Table of contents

The Baby Skyrmion model is a two-dimensional analogue of the full three-dimensional Skyrme model. It is not just useful for guiding investigations in the Skyrme model, it also has applications in condensed matter physics. Previous results on multi-charged Baby Skyrmion solutions have pointed to a modular structure, comprised of charge two rings and single charge one Skyrmions, which combine to form higher charged structures. In this paper we present alternative numerical solutions that correspond to new finite Baby Skyrmion chains, which have lower energy than those found previously, and are also good candidates for the global minimum energy solutions. We then proceed from the infinite plane, to Baby Skyrmions on a cylinder and then a torus, to obtain the solutions of periodic Baby Skyrmions, of which periodic segments will correspond to sections of large charge Baby Skyrmions in the plane.

Noise-induced transition in the solutions of the Kuramoto–Sivashinsky (K–S) equation is investigated using the minimum action method derived from the large deviation theory. This is then used as a starting point for exploring the configuration space of the K–S equation. The particular example considered here is the transition between a stable fixed point and a stable travelling wave. Five saddle points, up to constants due to translational invariance, are identified based on the information given by the minimum action path. Heteroclinic orbits between the saddle points are identified. Relations between noise-induced transitions and the saddle points are examined.

The well-known self-affine Sierpinski carpets, first studied by McMullen and Bedford independently, are constructed geometrically by repeating a single action according to a given pattern. In this paper, we extend them by randomly choosing a pattern from a set of patterns with different scales in each step of their construction process. The Hausdorff and box dimensions of the resulting limit sets are determined explicitly and the sufficient conditions for the corresponding Hausdorff measures to be positive finite are also obtained.

Let be a Hölder-continuous linear cocycle whose driving semiflow preserves a probability μ on a compact metric space X. Using the concept of two-sided quasi-Pesin orbits introduced in this paper, we show that the Lyapunov characteristic spectrum of μ can be approached arbitrarily by that of periodic points of f. So, if all periodic points of f have only positive Lyapunov exponents and such exponents are uniformly bounded away from zero, then μ also has only positive exponents. In our arguments, the exponential closing property of the driving semiflow is a basic condition, and it is easy to check that every C¹-expanding map of a closed manifold obeys this closing property.

Consequently, if f is a C^1+Hölder local diffeomorphism of a closed manifold and if it is Hölder conjugated to some C¹-expanding map, then f is itself expanding. This gives a positive answer to a question suggested by Anatole Katok.

We consider a nonlinear Neumann problem driven by the p-Laplace differential operator and with a nonsmooth locally Lipschitz potential (hemivariational inequality). We assume that the potential is asymptotically p-linear and crossing. Combining the nonsmooth critical point theory with suitable truncation and perturbation techniques, we show that the problem has at least two nontrivial smooth solutions, of which one is strictly positive.

We consider the surface quasi-geostrophic equation with dispersive forcing and critical dissipation. We prove the global existence of smooth solutions given sufficiently smooth initial data. This is done using a maximum principle for the solutions involving conservation of a certain family of moduli of continuity.

In this paper we investigate the existence of quasi-periodic solutions of non-autonomous two-dimensional reversible and Hamiltonian systems under the Bruno condition. As an application we study the dynamical stability of the trivial solution at the origin of a quasi-periodically forced planar system. Under a mild non-degeneracy condition we give a criterion that is necessary and sufficient for a large class of systems.

Herrmann et al (2009 C. R. Math.347 909–14) established the existence of self-similar solutions with algebraic decay at infinity for a coagulation equation with non-local drift. In this paper we obtain some qualitative properties of these solutions, including estimates for the coefficient of their algebraic decay.

Let M be a closed 3-manifold that supports a partially hyperbolic diffeomorphism f. If π₁(M) is nilpotent, the induced action of f_* on is partially hyperbolic. If π₁(M) is almost nilpotent or if π₁(M) has subexponential growth, M is finitely covered by a circle bundle over the torus. If π₁(M) is almost solvable, M is finitely covered by a torus bundle over the circle. Furthermore, there exist infinitely many hyperbolic 3-manifolds that do not support dynamically coherent partially hyperbolic diffeomorphisms; this list includes the Weeks manifold.

