Table of contents

Upper bounds for the Hausdorff dimension of compact invariant sets of C¹-maps on smooth Riemannian manifolds are given in terms of the singular values of the tangent map and the multiplicity function of the map, describing the number of pre-images of a certain point in a given set. For non-injective maps this improves previous results using only the singular values.

The probabilities for gaps in the eigenvalue spectrum of the finite-dimensional N × N random matrix Hermite and Jacobi unitary ensembles on some single and disconnected double intervals are found. These are cases where a reflection symmetry exists and the probability factors into two other related probabilities, defined on single intervals. Our investigation uses the system of partial differential equations arising from the Fredholm determinant expression for the gap probability and the differential-recurrence equations satisfied by Hermite and Jacobi orthogonal polynomials. In our study we find second- and third-order nonlinear ordinary differential equations defining the probabilities in the general N case. For N = 1 and 2 the probabilities and thus the solution of the equations are given explicitly. An asymptotic expansion for large gap size is obtained from the equation in the Hermite case, and also studied is the scaling at the edge of the Hermite spectrum as N → ∞, and the Jacobi to Hermite limit; these last two studies make correspondence to other cases reported here or known previously. Moreover, the differential equation arising in the Hermite ensemble is solved in terms of an explicit rational function of a Painlevé-V transcendent and its derivative, and an analogous solution is provided in the two Jacobi cases but this time involving a Painlevé-VI transcendent.

We investigate the stability of pulses that are created at T-points in reaction-diffusion systems on the real line. The pulses are formed by gluing unstable fronts and backs together. We demonstrate that the bifurcating pulses can nevertheless be stable, and establish necessary and sufficient conditions that involve only the front and the back for the stability of the bifurcating pulses.

Let be the zero-section of the cotangent bundle T* of a real analytic manifold . Let F:(T*,)→(T*,) be a real analytic local diffeomorphism preserving the canonical symplectic form ω = dα of T*, where α denotes the Liouville form. Suppose that F*α - α is an exact form dS. We prove that we can reconstruct F from S and f = F_|; if f is included into a flow, then F can be included into a Hamiltonian flow.

We consider a class of ordinary differential equations of the following type: -ü(t) = α(t)u³(t) + mu(t) + h(t) where α is bounded and changes sign. We study the effect of such a coefficient on the existence of oscillating solutions on bounded and unbounded domains.

We determine the properties of locking of inhibitory and of excitatory synaptic connections with neocortical pyramidal cells. We are able to give an overview of the emerging periodic behaviour, as a function of the perturbation strength. We show that a chaotic response emerges for inhibitory connections on an open set of positive measure of the parameter space. This implies that synchronization on the set of inhibitory connections is possible, with positive probability.

We give a definition of orbit complexity for points in topological dynamical systems over separable metric spaces. Our definition extends Brudno's definition meaningfully to the non-compact case. Interacting with constructive mathematics we investigate questions about orbit complexity from a new point of view. In particular, we show a relation between constructivity, entropy and orbit complexity.

The first definition of Lyapunov exponents (depending on a probability measure) for a one-dimensional cellular automaton was introduced by Shereshevsky in 1991. The existence of an almost everywhere constant value for each of the two exponents (left and right), requires particular conditions for the measure. Shereshevsky establishes an inequality involving these two constants and the metric entropies of both the shift and the cellular automaton. In this paper we first prove that Shereshevsky's two exponents exist for a more suitable class of measures, then, keeping the same conditions, we define new exponents, called average Lyapunov exponents which are smaller than or equal to the former. We obtain two inequalities: the first one is analogous to Shereshevsky's but concerns the average exponents; the second is the Shereshevsky inequality but with more suitable assumptions. These results are illustrated by two non-trivial examples, both proving that average exponents provide a better bound for the entropy, and one showing that the inequalities are strict in general.

We consider differentiable maps having an invariant torus with normal behaviour and a central part. We prove the existence and regularity of strong-stable manifolds and regularity with respect to the parameters. Then we prove a lambda lemma in this setting for C² maps. These results have applications in the study of diffusion problems, including Arnol'd diffusion.

We study the existence and some properties of solitary-wave solutions for an interaction equation between a long internal wave and a short surface wave in a two-layer fluid. We obtain the existence of solitary-wave solutions using the concentration compactness method developed by Lions. We also show that solutions of the minimization problem are analytic and they are translations of the symmetric decreasing rearrangement of themselves.

We derive semiclassical expressions for spectra, weighted by matrix elements of a Gaussian observable, relevant to a range of molecular and mesoscopic systems. We apply the formalism to the particular example of the resonant tunnelling diode (RTD) in tilted fields. The RTD is an experimental realization of a mesoscopic system exhibiting a transition to chaos. It has generated much interest and several different semiclassical theories for the RTD have been proposed recently.

Our formalism clarifies the relationship between the different approaches and to previous work on semiclassical theories of matrix elements. We introduce three possible levels of approximation in the application of the stationary phase approximation, depending on typical length scales of oscillations of the semiclassical Green function, relative to the degree of localization of the observable. Different types of trajectories (periodic, normal, closed and saddle orbits) are shown to arise from such considerations. We propose here for the first time a new type of trajectory (`minimal orbits') and show they provide the best real approximation to the complex saddle points of the stationary phase approximation.

We test the semiclassical formulae on quantum calculations and experimental data. We successfully treat phenomena beyond standard periodic orbit (PO) theory: `ghost regions' where no real PO can be found and regions with contributions from non-isolated POs. We show that the new types of trajectories (saddle and minimal orbits) provide accurate results. We discuss a divergence of the contribution of saddle orbits, which suggests the existence of bifurcation-type phenomena affecting the complex and non-periodic saddle orbits.

