arXiv subjects

Xijun Hu

Publications and source records attributed to Xijun Hu.

At least 37 records · Page 2Linked to original sources

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

On the $j$-th Eigenvalue of Sturm-Liouville Problem and the Maslov Index

In the previous papers \cite{HLWZ,KWZ}, the jump phenomena of the $j$-th eigenvalue were completely characterized for Sturm-Liouville problems. In this paper, we show that the jump number of these eigenvalue branches is exactly the Maslov index for the path of corresponding boundary conditions. Then we determine the sharp range of the $j$-th eigenvalue on each layer of the space of boundary conditions. Finally, we prove that the graph of monodromy matrix tends to the Dirichlet boundary condition as the spectral parameter goes to $-\infty$.

math.FA↗

arXiv2018

Decomposition of Spectral flow and Bott-type Iteration Formula

Let $A(t)$ be a continuous path of Fredhom operators, we first prove that the spectral flow $sf(A(t))$ is cogredient invariant. Based on this property, we give a decomposition formula of spectral flow if the path is invariant under a matrix-like cogredient. As applications, we give the generalized Bott-type iteration formula for linear Hamiltonian systems.

math.FA↗

arXiv2018

Singularity of the $n$-th eigenvalue of high dimensional Sturm-Liouville problems

It is natural to consider continuous dependence of the $n$-th eigenvalue on $d$-dimensional ($d\geq2$) Sturm-Liouville problems after the results on $1$-dimensional case by Kong, Wu and Zettl [14]. In this paper, we find all the boundary conditions such that the $n$-th eigenvalue is not continuous, and give complete characterization of asymptotic behavior of the $n$-th eigenvalue. This renders a precise description of the jump phenomena of the $n$-th eigenvalue near such a boundary condition. Furthermore, we divide the space of boundary conditions into $2d+1$ layers and show that the $n$-th eigenvalue is continuously dependent on Sturm-Liouville equations and on boundary conditions when restricted into each layer. In addition, we prove that the analytic and geometric multiplicities of an eigenvalue are equal. Finally, we obtain derivative formula and positive direction of eigenvalues with respect to boundary conditions.

math.SP↗

arXiv2018

Morse Index Theorem of Lagrangian Systems and Stability of Brake Orbit

In this paper, we prove Morse index theorem of Lagrangian system with any self-adjoint boundary conditions. Based on it, we give some nontrivial estimation on the difference of Morse indices. As an application, we get a new criterion for the stability problem of brake periodic orbit.

math.DS↗

arXiv2018

An Index theory for asymptotic motions under singular potentials

We develop an index theory for parabolic and collision solutions to the classical n-body problem and we prove sufficient conditions for the finiteness of the spectral index valid in a large class of trajectories ending with a total collapse or expanding with vanishing limiting velocities. Both problems suffer from a lack of compactness and can be brought in a similar form of a Lagrangian System on the half time line by a regularising change of coordinates which preserve the Lagrangian structure. We then introduce a Maslov-type index which is suitable to capture the asymptotic nature of these trajectories as half-clinic orbits: by taking into account the underlying Hamiltonian structure we define the appropriate notion of geometric index for this class of solutions and we develop the relative index theory.

math.DS↗

arXiv2018

Index Theory for Zero Energy Solutions of the Planar Anisotropic Kepler Problem

In the variational study of singular Lagrange systems, the zero energy solutions play an important role. In this paper we find a simple way of computing the Morse indices of these solutions for the planar anisotropic Kepler problem. In particular an interesting connection between the Morse indices and the oscillating behaviors of these solutions discovered by the physicist M. Gutzwiller is established.

math.DS↗

arXiv2017

Hill-type formula for Hamiltonian system with Lagrangian boundary conditions

In this paper, we build up Hill-type formula for linear Hamiltonian systems with Lagrangian boundary conditions, which include standard Neumann, Dirichlet boundary conditions. Such a kind of boundary conditions comes from the brake symmetry periodic orbits in $n$-body problem naturally. The Hill-type formula connects the infinite determinant of the Hessian of the action functional with the determinant of matrices which depend on the monodromy matrix and boundary conditions. Consequently, we derive the Krein-type trace formula and give nontrivial estimation for the eigenvalue problem. Combined with the Maslov-type index theory, we give some new stability criteria for the brake symmetry periodic solutions of Hamiltonian systems. As an application, we study the linear stability of elliptic relative equilibria in planar $3$-body problem.