If f is a strong partially hyperbolic diffeomorphism on a closed 3-manifold M and if π₁(M) is nilpotent, then the lifts of the stable and unstable foliations are quasi-isometric in the universal cover of M. It then follows that f is dynamically coherent.

We also provide a sufficient condition for dynamical coherence in any dimension. If f is centre-bunched and if the centre-stable and centre-unstable distributions are Lipschitz, then the partially hyperbolic diffeomorphism f must be dynamically coherent.

We prove that sufficiently smooth solutions of equations of a certain class have two interesting properties. These evolution equations are in a sense degenerate, in that every term on the right-hand side of the evolution equation has either the unknown or its first spatial derivative as a factor. We first find a conserved quantity for the equation: the measure of the set on which the solution is non-zero. Second, we show that solutions which are initially non-negative remain non-negative for all times. These properties rely heavily upon the degeneracy of the leading order term. When the equation is more degenerate, we are able to prove that there are additional conserved quantities: the measure of the set on which the solution is positive and the measure of the set on which the solution is negative. To illustrate these results, we give examples of equations with nonlinear dispersion which have solutions in spaces with sufficient regularity to satisfy the hypotheses of the support and positivity theorems. An important family of equations with nonlinear dispersion are the Rosenau–Hyman compacton equations; there is no existence theory yet for these equations, but the known solutions of the compacton equations are of lower regularity than is needed for the preceding theorems. We prove an additional positivity theorem which applies to solutions of the same family of equations in a function space which includes some solutions of compacton equations.

We present two calculations for a class of robust homoclinic cycles with symmetry , for which the sufficient conditions for asymptotic stability given by Krupa and Melbourne are not optimal.

Firstly, we compute optimal conditions for asymptotic stability using transition matrix techniques which make explicit use of the geometry of the group action.

Secondly we consider a specific polynomial vector field that contains a robust heteroclinic cycle with this symmetry. Through an explicit computation of the global parts of the Poincaré map near the cycle we show that, generically, the resonance bifurcation from the cycle is supercritical: a unique branch of asymptotically stable periodic orbits emerges from the resonance bifurcation and exists for coefficient values where the cycle has lost stability.

This second calculation is of a novel kind: it is the first calculation that explicitly computes the criticality of a resonance bifurcation, and it answers a conjecture of Field and Swift in a particular limiting case. Moreover, we are able to obtain an asymptotically correct analytic expression for the period of the bifurcating orbit, with no adjustable parameters, which has not previously been achieved. We show that the asymptotic analysis compares very favourably with numerical results.

We study the process of escape of orbits through a hole in the phase space of a dynamical system generated by a chaotic map of an interval. If this hole is an element of Markov partition we are able to use the machinery of symbolic dynamics and estimate the probability of an orbit to escape at the instant of time n in terms of the topological pressure over corresponding symbolic dynamical systems. We obtain exact formulae allowing us to compare different holes according to their ability to support escaping flow of orbits. These results are applicable to the classification of vertices of dynamical networks in terms of the loads on nodes and edges of a network.

We study the dead-core problem for the fast diffusion equation with strong absorption. Unlike in many other related problems of singularity formation, we show that the temporal rate of formation of the dead-core is not self-similar. We moreover obtain precise estimates on rescaled solutions and on the single-point final dead-core profile. Results of this type were up to now known only for problems with linear diffusion. The proofs rely on self-similar variables and require a delicate use of the Zelenyak method.

This paper is devoted to the analysis of some uniqueness properties of a classical reaction–diffusion equation of the Fisher-KPP type, coming from population dynamics in heterogeneous environments. We work in a one-dimensional interval (a, b) and we assume a nonlinear term of the form u (μ(x) − γu) where μ belongs to a fixed subset of C⁰([a, b]). We prove that the knowledge of u at t = 0 and of u, u_x at a single point x₀ and for small times t ∊ (0, ε) is sufficient to completely determine the couple (u(t, x), μ(x)) provided γ is known. Additionally, if u_xx(t, x₀) is also measured for t ∊ (0, ε), the triplet (u(t, x), μ(x), γ) is also completely determined. Those analytical results are completed with numerical simulations which show that, in practice, measurements of u and u_x at a single point x₀ (and for t ∊ (0, ε)) are sufficient to obtain a good approximation of the coefficient μ(x). These numerical simulations also show that the measurement of the derivative u_x is essential in order to accurately determine μ(x).

