Table of contents

We construct a renormalization operator acting on the space of analytic Hamiltonians defined on , d ⩾ 2, based on the multidimensional continued fractions algorithm developed by the authors. We show convergence of orbits of the operator around integrable Hamiltonians satisfying a non-degeneracy condition. This in turn yields a new proof of a KAM-type theorem on the stability of diophantine invariant tori.

Conventionally, bright solitary wave solutions can be obtained in self-focusing nonlinear Schrödinger equations with attractive self-interaction. However, when self-interaction becomes repulsive, it seems impossible to have bright solitary wave solutions. Here we show that there exist symbiotic bright solitary wave solutions of coupled nonlinear Schrödinger equations with repulsive self-interaction but strongly attractive interspecies interaction. For such coupled nonlinear Schrödinger equations in two- and three-dimensional domains, we prove the existence of least energy solutions and study the location and configuration of symbiotic bright solitons. We use Nehari's manifold to construct least energy solutions and derive their asymptotic behaviour by some techniques of singular perturbation problems.

A theory for the fluid flux generated across heteroclinic separatrices under the influence of time-aperiodic perturbations is presented. The flux is explicitly defined as the amount of fluid transferred per unit time, and its detailed time-dependence monitored. The perturbations are allowed to be significantly discontinuous in time, including for example impulsive (Dirac delta type) discontinuities. The flux is characterized in terms of time-varying separatrices, with easily computable formulae (directly related to Melnikov functions) provided.

In this paper, a novel coupling scheme with different coupling delays is presented to achieve generalized synchronization (complete synchronization (CS), anticipating synchronization (AS) and lag synchronization (LS)). The Lyapunov–Krasovskii functional method is employed to investigate the global asymptotic stability of the error dynamical system. The theoretical analysis indicates that delayed chaotic neural networks coupled in this way can switch arbitrarily among CS, AS and LS under the newly proposed coupling configuration. Switching among different kinds of synchronization can be done by changing the transformation time of the coupling signal. Numerical simulations agree with the theoretical analysis.

We investigate the stability of the three-phase contact line of a thin liquid ridge on a hydrophobic substrate for flow driven by surface tension and van der Waals forces. We study the role of slippage in the emerging instability at the three-phase contact line by comparing the lubrication models for no-slip and slip conditions at the liquid/substrate interface.

For both cases we derive a sharp-interface model via matched asymptotic expansions and derive the eigenvalues from a linear stability analysis of the respective reduced models. We compare our asymptotic results with the eigenvalues obtained numerically for the full lubrication models.

We explore the complicated dynamics arising in a neighbourhood of a homoclinic point associated with a homoclinic bifurcation of a two-parameter family of three-dimensional dissipative diffeomorphisms. We address the case in which the unstable manifold of the periodic saddle involved in the homoclinic bifurcation has dimension two. Besides proving the existence of strange attractors with two positive Lyapounov exponents for the associated limit return map, we also select a curve in the space of parameters in order to numerically detect the presence of possible new families of one-dimensional and two-dimensional strange attractors. The end of this curve of parameters corresponds to a return map which is conjugate to a 'bidimensional tent map'.

We study here the asymptotic behaviour of weakly transverse water-waves in the long waves regime. It is well-known that the Kadomtsev–Petviashvili (KP) approximation describes formally the dynamics of the exact solutions of the water-waves equations. We provide here a rigorous justification of this approximation, showing that if solutions of the water-waves equations exist over a relevant time scale, then they are well approximated by the KP approximation. A nonphysical zero mass assumption, inherent to the structure of the KP equation, is however needed to obtain this result; this is the reason why we introduce a class of weakly transverse Boussinesq systems. These new systems provide a much more precise approximation than the KP equation and do not require any zero mass assumption.

The hyperbolic and Hausdorff dimensions are shown to coincide for C² maps without recurrent critical points. The maps may have parabolic periodic points. The Julia set for certain such maps may have hyperbolic dimension equal to 1 but Lebesgue measure equal to 0.

