Related papers: Intersections of Lagrangian submanifolds and the M…
We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds. We do not need to know the explicit formulas for the homoclinic orbits prior…
We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds (NHIMs). The method is based on a new geometric proof of the normally…
In this work we consider a two-dimensional piecewise smooth system, defined in two domains separated by the switching manifold $x=0$. We assume that there exists a piecewise-defined continuous Hamiltonian that is a first integral of the…
We develop a Melnikov method for volume-preserving maps with codimension one invariant manifolds. The Melnikov function is shown to be related to the flux of the perturbation through the unperturbed invariant surface. As an example, we…
We define a new family of open Gromov-Witten type invariants based on intersection theory on the moduli space of pseudoholomorphic curves of arbitrary genus with boundary in a Lagrangian submanifold. We assume the Lagrangian submanifold…
This paper is Part I of a two-part series. We investigate bifurcation phenomena in Lagrangian systems with various boundary conditions and constraints, focusing on the interplay between Morse theory and the existence of multiple solutions…
We describe and characterize rigorously the chaotic behavior of the sine-Gordon equation. The existence of invariant manifolds and the persistence of homoclinic orbits for a perturbed sine--Gordon equation are established. We apply a…
In this paper we present a computer-assisted procedure for proving the existence of transverse heteroclinic orbits connecting hyperbolic equilibria of polynomial vector fields. The idea is to compute high-order Taylor approximations of…
In this paper, we extend Rabinowitz Floer homology theory which has been established and extensively studied for hypersurfaces to coisotropic submanifolds of higher codimension. With this generalized version of Rabinowitz Floer homology…
We prove an analog of Shilnikov Lemma for a normally hyperbolic symplectic critical manifold $M\subset H^{-1}(0)$ of a Hamiltonian system. Using this result, trajectories with small energy $H=\mu>0$ shadowing chains of homoclinic orbits to…
We found some Lagrangian submanifolds of the adjoint semisimple orbit in two cases. For the first, the compact case, also known as the Generalized flag manifolds, we prove that the real flags can be seen as infinitesimally tight Lagrangian…
Given a compact Lagrangian $L$ in a semipositive convex-at-infinity symplectic manifold $W$, we establish a cup-length estimate for the action values of $L$ associated to a Hamiltonian isotopy whose spectral norm is smaller than some…
We illustrate a completely analytic approach to Mel'nikov theory, which is based on a suitable extension of a classical method, and which is parallel and -- at least in part -- complementary to the standard procedure. This approach can be…
In this paper we study a beyond all order phenomenon which appears in the analytic unfoldings of the Hopf-zero singularity. It consists in the breakdown of a two-dimensional heteroclinic surface which exists in the truncated normal form of…
This dissertation describes the space of heteroclinic orbits for a class of semilinear parabolic equations, focusing primarily on the case where the nonlinearity is a second degree polynomial with variable coefficients. Along the way, a new…
We analyse the intersection of positively and negatively sectional-hyperbolic sets for flows on compact manifolds. First we prove that such an intersection is hyperbolic if the intersecting sets are both transitive (this is false without…
We investigate the extrinsic topology of Lagrangian submanifolds and of their submanifolds in closed symplectic manifolds using Floer homological methods. The first result asserts that the homology class of a displaceable monotone…
This article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits. In addition, tangencies in the intersection of un/stable manifolds are…
We propose a general framework governing the intersection properties of extremal rays of irreducible holomorphic symplectic manifolds under the Beauville-Bogomolov form. Our main thesis is that extremal rays associated to Lagrangian…
We consider a class of two-degree-of-freedom Hamiltonian systems with saddle-centers connected by heteroclinic orbits and discuss some relationships between the existence of transverse heteroclinic orbits and nonintegrability. By the…