We prove that there is a residual subset in Diff ¹(M) such that, for every , any homoclinic class of f with invariant one-dimensional central bundle containing saddles of different indices (i.e. with different dimensions of the stable invariant manifold) coincides with the support of some invariant ergodic non-hyperbolic (one of the Lyapunov exponents is equal to zero) measure of f.

This paper is devoted to a modification of the classical Cahn–Hilliard equation proposed by some physicists. This modification is obtained by adding the second time derivative of the order parameter multiplied by an inertial coefficient ε > 0, which is usually small in comparison with the other physical constants. The main feature of this equation is the fact that even a globally bounded nonlinearity is 'supercritical' in the case of two and three space dimensions. Thus, the standard methods used for studying semilinear hyperbolic equations are not very effective in the present case. Nevertheless, we have recently proven the global existence and dissipativity of strong solutions in the 2D case (with a cubic controlled growth nonlinearity) and for the 3D case with small ε and arbitrary growth rate of the nonlinearity (see (Grasselli et al 2009 J. Evol. Eqns9 371–404, Grasselli et al 2009 Commun. Partial Diff. Eqns34 137–70)). The present contribution studies the long-time behaviour of rather weak (energy) solutions of that equation and it is a natural complement of the results of our previous papers (Grasselli et al 2009 J. Evol. Eqns9 371–404, Grasselli et al 2009 Commun. Partial Diff. Eqns34 137–70). In particular, we prove here that the attractors for energy and strong solutions coincide for both the cases mentioned above. Thus, the energy solutions are asymptotically smooth. In addition, we show that the non-smooth part of any energy solution decays exponentially in time and deduce that the (smooth) exponential attractor for the strong solutions constructed previously is simultaneously the exponential attractor for the energy solutions as well. It is worth noting that the uniqueness of energy solutions in the 3D case is not known yet, so we have to use the so-called trajectory approach which does not require uniqueness. Finally, we apply the obtained exponential regularization of the energy solutions for verifying the dissipativity of solutions of the 2D modified Cahn–Hilliard equation in the intermediate phase space of weak solutions (in between energy and strong solutions) without any restriction on ε.

A trajectory of a system with two clearly separated time scales generally consists of fast segments (or jumps) followed by slow segments where the trajectory follows an attracting part of a slow manifold. The switch back to fast dynamics typically occurs when the trajectory passes a fold with respect to a fast direction. A special role is played by trajectories known as canard orbits, which do not jump at a fold but, instead, follow a repelling slow manifold for some time. We concentrate here on the case of a slow–fast system with two slow and one fast variable, where canard orbits arise geometrically as intersection curves of two-dimensional attracting and repelling slow manifolds. Canard orbits are intimately related to the dynamics near special points known as folded singularities, which in turn have been shown to explain small-amplitude oscillations that can be found as part of so-called mixed-mode oscillations.

In this paper we present a numerical method to detect and then follow branches of canard orbits in a system parameter. More specifically, we define well-posed two-point boundary value problems (BVPs) that represent orbit segments on the slow manifolds, and we continue their solution families with the package AUTO. In this way, we are able to deal effectively with the numerical challenge of strong attraction to and strong repulsion from the slow manifolds. Canard orbits are detected as the transverse intersection points of the curves along which attracting and repelling slow manifolds intersect a suitable section (near a folded node). These intersection points correspond to a unique pair of orbits segments, one on the attracting and one on the repelling slow manifold. After concatenation of the respective pairs of orbit segments, all detected canard orbits are represented as solutions of a single BVP, which allows us to continue them in system parameters. We demonstrate with two examples—the self-coupled FitzHugh–Nagumo system and a three-dimensional reduced Hodgkin–Huxley model—that branches of canard orbits can be computed reliably. Furthermore, our computations illustrate that the continuation of canard orbits allows one to find and investigate new types of dynamics, such as the interaction between canard orbits and a saddle periodic orbit that is generated in a singular Hopf bifurcation.