index and stability of symmetric periodic orbits in hamiltonian systems with application to figure eight orbit in this paper using the maslov index theory in symplectic geometry we build up some stability criteria for symmetric periodic orbits in a hamiltonian system which is motivated by the recent discoveries in the n body problem the key ingredient is a generalized bott type iteration formula for periodic solution in the presence of finite group action on the orbit for second order system we prove under general boundary conditions the close formula for the relationship between the morse index of an orbit in a lagrangian system and the maslov index of the fundamental solution for the corresponding orbit in its hamiltonian system counterpart and the boundary conditions cover the cases which appeared in the n body problem as an application we consider the stability problem of the celebrated figure eight orbit due to chenciner and montgomery in the planar three body problem with equal masses and we clarify the relationship between linear stability and its variational nature on various loop spaces the basic idea is as follows the variational characterization of the figure eight orbit provides information about its morse index based on its relation to the maslov index our stability criteria come into play

continuous dependence and estimates of eigenvalues for periodic generalized camassa holm equations abstract in this paper we are concerned with the spectral problem y 1 y 2 x 1 2 1 2 m 1 2 m 1 2 y 1 y 2 for the periodic generalized camassa holm equations the first aim is to study the structure of eigenvalues and prove the continuous dependence of eigenvalues on potentials we prove that as nonlinear functionals of potentials eigenvalues are continuous in potentials with respect to the weak topologies in the lebesgue space l 1 0 t the second aim is to find the estimates of the eigenvalues when the l 1 norm of potentials is given based on an application of trace formulas

a remarkable periodic solution of the three body problem in the case of equal masses using a variational method we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane the orbit has zero angular momentum and a very rich symmetry pattern its most surprising feature is that the three bodies chase each other around a flxed eight shaped curve setting aside collinear motions the only other known motion along a flxed curve in the inertial plane is the lagrange relative equilibrium in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle our orbit visits in turns every euler conflguration in which one of the bodies sits at the midpoint of the segment deflned by the other two figure 1 numerical computations

analysis of the stability of a family of singular limit linear periodic systems in r4 applications abstract in this paper we consider a 4d periodic linear system depending on a small parameter 0 we assume that the limit system has a singularity at t 0 of the form 1 c 1 c 2 t 2 with c 1 c 2 0 and c 1 0 as 0 using a blow up technique we develop an asymptotic formula for the stability parameters as goes to zero as an example we consider the homographic solutions of the planar three body problem for an homogeneous potential of degree 0 2 newtonian three body problem is obtained for 1 the parameter can be taken as 1 e 2 being e the eccentricity or a generalised eccentricity if 1 the behaviour of the stability parameters predicted by the formula is checked against numerical computations and some results of a global numerical exploration are displayed

eigenvalue problem of sturm liouville systems with separated boundary conditions let lambda j be the jth eigenvalue of sturm liouville systems with separated boundary conditions we build up the hill type formula which represent prod nolimits j 1 lambda j 1 as a determinant of finite matrix consequently we get the krein type trace formula based on the hill type formula which express sum nolimits j 1 over lambda 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

trace formula for linear hamiltonian systems with its applications to elliptic lagrangian solutions in the present paper we build up trace formulas for both the linear hamiltonian systems and sturm liouville systems the formula connects the monodromy matrix of a symmetric periodic orbit with the infinite sum of eigenvalues of the hessian of the action functional a natural application is to study the non degeneracy of linear hamiltonian systems precisely by the trace formula we can give an estimation for the upper bound such that the non degeneracy preserves moreover we could estimate the relative morse index by the trace formula consequently a series of new stability criteria for the symmetric periodic orbits is given as a concrete application the trace formula is used to study the linear stability of elliptic lagrangian solutions of the classical planar three body problem which depends on the mass parameter beta in 0 9 and the eccentricity e in 0 1 based on the trace formula we estimate the stable region and hyperbolic region of the elliptic lagrangian solutions

