CONTENTS -- Preface -- List of members of the scientific committee, the organizing committee and the invited speakers of ICVAM-2 -- On 2-tori having a pole V. Bangert -- 1. Introduction -- 2. Poles in n-dimensional manifolds -- 3. Proof of the theorems -- References -- The nonlinear Schroedinger equation: Existence, stability and dynamics of solitons V. Benci, M. Ghimenti and A. M. Micheletti -- 1. Introduction -- 2. Main assumptions -- 2.1. Notations -- 3. General features of NSE -- 3.1. The first integrals of NSE -- 4. The case h = 1 -- V = 0 -- 5. The case h small enough -- 5.1. Existence and dynamics of barycenter -- 6. The final result -- 6.1. Barycenter and concentration point -- 6.2. Equation of the traveling soliton -- 7. The swarm interpretation -- 7.1. An heuristic proof -- References -- Turing patterns and standing waves in Fitzhugh-Nagumo type systems C.-N. Chen and S.-Y. Kung -- Acknowledgments -- References -- Variational principle and linear stability of periodic orbits in celestial mechanics X. Hu and S. Sun -- 1. Stability -- 2. Calculus of Variations -- 2.1. Topological Constraints -- 2.2. Symmetry Constraints -- 3. From Variations to Stability via Index -- 3.1. General Idea from Variational Methods to Stability -- 3.2. Elliptic Lagrangian Homographic Triangles13 -- 3.3. Figure-Eight14 -- 4. Symmetry vs. Index -- Aknowledgments -- References -- Remarks on mean value properties Y. Y. Li and L. Nguyen -- 1. Introduction -- 2. Mean value properties in Euclidean spaces -- 2.1. Elliptic case -- 2.2. Parabolic case -- 3. Mean value properties on hyperbolic spaces -- 3.1. Elliptic case -- 3.2. Parabolic case -- Acknowledgments -- References -- Brake orbits in bounded convex symmetric domains C. Liu and D. Zhang -- 1. Introduction -- 1.1. Background for brake orbits -- 1.2. Main results -- 1.3. Problems for further study.

2. Iteration formulas for Maslov-type index theory associated with a Lagrangian subspace -- 3. Variational setup -- 4. Sketch of the proofs of Theorem 1.1 -- Acknowledgments -- References -- Elliptic systems on RN with nonlinearities of linear growth Z. Liu, J. Su and Z.-Q. Wang -- 1. Introduction -- 2. Proof of Theorem 1.1 -- Acknowledgments -- References -- Recent progress on closed geodesics in some compact simply connected manifolds Y. Long -- 1. A brief history on closed geodesics -- 2. New multiplicity results on closed geodesics and main ideas -- 3. Classi cation of closed geodesics -- 4. Studies on rational closed geodesics -- 5. Studies on irrational closed geodesics -- 6. On 3 or 4 dimensional compact simply connected manifolds -- 7. More open problems -- Acknowledgments -- References -- Shortest curves associated to a degenerate Jacobi metric on T2 J. N. Mather -- 1. Introduction -- 2. The Main Results -- 3. Linearly Independent Homology Classes -- 4. Consequences of (?) -- 5. The Stable Norm -- 6. Proof of Theorems 2 and 3 -- 7. Shortest Curves in a Neighborhood of m0 -- 8. Distance from m0 -- 9. An Example -- Acknowledgment -- References -- Topological bifurcation theory: Old and new J. Mawhin -- 1. Introduction -- 2. Rabinowitz' global bifurcation theorem -- 3. Poincar e's topological approach to local bifurcation theory -- 4. A modern version of Poincar e's approach -- 5. Leray-Schauder continuation theorem -- 6. Topological bifurcation without degree -- 7. Extensions of Rabinowitz theorem -- References -- An equivariant CW-complex for the free loop space of a Finsler manifold H.-B. Rademacher -- 1. Statement of the Result -- 2. Proofs -- References -- Floer homology, relative braid classes, and low dimensional dynamics R. Vandervorst -- 1. Prelude -- 2. Relative braid classes -- 2.1. Braids on the 2-disc.

2.3. Discrete braid diagrams -- 3. An invariant for discrete relative braid classes -- 4. Parabolic dynamics -- 5. Braids and dynamics -- 5.1. The Cauchy-Riemann equations -- 5.2. The Heat ow -- 5.3. Discrete parabolic equations -- 6. Invariants -- 6.1. Basic ingredients -- 6.2. Floer homology, Morse homology and the Conley index -- 6.3. Basic properties -- 7. An example -- References -- Exponential growth rate of paths and its connection with dynamics Z. Xia and P. Zhang -- 1. Introduction -- 2. Exponential Growth Rate of Directed Graph with Length Information -- 2.1. Case 1 -- 2.2. Case 2 -- 2.3. Case 3 -- 3. Exponential Growth Rate, Topological Entropy and Principal Eigenvalue -- Acknowledgments -- References -- Rabinowitz's theorems revisited W. Zou -- 1. Introduction -- 2. Symmetric Case -- 3. Non-symmetric Case-1 -- 4. Non-symmetric Case-2 -- References.