math.SP↗

arXiv2017

A dihedral Bott-type iteration formula and stability of symmetric periodic orbits

In 1956, Bott in his celebrated paper on closed geodesics and Sturm intersection theory, proved an Index Iteration Formula for closed geodesics on Riemannian manifolds. Some years later, Ekeland improved this formula in the case of convex Hamiltonians and, in 1999, Long generalized the Bott iteration formula by putting in its natural symplectic context and constructing a very effective Index Theory. The literature about this formula is quite broad and the dynamical implications in the Hamiltonian world (e.g. existence, multiplicity, linear stability etc.) are enormous. Motivated by the recent discoveries on the stability properties of symmetric periodic solutions of singular Lagrangian systems, we establish a Bott-type iteration formula for dihedrally equivariant Lagrangian and Hamiltonian systems. We finally apply our theory for computing the Morse indices of the celebrated Chenciner and Montgomery figure-eight orbit for the planar three body problem in different equivariant spaces. Our last dynamical consequence is an hyperbolicity criterion for reversible Lagrangian systems.

math.DS↗

arXiv2017

The role of the coherence length for the establishment of global phase coherence in three-dimensional arrays of ultra-thin quasi-one-dimensional superconducting Pb and NbN nanowires

We have fabricated 5 nm ultra-thin NbN nanowires that form a dense and regular array in the linear channels of mesoporous SBA-15 silica substrates. Bulk NbN is a well-known classical superconductor with Tc of 16 K. We show that, by being incorporated into this nanostructure, the composite material exhibits typical quasi-one-dimensional characteristics. We compare the superconducting properties with those of superconducting Pb nanowires of same dimensionality in identical configuration within the linear SBA-15 pores. While Pb nanowire arrays show a pronounced crossover from 1D superconductivity at high temperatures to a 3D bulk superconducting state in the low temperature regime with true zero resistance triggered by transversal Josephson interaction, this transition appears to be completely absent in the NbN nanowire array. The small coherence length in NbN, which strongly suppresses the Josephson coupling is discussed as the origin of this difference.

cond-mat.supr-con↗

arXiv2017

Bifurcation of heteroclinic orbits via an index theory

Heteroclinic orbits for one-parameter families of nonautonomous vectorfields appear in a very natural way in many physical applications. Inspired by some recent bifurcation results for homoclinic trajectories of nonautonomous vectorfield, we define a new $\mathbf Z_2$-index and we construct a index theory for heteroclinic orbits of nonautonomous vectorfield. We prove an index theorem, by showing that, under some standard transversality assumptions, the $\mathbf Z_2$-index is equal to the parity, a homotopy invariant for paths of Fredholm operators of index 0. As a direct consequence of the index theory developed in this paper, we get a new bifurcation result for heteroclinic orbits.

math.DS↗

arXiv2017

Index theory for heteroclinic orbits of Hamiltonian systems

Index theory revealed its outstanding role in the study of periodic orbits of Hamiltonian systems and the dynamical consequences of this theory are enormous. Although the index theory in the periodic case is well-established, very few results are known in the case of homoclinic orbits of Hamiltonian systems. Moreover, to the authors' knowledge, no results have been yet proved in the case of heteroclinic and halfclinic (i.e. parametrised by a half-line) orbits. Motivated by the importance played by these motions in understanding several challenging problems in Classical Mechanics, we develop a new index theory and we prove at once a general spectral flow formula for heteroclinic, homoclinic and halfclinic trajectories. Finally we show how this index theory can be used to recover all the (classical) existing results on orbits parametrised by bounded intervals.

math.DS↗

arXiv2016

Collision index and stability of elliptic relative equilibria in planar n-body problem

