Subharmonic solutions of Hamiltonian systems and the Maslov-type index theory

This paper deals with the subharmonic solutions of Hamiltonian systems

(H)     

The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems

In this paper, the iteration formula of the Maslov-type index theory for linear Hamiltonian systems with continuous, periodic, and symmetric coefficients is established. This formula yields a new method to determine the minimality of the period for solutions of nonlinear autonomous Hamiltonian systems via their Maslov-type indices. Applications of this formula give new results on the existence of periodic solutions with prescribed minimal period for such systems, and unify known results under various convexity conditions.

     

HOMOCLINIC ORBITS FOR SECOND ORDER HAMILTONIAN SYSTEM WITH QUADRATIC GROWTH

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Variational construction of unbounded orbits in Lagrangian systems

We show the existence of unbounded orbits in perturbations of generic geodesic flow in T2 by a generic periodic potential. Different from previous work such as in Mather (1997), the initial values of the orbits obtained here are not required sufficiently large.     

An index theory and existence of multiple brake orbits for star-shaped Hamiltonian systems

Supported in part by the Swedish Natural Science Research Council     

The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems

In this paper, the Conley conjecture, which was recently proved by Franks and Handel [J. Franks, M. Handel, Periodic points of Hamiltonian surface diffeomorphism, Geom. Topol. 7 (2003) 713-756] (for surfaces of positive genus), Hingston [N. Hingston, Subharmonic solutions of Hamiltonian equations on tori, Ann. Math., in press] (for tori) and Ginzburg [V.L. Ginzburg, The Conley conjecture, arXiv: math.SG/0610956v1] (for closed symplectically aspherical manifolds), is proved for C¹-Hamiltonian systems on the cotangent bundle of a C³-smooth compact manifold M without boundary, of a time 1-periodic C²-smooth Hamiltonian H:R×T^*M→R which is strongly convex and has quadratic growth on the fibers. Namely, we show that such a Hamiltonian system has an infinite sequence of contractible integral periodic solutions such that any one of them cannot be obtained from others by iterations. If H also satisfies H(−t,q,−p)=H(t,q,p) for any (t,q,p)∈R×T^*M, it is shown that the time-1-map of the Hamiltonian system (if exists) has infinitely many periodic points siting in the zero section of T^*M. If M is C⁵-smooth and dimM>1, H is of C⁴ class and independent of time t, then for any τ>0 the corresponding system has an infinite sequence of contractible periodic solutions of periods of integral multiple of τ such that any one of them cannot be obtained from others by iterations or rotations. These results are obtained by proving similar results for the Lagrangian system of the Fenchel transform of H, L:R×TM→R, which is proved to be strongly convex and to have quadratic growth in the velocities yet.     

Conditional Fredholm determinant for the S-periodic orbits in Hamiltonian systems

For S being a symplectic orthogonal matrix on R2n, the S-periodic orbits in Hamiltonian systems are a solution which satisfies x(0)=Sx(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 Fréchet differentiability, the splittingness for the cyclic type symmetric solutions. Also, we generalize the Hill formula originally gotten by Hill and Poincaré. 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.     

Existence and multiplicity results for nonlinear Schrödinger-Poisson equation with general potential

This paper is concerned with the Schrödinger-Poisson equation-\mathrm{\Delta }u+V(x)u+\phi (x)u=f(x,u),x\in {ℝ}^{\mathrm{3}},-\mathrm{\Delta }\phi =u^{\mathrm{2}},\underset{\left|x\right|\to +\infty }{\lim }\phi (x)=\mathrm{0.}Under certain hypotheses on V and a general spectral assumption, the existence and multiplicity of solutions are obtained via variational methods.     

Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem

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.     

The study of minimal period estimates for brake orbits of autonomous subquadratic Hamiltonian systems

In this paper, we consider the minimal period estimates for brake orbits of autonomous subquadratic Hamiltonian systems. We prove that if the Hamiltonian function H ∈ C2(R2n, R) is unbounded and not uniformly coercive, there exists at least one nonconstant T-periodic brake orbit(z, T) with minimal period T or T /2 for every number T 0.     

An index for periodic orbits of local semidynamical systems

We define an index of Fuller type counting the number of periodic orbits of a semiflow on an ANR by a suitable approximation process.

