Table of contents

In this paper, we study a class of Schrödinger equations

$$\begin{eqnarray}\begin{array}{*{35}{l}} -\bigtriangleup u=k(u), & x\in {{\mathbb{R}}^{N}}, \end{array}\end{eqnarray}$$

$N\geqslant 3$

We study properties of the stable norm on the first homology group of the 2-torus with respect to Riemannian or Finsler metrics, focusing on points with irrational slope. Our results show that the stable norm detects KAM-tori and hyperbolicity in the geodesic flow. Along the way, we shall prove new inequalities for the stable norm near rational directions. Moreover, we study the stable norm in some natural examples reflecting the new results in this paper.

We consider a class of stage-structured differential equations with unimodal feedback. By using the time delay as a bifurcation parameter, we show that the number of local Hopf bifurcation values is finite. Furthermore, we analytically prove that these local Hopf bifurcation values are neatly paired, and each pair is jointed by a bounded global Hopf branch. We use the well-known Mackey–Glass equation with a stage structure as an illustrative example to demonstrate that bounded global Hopf branches can induce interesting and rich dynamics. As the delay increases over a finite interval, the stage-structured Mackey–Glass equation exhibits certain symmetric dynamic patterns: the solutions evolve from a stable equilibrium to sustained stable periodic oscillations, to chaotic-like aperiodic oscillations and back to sustained stable periodic oscillations, to a stable equilibrium.

We set up a methodology for computer assisted proofs of the existence and the KAM stability of an arbitrary periodic orbit for Hamiltonian systems. We give two examples of application for systems with two and three degrees of freedom. The first example verifies the existence of tiny elliptic islands inside large chaotic domains for a quartic potential. In the 3-body problem we prove the KAM stability of the well-known figure eight orbit and two selected orbits of the so called family of rotating eights. Some additional theoretical and numerical information is also given for the dynamics of both examples.

In this paper we study relations of various types of sensitivity between a t.d.s. (X, T) and t.d.s. (M(X), T_M) induced by (X, T) on the space of probability measures. Among other results, we prove that $\mathcal{F}$-sensitivity of (M(X), T_M) implies the same of (X, T) and the converse is also true when $\mathcal{F}$ is a filter. We show that (X, T) is multi-sensitive if and only if so is (M(X), T_M) and that (X, T) is $\mathcal{F}$-sensitive if and only if $\left({{M}_{n}}(X),{{T}_{M}}\right)$ is $\mathcal{F}$-sensitive (for some/all $n\in \mathbb{N}$). We finish the paper providing an example of a minimal syndetically sensitive t.d.s. or a Li–Yorke sensitive t.d.s. such that induced t.d.s. fails to be sensitive.

Parker's classical stellar wind solution [20] describing steady spherically symmetric outflow from the surface of a star is revisited. Viscous dissipation is retained. The resulting system of equations has slow-fast structure and is amenable to analysis using geometric singular perturbation theory. This technique leads to a reinterpretation of the sonic point as a folded saddle and the identification of shock solutions as canard trajectories in space [22]. The results shed light on the location of the shock and its sensitivity to the system parameters. The related spherically symmetric stellar accretion solution of Bondi [4] is described by the same theory.

We compute explicit formulae for the moments of the densities of the eigenvalues of the classical β-ensembles for finite matrix dimension as well as the expectation values of the coefficients of the characteristic polynomials. In particular, the moments are linear combinations of averages of Jack polynomials, whose coefficients are related to specific examples of Jack characters.

The main purpose of this article is to study the problem of limit cycle bifurcations of perturbed completely integrable systems from a geometric point of view.

We show that if the rotation set of a homeomorphism of the torus is stable under small perturbations of the dynamics, then it is a convex polygon with rational vertices. We also show that such homeomorphisms are C⁰-generic and have bounded rotational deviations (even for pseudo-orbits). The results hold both in the area-preserving setting and in the general setting. When the rotation set is stable, we give explicit estimates on the type of rationals that may appear as vertices of rotation sets in terms of the stability constants.