It is well known that a planar central configuration of the $n$-body problem gives rise to solutions where each particle moves on a specific Keplerian orbit while the totality of the particles move on a homographic motion. When the eccentricity $e$ of the Keplerian orbit belongs in $[0,1)$, following Meyer and Schmidt, we call such solutions elliptic relative equilibria (shortly, ERE). In order to study the linear stability of ERE in the near-collision case, namely when $1-e$ is small enough, we introduce the collision index for planar central configurations. The collision index is a Maslov-type index for heteroclinic orbits and orbits parametrised by half-lines that, according to the Definition given by authors in [16], we shall refer to as half-clinic orbits and whose Definition in this context, is essentially based on a blow up technique in the case $e=1$. We get the fundamental properties of collision index and approximation theorems. As applications, we give some new hyperbolic criteria and prove that, generically, the ERE of minimal central configurations are hyperbolic in the near-collision case, and we give detailed analysis of Euler collinear orbits in the near-collision case.

math.DS↗

arXiv2016

Dramatic increase of the onset critical temperature and critical field of elemental Sn in the form of thin nanowires

Sn is a well-known classical superconductor on the border between type I and type II with critical temperature of 3.722K and critical field of 0.031T. We show by means of specific heat and electric magneto-transport data that its critical parameters can be dramatically increased if it is brought in the form of loosely bound bundles of thin nanowires. The specific heat displays a pronounced double phase transition at 3.7K and 5.5K, which we attribute to the inner 'bulk' contribution of the nanowires and to the surface contribution, respectively. The latter is visible only because of the large volume fraction of the surface layer in relation to their bulk inner volume. The upper transition coincides with the onset of the resistive transition, while zero resistance is gradually approached below the lower transition. The large coherence length of 230nm at 0K likely actuates a Josephson coupling between adjacent neighboring nanowires and thus suppresses the effect of 1D phase fluctuations along the nanowires, and stabilizes 3D phase coherence throughout the entire network with zero resistance. A magnetic field of more than 3T is required to restore the normal state, which means that the critical field is enhanced by about two orders of magnitude with respect to Sn in its bulk form.

cond-mat.supr-con↗

arXiv2015

Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs

The present paper is devoted to studying the Hill-type formula and Krein-type trace formula for ODE, which is a continuous work of our previous work for Hamiltonian systems \cite{HOW}. Hill-type formula and Krein-type trace formula are given by Hill at 1877 and Krein in 1950's separately. Recently, we find that there is a closed relationship between them \cite{HOW}. In this paper, we will obtain the Hill-type formula for the $S$-periodic orbits of the first order ODEs. Such a kind of orbits is considered naturally to study the symmetric periodic and quasi-periodic solutions. By some similar idea in \cite{HOW}, based on the Hill-type formula, we will build up the Krein-type trace formula for the first order ODEs, which can be seen as a non-self-adjoint version of the case of Hamiltonian system.

math.SP↗

arXiv2015

Eigenvalue problem of Sturm-Liouville systems with separated boundary conditions

Let $λ_j$ be the $j$-th eigenvalue of Sturm-Liouville systems with separated boundary conditions, we build up the Hill-type formula, which represent $\prod\limits_{j}(1-λ_j^{-1})$ as a determinant of finite matrix. This is the first attack on such a formula under non-periodic type boundary conditions. Consequently, we get the Krein-type trace formula based on the Hill-type formula, which express $\sum\limits_{j}{1\over λ_j^m}$ as trace of finite matrices. The trace formula can be used to estimate the conjugate point alone a geodesic in Riemannian manifold and to get some infinite sum identities.

math.SP↗

arXiv2014

Stability of closed characteristics on compact convex hypersurfaces in R^{2n}

Let $Σ\subset \R^{2n}$ with $n\geq2$ be any $C^2$ compact convex hypersurface and only has finitely geometrically distinct closed characteristics. Based on Y.Long and C.Zhu 's index jump methods \cite{LoZ1}, we prove that there are at least two geometrically distinct elliptic closed characteristics, and moreover, there exist at least $\varrho_{n} (Σ)$ ($\varrho_{n}(Σ)\geq[\frac{n}{2}]+1$) geometrically distinct closed characteristics such that for any two elements among them, the ratio of their mean indices is irrational number.

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↗