linear stability of the elliptic lagrangian triangle solutions in the three body problem abstract this paper concerns the linear stability of the well known periodic orbits of lagrange in the three body problem given any three masses there exists a family of periodic solutions for which each body is at the vertex of an equilateral triangle and travels along an elliptic kepler orbit reductions are performed to derive equations which determine the linear stability of the periodic solutions these equations depend on two parameters the eccentricity e of the orbit and the mass parameter 27 m1m2 m1m3 m2m3 m1 m2 m3 2 a combination of numerical and analytic methods is used to find the regions of stability in the e plane in particular using perturbation techniques it is rigorously proven that there are mass values where the truly elliptic orbits are linearly stable even though the circular orbits are not

conditioning likelihood and coherence a review of some foundational concepts 2000 conditioning likelihood and coherence a review of some foundational concepts journal of the american statistical association vol 95 no 452 pp 1340 1346

conditional fredholm determinant for the s periodic orbits in hamiltonian systems abstract for s being a symplectic orthogonal matrix on r 2 n the s periodic orbits in hamiltonian systems are a solution which satisfies x 0 s x t for some period t this paper is devoted to establishing the theory of conditional fredholm determinant in studying the s periodic orbits in hamiltonian systems first we study the property of the conditional fredholm determinant such as the frechet differentiability the splittingness for the cyclic type symmetric solutions also we generalize the hill formula originally gotten by hill and poincare more precisely let m be the monodromy matrix of the s periodic orbits then we get the formula relating the characteristic polynomial of the matrix sm and the conditional fredhom determinant moreover we study the relation of the conditional fredholm determinant and the relative morse index applications to the problem of linear stability for the s periodic orbits are given

morse index and stability of elliptic lagrangian solutions in the planar three body problem abstract we illustrate a new way to study the stability problem in celestial mechanics in this paper using the variational nature of elliptic lagrangian solutions in the planar three body problem we study the relation between morse index and its stability via maslov type index theory of periodic solutions of hamiltonian system for elliptic lagrangian solutions we get an estimate of the algebraic multiplicity of unit eigenvalues of its monodromy matrix in terms of the morse index which is the key to understand the stability problem as a special case we provide a criterion to spectral stability of relative equilibrium

stability diagram for 4d linear periodic systems with applications to homographic solutions we consider a family of 4 dimensional hamiltonian time periodic linear systems depending on three parameters 1 2 and e such that for e 0 the system becomes autonomous using normal form techniques we study stability and bifurcations for e 0 small enough we pay special attention to the d alembert case the results are applied to the study of the linear stability of homographic solutions of the planar three body problem for some homogeneous potential of degree 0 2 including the newtonian case

existence and minimizing properties of retrograde orbits to the three body problem with various choices of masses poincare made the first attempt in 1896 on applying variational calculus to the three body problem and observed that collision orbits do not necessarily have higher values of action than classical solutions little progress had been made on resolving this difficulty until a recent breakthrough by chenciner and montgomery afterward variational methods were successfully applied to the jv body problem to construct new classes of solutions in order to avoid collisions the problem is confined to symmetric path spaces and all new planar solutions were constructed under the assumption that some masses are equal a question for the variational approach on planar problems naturally arises are minimizing methods useful only when some masses are identical this article addresses this question for the three body problem for various choices of masses it is proved that there exist infinitely many solutions with a certain topological type called retrograde orbits that minimize the action functional on certain path spaces cases covered in our work include triple stars in retrograde motions double stars with one outer planet and some double stars with one planet orbiting around one primary mass our results largely complement the classical results by the poincare continuation method and conley s geometric approach

trace estimation of a family of periodic sturm liouville operators with application to robe s restricted three body problem in this paper we consider a family of sturm liouville operators on the periodic domain the bifurcation with respect to the parameter region is studied and the elliptic regions are estimated by trace formula at last these results are used to study the linear stability of the elliptic equilibrium point along the z axis in robe s restricted three body problem

