Table of contents

The symmetric dynamics of two kinks and one antikink in classical (1 + 1)-dimensional theory is investigated. Gradient flow is used to construct a collective coordinate model of the system. The relationship between the discrete vibrational mode of a single kink, and the process of kink - antikink pair production is explored.

Poursuivant les travaux entrepris par Mourtada, on montre qu'une famille de champs de vecteurs avec un polycycle algébrique générique à quatre sommets hyperboliques possède une cyclicité maximale de cinq cycles limites. Cette cyclicité est atteinte dans un ouvert connexe en les paramètres dont le bord contient en particulier une ligne générique de singularités de type queue d'aronde. On donne également une estimation asymptotique du volume de cet ouvert, ainsi qu'un exemple explicite de famille polynomiale de champs de vecteurs remplissant les conditions ci-dessus et possèdant cinq cycles limites. Les méthodes employées sont très diverses: raisonnements géométriques (la théorie des catastrophes de Thom et la théorie des singularités algébriques), développements de Puiseux, le nombre de racines majoré par la règle de Descartes et calculé exactement par des suites de Sturm, et d'autres méthodes spécifiques au calcul formel comme par exemple la décomposition cylindrique algébrique et les résolutions de systèmes algébriques via la construction de bases de Gröbner. Les calculs ont été effectués formellement, c'est-à-dire sans faire appel à la moindre approximation numérique, en utilisant le système de calcul formel AXIOM.

Spatiotemporally chaotic dynamics of a Kuramoto - Sivashinsky system is described by means of an infinite hierarchy of its unstable spatiotemporally periodic solutions. An intrinsic parametrization of the corresponding invariant set serves as an accurate guide to the high-dimensional dynamics, and the periodic orbit theory yields several global averages characterizing the chaotic dynamics.

In this paper we obtain explicit lower bounds for the radius of convergence of the Painlevé expansions of the Korteweg - de Vries equation around a movable singularity manifold in terms of the sup norms of the arbitrary functions involved. We use this estimate to prove the well-posedness of the singular Cauchy problem on in the form of continuous dependence of the meromorphic solution on the arbitrary data.

We prove stochastic stability of chaotic maps for a general class of Markov random perturbations (including singular ones) satisfying some kind of mixing conditions. One of the consequences of this statement is the proof of Ulam's conjecture about the approximation of the dynamics of a chaotic system by a finite state Markov chain. Conditions under which the localization phenomenon (i.e. stabilization of singular invariant measures) takes place are also considered. Our main tools are the so-called bounded variation approach combined with the ergodic theorem of Ionescu - Tulcea and Marinescu, and a random walk argument that we apply to prove the absence of `traps' under the action of random perturbations.

It is shown that a rigorous estimate for the fractal dimension of the global attractor of the three-dimensional incompressible Navier - Stokes equations on periodic boundary conditions is given by where L is the box length and is a Kolmogorov length defined by with the energy dissipation rate given by . By interpreting the dimension of the attractor as the number of degrees of freedom of the system, we obtain an estimate for an average natural length scale for the flow which is given by

In this paper we study the topological entropy of the motion of a particle of charge on a closed manifold M under the effect of a magnetic field. The function is an even function and we have shown in [6] that the considered system is not Anosov if is sufficiently large. Let be a maximal interval such that the motion of the particle of charge is Anosov. We show that strictly decreases on . It follows that is an absolute maximum of h on the interval . We also obtain upper estimates for the first derivative of h involving the magnetic field and the curvature of M.

This paper proves the existence of solitary waves for several fifth-order models for water waves. The method relies on a variational characterization of these solitary waves. Conclusions include: (1) there are solitary waves with negative speed and small amplitude for fifth-order, Korteweg - de Vries-like models; (2) the minimizing solutions, in the sense of the variational principle considered, are not close to the third-order Korteweg - de Vries one-soliton, which may explain difficulties with the perturbation of the latter solution.

