Table of contents

We consider projections of a smooth and regular surface M in the Minkowski 3-space $\mathbb R^3_1$ along lightlike directions to a fixed transverse plane. The lightlike directions in $\mathbb R^3_1$ can be parametrized by a circle on the lightcone and the resulting 1-parameter family of projections can be considered as viewing M along a special 'camera motion'. The associated 1-parameter families of contour generators and apparent contours reveal some aspects of the extrinsic and intrinsic geometry of M. We characterize geometrically the generic $\mathcal A_e$-codimension ⩽1 singularities of a given projection and consider their bifurcations in the family of projections. We show that the families of contour generators and apparent contours are solutions of certain first order ordinary differential equations and obtain their generic local configurations.

In this paper we prove the existence of a trajectory attractor (in the sense of Chepyzhov and Vishik) for a nonlinear PDE system obtained from a 3D liquid crystal model accounting for stretching effects. The system couples a nonlinear evolution equation for the director d (introduced in order to describe the preferred orientation of the molecules) with an incompressible Navier–Stokes equation for the evolution of the velocity field u. The technique is based on the introduction of a suitable trajectory space and of a metric accounting for the double-well type nonlinearity contained in the director equation. Finally, a dissipative estimate is obtained by using a proper integrated energy inequality. Both the cases of (homogeneous) Neumann and (non-homogeneous) Dirichlet boundary conditions for d are considered.

In this second paper, we establish the existence of ground state solutions of the nonlinear Schrödinger system studied in the first paper (Hajaiej H 2012 Math. Models Methods Appl. Sci.22 1250010) when the diamagnetic field is nul. We also prove some symmetry properties of these kinds of solutions.

An integrable generalization on the 2D sphere S² and the hyperbolic plane H² of the Euclidean anisotropic oscillator Hamiltonian with 'centrifugal' terms given by

$$\begin{equation*} \mathcal{H}=\frac{1}{2}(p_1^2+p_2^2)+ \delta q_{1}^{2}+(\delta + \Omega)q_{2}^{2} +\frac{\lambda_1}{q_{1}^{2}}+\frac{\lambda_2}{q_{2}^{2}} \end{equation*}$$

${\cal H}_\kappa$

${\cal H}_\kappa$

${\cal H}_\kappa$

${\cal H}_\kappa$

${\cal H}_\kappa$

The superintegrability issue for ${\cal H}_\kappa$ is discussed by focusing on the value Ω = 3δ, which is one of the cases for which the Euclidean Hamiltonian $\mathcal{H}$ is known to be superintegrable (the 1 : 2 oscillator). We show numerically that for Ω = 3δ the curved Hamiltonian ${\cal H}_\kappa$ presents nonperiodic bounded trajectories, which seems to indicate that ${\cal H}_\kappa$ provides a non-superintegrable generalization of ${\cal H}$ even for values of Ω that lead to commensurate frequencies in the Euclidean case. We compare this result with a previously known superintegrable curved analogue ${\cal H}_\kappa'$ of the 1 : 2 Euclidean oscillator, which is described in detail, showing that the Ω = 3δ specialization of ${\cal H}_\kappa$ does not coincide with ${\cal H}_\kappa'$. Hence we conjecture that ${\cal H}_\kappa$ would be an integrable (but not superintegrable) curved generalization of the anisotropic oscillator that exists for any value of Ω and has constants of the motion that are quadratic in the momenta. Thus each commensurate Euclidean oscillator could admit another specific superintegrable curved Hamiltonian which would be different from ${\cal H}_\kappa$ and endowed with higher order integrals. Finally, the geometrical interpretation of the curved 'centrifugal' terms appearing in ${\cal H}_\kappa$ is also discussed in detail.

We prove that every self-homeomorphism h : K_s → K_s on the inverse limit space K_s of the tent map T_s with slope $s \in (\sqrt 2, 2]$ has topological entropy h_top(h) = |R| log s, where $R \in {\mathbb Z}$ is such that h and σ^R are isotopic. Conclusions on all possible values of the entropy of homeomorphisms of the inverse limit space of a (renormalizable) quadratic map are also drawn.

We investigate the existence, uniqueness and Gaussian curvature of the invariant carrying simplices of 3 species autonomous totally competitive Lotka–Volterra systems. Explicit examples are given where the carrying simplex is convex or concave, but also where the curvature is not single-signed. Our method monitors the curvature of an evolving surface that converges uniformly to the carrying simplex, and generally relies on establishing that the Gaussian image of the evolving surface is confined to an invariant cone. We also discuss the relationship between the curvature of the carrying simplex near an interior fixed point and its Split Lyapunov stability. Finally we comment on extensions to general Lotka–Volterra systems that are not competitive.

We prove some regularity results for a class of two-dimensional non-Newtonian fluids. Then, by applying results from Dashti and Robinson (2009 Nonlinearity22 735–46), we show uniqueness of particle trajectories.

The entropy h(T_α) of α-continued fraction transformations is known to be locally monotone outside a closed, totally disconnected set $\mathcal{E}$. We will exploit the explicit description of the fractal structure of $\mathcal{E}$ to investigate the self-similarities displayed by the graph of the function α ↦ h(T_α). Finally, we completely characterize the plateaux occurring in this graph, and classify the local monotonic behaviour.

In this work we study the relation between the regularity of invariant foliations and the Lyapunov exponents of partially hyperbolic diffeomorphisms. We suggest a new regularity condition for foliations in terms of disintegration of the Lebesgue measure which can be considered to be a criterium for the rigidity of Lyapunov exponents.

In this paper, we study a system of partial differential equations describing the evolution of a population under chemotactic effects with non-local reaction terms. We consider an external application of chemoattractant in the system and study the cases of one and two populations in competition. By introducing global competitive/cooperative factors in terms of the total mass of the populations, we obtain, for a range of parameters, that any solution with positive and bounded initial data converges to a spatially homogeneous state with positive components. The proofs rely on the maximum principle for spatially homogeneous sub- and super-solutions.

We analyse the blowup (finite-time singularity) in inviscid shell models of convective turbulence. We show that the blowup exists and its internal structure undergoes a series of bifurcations under a change of shell model parameter. Various blowup structures are observed and explained, which vary from self-similar to periodic, quasi-periodic and chaotic regimes. Though the blowup takes sophisticated forms, its asymptotic small-scale structure is independent of the initial conditions, i.e. universal. Finally, we discuss the implications of the obtained results for the open problems of blowup in inviscid flows and for the theory of turbulence.

Starting with the work of Lapidus and van Frankenhuysen a number of papers have introduced zeta functions as a way of capturing multifractal information. In this paper we propose a new multifractal zeta function and show that under certain conditions the abscissa of convergence yields the Hausdorff multifractal spectrum for a class of measures.