turing pattern dynamics in an activator inhibitor system with superdiffusion the fractional operator is introduced to an activator inhibitor system to describe species anomalous superdiffusion the effects of the superdiffusive exponent on pattern formation and pattern selection are studied our linear stability analysis shows that the wave number of the turing pattern increases with the superdiffusive exponent a weakly nonlinear analysis yields a system of amplitude equations and the analysis of these amplitude equations predicts parameter regimes where hexagons stripes and their coexistence are expected numerical simulations of the activator inhibitor model near the stability boundaries confirm our analytical results since diffusion rate manifests in both diffusion constant and diffusion exponent we numerically explore their interactions on the emergence of turing patterns when the activator and inhibitor have different superdiffusive exponents we find that the critical ratio of the diffusion rate of the inhibitor to the activator required for the formation of the turing pattern increases monotonically with the superdiffusive exponent we conclude that small ratio than unity of anomalous diffusion exponent between the inhibitor and activator is more likely to promote the emergence of the turing pattern relative to the normal diffusion

hill type formula for hamiltonian system with lagrangian boundary conditions abstract 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 n reversible symmetry periodic orbits in n body problem naturally where n is an anti symplectic orthogonal matrix with n 2 i 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 n reversible symmetry periodic solutions of hamiltonian systems as an application we study the linear stability of elliptic relative equilibria in planar 3 body problem

index iteration theories for periodic orbits old and new abstract in this paper we give a survey of the index iteration theories developed for periodic solution orbits of hamiltonian systems in treating various problems from 1950s up to now including some recent developments and some of their applications to periodic solution problems

on the existence of collisionless equivariant minimizers for the classical n body problem we show that the minimization of the lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n body problem provided that some simple conditions on the symmetry group are satisfied more precisely we give a fairly general condition on symmetry groups g of the loop space for the n body problem with potential of homogeneous degree with 0 which ensures that the restriction of the lagrangian action mathcal a to the space g of g equivariant loops is coercive and its minimizers are collisionless without any strong force assumption in proving that local minima of g are free of collisions we develop an averaging technique based on marchal s idea of replacing some of the point masses with suitable shapes see 10 as an application several new orbits can be found with some appropriate choice of g furthermore the result can be used to give a simplified and unitary proof of the existence of many already known minimizing periodic orbits

existence and multiplicity of rotating periodic solutions for resonant hamiltonian systems abstract in the present paper we consider a class of resonant hamiltonian systems x j h x t x in r 2 n we will use saddle point reduction morse theory combining the technique of penalized functionals to obtain the existence of nontrivial rotating periodic solutions i e x t t q x t for any t r with t 0 and q an symplectic orthogonal matrix in the case q k i 2 n for any positive integer k such a rotating periodic solution is just a quasi periodic solution moreover if h is even in x we will give the multiplicity of nontrivial rotating periodic solutions by using two abstract critical theorems and previous techniques

elliptic relative equilibria in the n body problem a planar central configuration of the n body problem gives rise to a solution where each particle moves on a specific keplerian orbit while the totality of the particles move on a homothety motion if the keplerian orbit is elliptic then the solution is an equilibrium in pulsating coordinates so we call this solution an elliptic relative equilibrium the totality of such solutions forms a four dimensional symplectic subspace and we give a symplectic coordinate system which is adapted to this subspace and its symplectic complement in our coordinate system the linear variational equations of such a solution decouple into three subsystems one subsystem simply gives the motion of the center of mass another is kepler s problem and the third determines the nontrivial characteristic multipliers using these coordinates we study the linear stability of the elliptic relative equilibrium defined by the equilateral triangular central configuration of the three body problem we reproduce the analytic studies of g roberts we also study the linear stability of the four and five body problem where three or four bodies of unit mass are at the vertices of a equilateral triangle or

morse index and linear stability of the lagrangian circular orbit in a three body type problem via index theory it is well known that the linear stability of the lagrangian elliptic solutions in the classical planar three body problem depends on a mass parameter and on the eccentricity e of the orbit we consider only the circular case e 0 but under the action of a broader family of singular potentials homogeneous potentials for alpha in 0 2 and the logarithmic one it turns out indeed that the lagrangian circular orbit persists also in this more general setting we discover a region of linear stability expressed in terms of the homogeneity parameter and the mass parameter then we compute the morse index of this orbit and of its iterates and we find that the boundary of the stability region is the envelope of a family of curves on which the morse indices of the iterates jump in order to conduct our analysis we rely on a maslov type index theory devised and developed by y long x hu and s sun a key role is played by an appropriate index theorem and by some precise computations of suitable maslov type indices