In this paper, we consider germs at of diffeomorphisms which fix the origin and such that their linear part at zero generates a rotation. We provide a condition on the dynamics under which the angle of the rotation is a topological invariant. We give an application to germs of vector fields of whose linear part generates a rotation.

We show that the Lyapunov exponent for the Sinai billiard is with where R is the radius of the circular scatterer. We consider the disk-to-disk-map of the standard configuration where the disk is centred inside a unit square.

The investigation of exponentially small splitting of separatrices for high-frequency time-periodic perturbation of a Hamiltonian with one degree of freedom leads to a reference system in the complex phase space. The reference system is independent of a small parameter, e.g., the perturbation period of the original system. The splitting of the invariant manifolds is described for the system , which is a reference system for high-frequency perturbations of the pendulum.

This note presents a condition sufficient for the nonexistence of invariant curves of certain mappings defined in the cylinder. The result is applied to a concrete case to obtain a counterexample to certain tentative extensions of the small twist theorem.

Using weighted -norms we derive new bounds on the long-time behaviour of the solutions improving on the known results of the polynomial growth with respect to the instability parameter. These estimates are valid for quite arbitrary, possibly unbounded domains. We establish precise estimates on the maximal influence of the boundaries on the dynamics in the interior. For instance, the attractor for the domain with periodic boundary conditions is upper semicontinuous to .

We use so-called energy-dependent Schrödinger operators to establish a link between special classes of solutions of N-component systems of evolution equations and finite dimensional Hamiltonian systems on the moduli spaces of Riemann surfaces. We also investigate the phase-space geometry of these Hamiltonian systems and introduce deformations of the level sets associated to conserved quantities, which results in a new class of solutions with monodromy for N-component systems of PDEs.

After constructing a variety of mechanical systems related to the spatial flows of nonlinear evolution equations, we investigate their semiclassical limits. In particular, we obtain semiclassical asymptotics for the Bloch eigenfunctions of the energy dependent Schrödinger operators, which is of importance in investigating zero-dispersion limits of N-component systems of PDEs.

We introduce horseshoe-type mappings which are geometrically similar to Smale's horseshoes. For such mappings we prove by means of the fixed point index the existence of chaotic dynamics - the semi-conjugacy to the shift on a finite number of symbols. Our theorem does not require any assumptions concerning derivatives, it is a purely topological result. The assumptions of our theorem are then rigorously verified by computer assisted computations for the classical Hénon map and for classical Rössler equations.

The classification of map-germs (up to a variety of equivalences) has many applications in differential geometry, to the study of wavefronts and caustics and to bifurcation theory. In a previous paper the first and last authors together with C T C Wall, gave some useful criteria for map-germs to be finitely determined with respect to a wide range of equivalence relations. The results presented in this paper, used in conjunction with these determinacy techniques, provide a very efficient classification procedure. Moreover, we show that the algebraic criteria involved in these calculations may be reduced to finite-dimensional symbolic problems which may be performed by a computer.

The dynamics of flow inside a cylinder at high Reynolds number are considered. A study of the viscous boundary layer near the walls is performed. In the case where there is no pressure gradient, a result is proven demonstrating that a regular perturbation expansion holds for the solution, even when a small discontinuity exists in the wall data. In addition, the characteristic decay rate of the flow in the viscous boundary layer is established. In the case where there is a pressure gradient, a result is proven demonstrating that an additional scale, related to the size of the disturbance and larger than the boundary-layer width, must be used in a multiple-scale expansion. Examination of the divergence of these multiple-scale expansions for finite disturbances leads to discussions of viscous flow and separation processes.

Using the formalism defined by Lauterbach and Roberts (1992 J. Diff. Eqns. 100 22 - 48), we develop a geometric approach for the problem of forced symmetry breaking for periodic orbits in G-equivariant systems of ODEs. We show that this problem can be studied as the perturbation of the identity mapping on the double coset space LG/K where K is the maximal subgroup of G acting on the periodic orbit and L the symmetry of the perturbation. We exhibit some examples where this kind of symmetry breaking allows us to show the existence of heteroclinic cycles between periodic solutions.