We compute the Hausdorff dimension for an open class of iterated function schemes in in the C² topology, in terms of the pressure function. This class is characterized by a domination condition that is not too strong, plus the non-overlapping condition. Also in this class there exists a unique invariant probability measure of full Hausdorff dimension. Finally we show that the Hausdorff dimension is a continuous function in this class.

The Keller–Segel model is a system of partial differential equations modelling chemotactic aggregation in cellular systems. This model has blowing-up solutions for large enough initial conditions in dimensions d ≥ 2, but all the solutions are regular in one dimension, a mathematical fact that crucially affects the patterns that can form in the biological system. One of the strongest assumptions of the Keller–Segel model is the diffusive character of the cellular motion, known to be false in many situations. We extend this model to such situations in which the cellular dispersal is better modelled by a fractional operator. We analyse this fractional Keller–Segel model and find that all solutions are again globally bounded in time in one dimension. This fact shows the robustness of the main biological conclusions obtained from the Keller–Segel model.

The asymptotic stability for the weak solution θ of the critical and supercritical dissipative quasi-geostrophic equation in the Serrin-type class ∇θ ∊ L^r(0, ∞;L^p(R²)) is examined. This equation is perturbed by large initial data and external functions. It is shown that every weak perturbed solution has the same asymptotic behaviour as that of θ. More precisely, the difference decays in the norm of L²(R²).

We consider complex polynomials f(z) = z^ℓ+c₁ for and and find some combinatorial types and values of ℓ such that there is no invariant probability measure equivalent to conformal measure on the Julia set. This holds for particular Fibonacci-like and Feigenbaum combinatorial types when ℓ is sufficiently large and also for a class of 'long-branched' maps of any critical order.

In this paper we consider the interaction of the Lorenz manifold—the two-dimensional stable manifold of the origin of the Lorenz equations—with the two-dimensional unstable manifolds of the secondary equilibria or bifurcating periodic orbits of saddle type. We compute these manifolds for varying values of the parameter ϱ in the Lorenz equations, which corresponds to the transition from simple to chaotic dynamics with the classic Lorenz butterfly attractor at ϱ = 28.

Furthermore, we find and continue in ϱ the first 512 generic heteroclinic orbits that are given as the intersection curves of these two-dimensional manifolds. The branch of each heteroclinic orbit emerges from the well-known first codimension-one homoclinic explosion point at ϱ ≈ 13.9265, has a fold and then ends at another homoclinic explosion point with a specific ϱ-value. We describe the combinatorial structure of which heteroclinic orbit ends at which homoclinic explosion point. This is verified with our data for the 512 branches from which we automatically extract (by means of a small computer program) the relevant symbolic information.

Our results on the manifold structure are complementary to previous work on the symbolic dynamics of periodic orbits in the Lorenz attractor. We point out the connections and discuss directions for future research.

The goal of the paper is to study the asymptotic behaviour of solutions to a high contrast quasilinear equation of the form where with n ⩾ 2, 1 < p ⩽ n, and the coefficient G^ε(x) is assumed to blow up as ε → 0 on a set of N_ε isolated inclusions of asymptotically small measure. Here as ε → 0. It is shown that the asymptotic behaviour, as ε → 0, of the solution u^ε is described in terms of a homogenized quasilinear equation of the form where the coefficient B(x) is calculated as a local energy characteristic of the microstructure associated with the potential G^ε(x) in the original problem. This result is then illustrated with a periodic example and a nonperiodic one.

Inter-relations between random matrix ensembles with different symmetry types provide inter-relations between generating functions for the gap probabilities at the spectrum edge. Combining these in the scaled limit with the exact evaluation of the gap probabilities for certain superimposed ensembles with orthogonal symmetry allows for the exact evaluation of the gap probabilities at the hard and soft spectrum edges in the cases of orthogonal and symplectic symmetry. These exact evaluations are given in terms of Painlevé transcendents and in terms of Fredholm determinants.

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