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].     

The Conley index for fast-slow systems II: Multidimensional slow variable

We use the Conley index theory to develop a general method to prove existence of periodic and heteroclinic orbits in a singularly perturbed system of ODEs. This is a continuation of the authors' earlier work [T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka, J. Reineck, The Conley index for fast-slow systems I: One-dimensional slow variable, J. Dynam. Differential Equations 11 (1999) 427-470] which is now extended to systems with multidimensional slow variables. The key new idea is the observation that the Conley index in fast-slow systems has a cohomological product structure. The factors in this product are the slow index, which captures information about the flow in the slow direction transverse to the slow flow, and the fast index, which is analogous to the Conley index for fast-slow systems with one-dimensional slow flow [T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka, J. Reineck, The Conley index for fast-slow systems I: One-dimensional slow variable, J. Dynam. Differential Equations 11 (1999) 427-470].     

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

This paper deals with the subharmonic solutions of Hamiltonian systems

(H)     

Bourgin-Yang Type Theorem and its application to -equivariant Hamiltonian systems

We will be concerned with the existence of multiple periodic solutions of asymptotically linear Hamiltonian systems with the presence of -action. To that purpose we prove a new version of the Bourgin-Yang theorem. Using the notion of the crossing number we also introduce a new definition of the Morse index for indefinite functionals.

     

Critical points theorems concerning strongly indefinite functionals and infinite many periodic solutions for a class of Hamiltonian systems

Based on new deformation theorems concerning strongly indefinite functionals, we give some new min-max theorems which are useful in looking for critical points of functionals which are strongly indefinite and satisfy Cerami condition instead of Palais-Smale condition. As one application of abstract results, we study existence of multiple periodic solutions for a class of non-autonomous first order Hamiltonian system

     

Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems

In this paper we study the following nonperiodic second order Hamiltonian system

     

Sobolev’s spaces on time scales and its applications to a class of second order Hamiltonian systems on time scales

In this paper, we study the Sobolev’s spaces on time scales and their properties. As applications, we present a recent approach via variational methods and the critical point theory to obtain the existence of solutions for a class of second order Hamiltonian systems on time scales. By establishing a proper variational setting, three existence results for systems under consideration are obtained. Finally, three examples are presented to illustrate the feasibility and effectiveness of the existence results.     

A continuation theorem on periodic solutions of regular nonlinear systems and its application to the exact tracking problem for the inverted spherical pendulum

We present a continuation method to obtain a family of T-periodic solutions for a family of T-periodic systems.In particular, we present some sufficient analytical conditions, which have the advantage of being an easy application to some systems of interest in physics or engineering. We apply these conditions to the exact tracking problem for the inverted spherical pendulum.