 can be applied to these solutions of the charged three-body problem to get their linear stability.

math.DS↗
arXiv2015

Resonance identities and stability of symmetric closed characteristics on symmetric compact star-shaped hypersurfaces

So far, it is still unknown whether all the closed characteristics on a symmetric compact star-shaped hypersurface $Σ$ in ${\bf R}^{2n}$ are symmetric. In order to understand behaviors of such orbits, in this paper we establish first two new resonance identities for symmetric closed characteristics on symmetric compact star-shaped hypersurface $Σ$ in ${\bf R}^{2n}$ when there exist only finitely many geometrically distinct symmetric closed characteristics on $Σ$, which extend the identity established by Liu and Long in \cite{LLo1} of 2013 for symmetric strictly convex hypersurfaces. Then as an application of these identities and the identities established by Liu, Long and Wang recently in \cite{LLW1} for all closed characteristics on the same hypersurface, we prove that if there exist exactly two geometrically distinct closed characteristics on a symmetric compact star-shaped hypersuface in ${\bf R}^4$, then both of them must be elliptic.

math.DS↗
arXiv2015

Topological structure of non-contractible loop space and closed geodesics on real projective spaces with odd dimensions

In this paper, we use Chas-Sullivan theory on loop homology and Leray-Serre spectral sequence to investigate the topological structure of the non-contractible component of the free loop space on the real projective spaces with odd dimensions. Then we apply the result to get the resonance identity of non-contractible homologically visible prime closed geodesics on such spaces provided the total number of distinct prime closed geodesics is finite.

math.GT↗
arXiv2014

The existence of two closed characteristics on every compact star-shaped hypersurface in ${\bf R}^4$

Recently, Cristofaro-Gardiner and Hutchings proved that there exist at least two closed characteristics on every compact star-shaped hypersuface in ${\bf R}^4$. Then Ginzburg, Hein, Hryniewicz, and Macarini gave this result a second proof. In this paper, we give it a third proof by using index iteration theory, resonance identities of closed characteristics and a remarkable theorem of Ginzburg et al.

math.DS↗
arXiv2014

Resonance Identities for Closed Characteristics on Compact Star-shaped Hypersurfaces in ${\bf R}^{2n}$

Resonance relations among periodic orbits on given energy hypersurfaces are very important for getting deeper understanding of the dynamics of the corresponding Hamiltonian systems. In this paper, we establish two new resonance identities for closed characteristics on every compact star-shaped hypersurface $Σ$ in ${\bf R}^{2n}$ when the number of geometrically distinct closed characteristics on $Σ$ is finite, which extend those identities established by C. Viterbo in 1989 for star-shaped hypersurfaces assuming in addition that all the closed characteristics and their iterates are non-degenerate, and that by W. Wang, X. Hu and Y. Long in 2007 for strictly convex hypersurfaces in ${\bf R}^{2n}$.

math.DS↗
arXiv2014

Linear stability of elliptic Lagrangian solutions of the planar three-body problem via index theory

It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter $\bb=27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2\in [0,9]$ and the eccentricity $e\in [0,1)$. We are not aware of any existing analytical method which relates the linear stability of these solutions to the two parameters directly in the full rectangle $[0,9]\times [0,1)$, besides perturbation methods for $e>0$ small enough, blow-up techniques for $e$ sufficiently close to 1, and numerical studies. In this paper, we introduce a new rigorous analytical method to study the linear stability of these solutions in terms of the two parameters in the full $(\bb,e)$ range $[0,9]\times [0,1)$ via the $\om$-index theory of symplectic paths for $\om$ belonging to the unit circle of the complex plane, and the theory of linear operators. After establishing the $\om$-index decreasing property of the solutions in $\bb$ for fixed $e\in [0,1)$, we prove the existence of three curves located from left to right in the rectangle $[0,9]\times [0,1)$, among which two are -1 degeneracy curves and the third one is the right envelop curve of the $\om$-degeneracy curves for $\om\not=1$, and show that the linear stability pattern of such elliptic Lagrangian solutions changes if and only if the parameter $(\bb,e)$ passes through each of these three curves. Interesting symmetries of these curves are also observed. The singular case when the eccentricity $e$ approaches to 1 is also analyzed in details concerning the linear stability.

math.DS↗