In this article, we present some interesting non-steady explicit solutions to the 2D Euler and Navier–Stokes equations. Explicit calculations on the explicit solutions show that Navier–Stokes (and Euler) equations have the novel property of rough dependence upon initial data in contrast to the sensitive dependence upon initial data found in chaos. Through the explicit calculations, we are able to obtain a lower bound on the norm of the Fréchet derivative of the solution operator at the explicit solutions to the Navier–Stokes equations. The lower bound approaches infinity as the Reynolds number approaches infinity. For Euler equations, this lower bound is indeed infinity. The rough dependence property in the inviscid case is closely related to the theorem of Cauchy. The viscous effect on the theorem of Cauchy and the rough dependence property is also studied.

Particles in rotating saddle potentials exhibit precessional motion which, up to now, has been explained by explicit computation. We show that this precession is due to a hidden Coriolis-like force which, unlike the standard Coriolis force, is present in the inertial frame. We do so by finding a hodograph-like 'guiding center' transformation using the method of normal form. We also point out that the transformation cannot be of contact type in principle, thus showing that the standard (in applied literature) heuristic averaging obscures the fact that the transformation of the position must involve the velocity.

The paper investigates the well-posedness and the longtime dynamics of the quasilinear wave equations with structural damping and supercritical nonlinearities: ${{u}_{tt}}- \Delta u+{{\left(- \Delta \right)}^{\alpha}}{{u}_{t}}-\nabla \cdot \left(\frac{\nabla u}{\sqrt{1+|\nabla u{{|}^{2}}}}\right)+g(u)=f(x)$, with $1/2<\alpha <1$. The main results are concerned with the quasilinear term $\nabla \cdot \left(\frac{\nabla u}{\sqrt{1+|\nabla u{{|}^{2}}}}\right)$ and the nonlinearities g(u) with supercritical growth. Under the rather mild conditions, the well-posedness and the existence of the global and exponential attractors (rather than the weak ones) are established in natural energy space. These results show that even for the supercritical nonlinearities, the regularity and the longtime dynamics of the solutions of the mentioned equations are of the characteristics of the parabolic equations because of the effectiveness of the structural damping.

We study real analytic perturbations of hyperbolic linear automorphisms on the 2-torus. The Koopman and the transfer operator are nuclear of order 0 when acting on a suitable Hilbert space. We show the generic existence of non-trivial Ruelle resonances for both operators. We prove that some of the perturbations preserve the volume and some of them do not.

We address the problem of long-time asymptotics for the solutions of the Korteweg–de Vries equation under low regularity assumptions. We consider decaying initial data admitting only a finite number of moments. For the so-called 'soliton region', an improved asymptotic estimate is provided, in comparison with the one in Grunert and Teschl (2009 Math. Phys. Anal. Geom. 12 287–324). Our analysis is based on the dbar steepest descent method proposed by Miller and McLaughlin.

We introduce a random dynamical system related to continued fraction expansions. It uses random combinations of the Gauss map and the Rényi (or backwards) continued fraction map. We explore the continued fraction expansions that this system produces, as well as the dynamical properties of the system.

We consider a smooth one-parameter family $t\mapsto \left(\,{{f}_{t}}:M\to M\right)$ of diffeomorphisms with compact transitive Axiom A attractors ${{ \Lambda }_{t}}$, denoting by $\text{d}{{\rho}_{t}}$ the SRB measure of ${{f}_{t}}{{|}_{{{ \Lambda }_{t}}}}$. Our first result is that for any function θ in the Sobolev space $H_{p}^{r}(M)$, with $1<p<\infty$ and 0  <  r  <  1/p, the map $t\mapsto {\int}^{}\theta \,\text{d}{{\rho}_{t}}$ is α-Hölder continuous for all $\alpha <r$. This applies to $\theta (x)=h(x) \Theta \left(g(x)-a\right)$ (for all $\alpha <1$) for h and g smooth and $\Theta$ the Heaviside function, if a is not a critical value of g. Our second result says that for any such function $\theta (x)=h(x) \Theta \left(g(x)-a\right)$ so that in addition the intersection of $\left\{x|g(x)=a\right\}$ with the support of h is foliated by 'admissible stable leaves' of f_t, the map $t\mapsto {\int}^{}\theta \,\text{d}{{\rho}_{t}}$ is differentiable. (We provide distributional linear response and fluctuation-dissipation formulas for the derivative.) Obtaining linear response or fractional response for such observables θ is motivated by extreme-value theory.