Extending the gauge-invariance principle for τ functions of the standard bilinear formalism to the supersymmetric case, we define = 1 supersymmetric Hirota operators. Using them, we bilinearize SUSY KdV-type equations (KdV, Sawada-Kotera-Ramani). The supersoliton solutions and extension to SUSY sine-Gordon are also discussed. It is shown that the Lax-integrable SUSY KdV of the Mathieu equation does not possess an N-supersoliton solution for N⩾3 for arbitrary parameters.The N-supersoliton solution only exists for a particular choice of parameters.

The problem of the existence of radial sine-Gordon breathers (i.e. radial solutions which are periodic in time and decay at infinity with respect to the spatial variable) in the (d + 1)-dimensional case (where d>1) is considered. We have constructed numerically such entities which have infinite energy; simultaneously, we give a numerical confirmation of the non-existence of such objects with finite energy. We offer an interpretation of the observed robustness of such entities as seen in numerical simulations. It is based on our numerical analysis of the problem in a bounded domain, 0<r<R, and with non-integer values of d.

We show that the natural invariant state for Manneville-Pomeau maps can be characterized as a weakly Gibbsian state. In this way we make a connection between the study of intermittency via non-uniformly expanding maps and the thermodynamic formalism for non-uniformly convergent interactions.

In this paper we consider the standard map, and we study the invariant curve obtained by analytical continuation, with respect to the perturbative parameter , of the invariant circle of rotation number equal to the golden mean, corresponding to the case = 0. We show that, if we consider the parametrization that conjugates the dynamics of this curve to an irrational rotation, the domain of definition of this conjugation has an asymptotic boundary of analyticity when →0 (in the sense of the singular perturbation theory). This boundary is obtained by studying the conjugation problem for the so-called semi-standard map.

To prove this result we have used KAM-like methods adapted to the framework of singular perturbation theory, as well as matching techniques to join different pieces of the conjugation, obtained in different parts of its domain of analyticity.

We give conditions which guarantee that a piecewise expanding map T of a domain in ^d has an invariant density of bounded variation, that random perturbations of T by nearby maps also have invariant densities of bounded variation and that these converge to invariant densities of T as the size of the perturbation tends to zero.

We study the problem of entropy rigidity in the sense of Gromov-Katok for deformations of the geodesic flow by twisting the symplectic form by a closed 2-form Ω in M corresponding to a magnetic term. We show that if M is a hyperbolic manifold, then the origin is a convex point of the function given by the difference of the topological and Liouville entropies and has vanishing second derivative iff Ω is harmonic, giving a dynamical characterization of such forms. In particular, we have strict convexity for Euler-Lagrange deformations.

The main objective of this paper is to provide an explicit and fairly accurate upper bound for the number of zeros of Abelian integrals defined by quadratic isochronous centres when we perturb them inside the class of all polynomial systems of degree n.

In this work we collect together our connection results for asymptotics as t→ + 0 and t→ + ∞ of general and some special solutions of the fifth Painlevé equation. Keeping in mind already known and possible applications, actually, a slightly more general object is studied, namely, a nonlinear system of ordinary differential equations (ODEs) governing isomonodromy deformations for the (2×2) matrix linear ODE, .

In generalizing the classical theory of circle maps, we study the rotation set for maps of the real line x↦f(x) with almost periodic displacement f(x) - x. Such maps are in one-to-one correspondence with maps of compact Abelian topological groups with the displacement taking values in a dense one-parameter subgroup. For homeomorphisms, we show the existence of the analogue of the Poincaré rotation number, which is the common rotation number of all orbits besides possibly those that have rotation zero. (The coexistence of zero and non-zero rotation numbers is the main new phenomenon compared with the classical circle case.) For non-invertible maps, we prove results concerning the realization of points of the rotation interval as the rotation numbers of orbits and ergodic measures. We also address the issue of practical computation of the rotation number.

A hyperbolic equation, analogous to the telegrapher's equation in one dimension, is introduced to describe turbulent diffusion of a passive additive in a turbulent flow. The predictions of this equation, and those of the usual advection-diffusion equation, are compared with data on smoke plumes in the atmosphere and on heat flow in a wind tunnel. The predictions of the hyperbolic equation fit the data at all distances from the source, whereas those of the advection-diffusion equation fit only at large distances. The hyperbolic equation is derived from an integrodifferential equation for the mean concentration which allows it to vary rapidly. If the mean concentration varies sufficiently slowly compared with the correlation time of the turbulence, the hyperbolic equation reduces to the advection-diffusion equation. However, if the mean concentration varies very rapidly, the hyperbolic equation should be replaced by the integrodifferential equation.

We consider a class of (2+1)-dimensional baby Skyrme models with potentials that have more than one vacuum. These potentials are generalizations of the old and new baby Skyrme models in that they involve a more complicated dependence on ϕ₃ (V(ϕ₃)⩾0). The boundary conditions are such that ϕ₃ = 1 corresponds to the vacuum and ϕ₃ = -1 at the position of each skyrmion. We find that when the potential vanishes at ϕ₃ = -1 the configurations corresponding to the baby skyrmions lying `on top of each other' are the minima of the energy. However, when V(ϕ₃ = -1)≠0 the lowest field configurations correspond to separated baby skyrmions. We determine the energy distributions for skyrmions of degrees between one and eight and discuss their geometrical shapes and binding energies. We also compare the two-skyrmion states for these potentials. Most of our work has been performed numerically with the model being formulated in terms of three real scalar fields (satisfying one constraint).