     

Relative Morse index and multiple brake orbits of asymptotically linear Hamiltonian systems in the presence of symmetries

In this paper, we define a relative Morse index for two continuous symmetric matrices paths in R2n satisfying condition (B1) and study its relation with the Maslov-type indices under brake orbit boundary value of these two symmetric matrices paths. As applications, using this relation we obtain a multiple existence of periodic brake orbit solutions of asymptotically linear Hamiltonian system in the presence of symmetries.     

Periodic orbits of Hamiltonian systems on symmetric positive-type hypersurfaces

This paper discusses the existence and multiplicity of periodic orbits of Hamiltonian systems on symmetric positive-type hypersurfaces. We prove that each such energy hypersurface carries at least one symmetric periodic orbit. Under some suitable pinching conditions, we also get an existence result of multiple symmetric periodic orbits.     

Homoclinic orbits for Hamiltonian systems induced by impulses

In this paper, we consider a class of impulsive Hamiltonian systems with a p‐Laplacian operator. Under certain conditions, we establish the existence of homoclinic orbits by means of the mountain pass theorem and an approximation technique. In some special cases, the homoclinic orbits are induced by the impulses in the sense that the associated non‐impulsive systems admit no non‐trivial homoclinic orbits. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence of even homoclinic orbits for second-order Hamiltonian systems

Some existence theorems for even homoclinic orbits are obtained for a class of second-order nonautonomous Hamiltonian systems with symmetric potentials under a class of new superquadratic conditions. A homoclinic orbit is obtained as a limit of solutions of a certain sequence of nil-boundary-value problems which are obtained by the minimax methods.     

Hill-Type Formula and Krein-Type Trace Formula for Hamiltonian Systems

In this paper, we give a survey on the Hill-type formula and its applications.Moreover, we generalize the Hill-type formula for linear Hamiltonian systems and Sturm-Liouville systems with any self-adjoint boundary conditions, which include the standard Neumann, Dirichlet and periodic boundary conditions. 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. Further, based on the Hill-type formula, we derive the Krein-type trace formula. As applications, we give nontrivial estimations for the eigenvalue problem and the relative Morse index.     

Continuation of relative periodic orbits in a class of triatomic Hamiltonian systems

We study relative periodic orbits (i.e. time-periodic orbits in a frame rotating at constant velocity) in a class of triatomic Euclidean-invariant (planar) Hamiltonian systems. The system consists of two identical heavy atoms and a light one, and the atomic mass ratio is treated as a continuation parameter. Under some nondegeneracy conditions, we show that a given family of relative periodic orbits existing at infinite mass ratio (and parametrized by phase, rotational degree of freedom and period) persists for sufficiently large mass ratio and for nearby angular velocities (this result is valid for small angular velocities). The proof is based on a method initially introduced by Sepulchre and MacKay [J.-A. Sepulchre, R.S. MacKay, Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators, Nonlinearity 10 (1997) 679–713] and further developed by Muñoz-Almaraz et al. [F.J. Muñoz-Almaraz, et al., Continuation of periodic orbits in conservative and Hamiltonian systems, Physica D 181 (2003) 1–38] for the continuation of normal periodic orbits in Hamiltonian systems. Our results provide several types of relative periodic orbits, which extend from small amplitude relative normal modes [J.-P. Ortega, Relative normal modes for nonlinear Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 665–704] up to large amplitude solutions which are not restrained to a small neighborhood of a stable relative equilibrium. In particular, we show the existence of large amplitude motions of inversion, where the light atom periodically crosses the segment between heavy atoms. This analysis is completed by numerical results on the stability and bifurcations of some inversion orbits as their angular velocity is varied.     

Corrigendum to “The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems” [J. Funct. Anal. 256 (9) (2009) 2967-3034]

In lines 8-11 of Lu (2009) [18, p. 2977] we wrote: “For integer m?3, if M is Cm-smooth and Cm−1-smooth L:R×TM→R satisfies the assumptions (L1)-(L3), then the functional Lτ is C²-smooth, bounded below, satisfies the Palais-Smale condition, and all critical points of it have finite Morse indexes and nullities (see [1, Prop. 4.1, 4.2] and [4])”. However, as proved in Abbondandolo and Schwarz (2009) [2] the claim that Lτ is C²-smooth is true if and only if for every (t,q) the function v?L(t,q,v) is a polynomial of degree at most 2. So the arguments in Lu (2009) [18] are only valid for the physical Hamiltonian in (1.2) and corresponding Lagrangian therein. In this note we shall correct our arguments in Lu (2009) [18] with a new splitting lemma obtained in Lu (2011) [20].