Table of contents

The subject area referred to as 'wave chaos', 'quantum chaos' or 'quantum chaology' has been investigated mostly by the theoretical physics community in the last 30 years. The questions it raises have more recently also attracted the attention of mathematicians and mathematical physicists, due to connections with number theory, graph theory, Riemannian, hyperbolic or complex geometry, classical dynamical systems, probability, etc. After giving a rough account on 'what is quantum chaos?', I intend to list some pending questions, some of them having been raised a long time ago, some others more recent. The choice of problems (and of references) is of course partial and personal.

One of the outstanding open questions in modern applied mathematics is whether solutions of the incompressible Euler equations develop a singularity in the vorticity field in a finite time. This paper briefly reviews some of the issues concerning this problem, together with some observations that may suggest that it may be more subtle than first thought.

Driven by a deluge of data, biology is undergoing a transition to a more quantitative science. Making sense of the data, building new models, asking the right questions and designing smart experiments to answer them are becoming ever more relevant. In this endeavour, nonlinear approaches can play a fundamental role. The biochemical reactions that underlie life are very often nonlinear. The functional features exhibited by biological systems at all levels (from the activity of an enzyme to the organization of a colony of ants, via the development of an organism or a functional module like the one responsible for chemotaxis in bacteria) are dynamically robust. They are often unaffected by order of magnitude variations in the dynamical parameters, in the number or concentrations of actors (molecules, cells, organisms) or external inputs (food, temperature, pH, etc). This type of structural robustness is also a common feature of nonlinear systems, exemplified by the fundamental role played by dynamical fixed points and attractors and by the use of generic equations (logistic map, Fisher–Kolmogorov equation, the Stefan problem, etc.) in the study of a plethora of nonlinear phenomena. However, biological systems differ from these examples in two important ways: the intrinsic stochasticity arising from the often very small number of actors and the role played by evolution. On an evolutionary time scale, nothing in biology is frozen. The systems observed today have evolved from solutions adopted in the past and they will have to adapt in response to future conditions. The evolvability of biological system uniquely characterizes them and is central to biology. As the great biologist T Dobzhansky once wrote: 'nothing in biology makes sense except in the light of evolution'.

We present a numerical method for finding and continuing heteroclinic connections of vector fields that involve periodic orbits. Specifically, we concentrate on the case of a codimension-d heteroclinic connection from a saddle equilibrium to a saddle periodic orbit, denoted EtoP connection for short. By employing a Lin's method approach we construct a boundary value problem that has as its solution two orbit segments, one from the equilibrium to a suitable section Σ and the other from Σ to the periodic orbit. The difference between their two end points in Σ can be chosen in a d-dimensional subspace, and this gives rise to d well-defined test functions that are called the Lin gaps. A connecting orbit can be found in a systematic way by closing the Lin gaps one by one in d consecutive continuation runs. Indeed, any common zero of the Lin gaps corresponds to an EtoP connection, which can then be continued in system parameters.

The performance of our method is demonstrated with a number of examples. Specifically, we computate different types of EtoP orbits in the Lorenz system, in a vector-field model of a saddle-node Hopf bifurcation with global reinjection and in a four-dimensional Duffing-type system. Finally, we demonstrate the versatility of our geometric approach by finding a codimension-zero heteroclinic connection between two saddle periodic orbits in a four-dimensional vector field.

This paper studies the pullback asymptotic behaviour of trajectories for evolution equations. We first combine the idea of trajectory attractor and pullback attractor to formulate a new type of attractor called pullback trajectory attractor. Then we prove a sufficient condition for the existence of a pullback trajectory attractor for the translation cocycle defined on the united trajectory space of the evolution equations. Finally, we take a three-dimensional incompressible non-Newtonian fluid as the applied example and prove its pullback trajectory asymptotic smoothing effect.

We consider families of dynamics that can be described in terms of Perron–Frobenius operators with exponential mixing properties. For piecewise C² expanding interval maps we rigorously prove continuity properties of the drift J(λ) and of the diffusion coefficient D(λ) under parameter variation. Our main result is that D(λ) has a modulus of continuity of order , i.e. D(λ) is Lipschitz continuous up to quadratic logarithmic corrections. For a special class of piecewise linear maps we provide more precise estimates at specific parameter values. Our analytical findings are quantified numerically for the latter class of maps by using exact series expansions for the transport coefficients that can be evaluated numerically. We numerically observe strong local variations of all continuity properties.

We consider a random version of the McMullen–Bedford general Sierpinski carpet which is constructed by randomly choosing patterns in each step instead of a single pattern in its original form. Their Hausdorff, packing and box-counting dimensions are determined. A sufficient condition and a necessary condition for the Hausdorff measures in their dimensions to be positive are given. As an application, we discuss the issue on the intersection of the general Sierpinski carpet with its translations.

In this work we consider a 1 : −1 non-semi-simple resonant periodic orbit of a three degrees of freedom real analytic Hamiltonian system. From the formal analysis of the normal form, we prove the branching off of a two-parameter family of two-dimensional invariant tori of the normalized system, whose normal behaviour depends intrinsically on the coefficients of its low-order terms. Thus, only elliptic or elliptic together with parabolic and hyperbolic tori may detach from the resonant periodic orbit. Both patterns are mentioned in the literature as the direct and inverse, respectively, periodic Hopf bifurcation. In this paper we focus on the direct case, which has many applications in several fields of science. Our target is to prove, in the framework of Kolmogorov–Arnold–Moser (KAM) theory, the persistence of most of the (normally) elliptic tori of the normal form, when the whole Hamiltonian is taken into account, and to give a very precise characterization of the parameters labelling them, which can be selected with a very clear dynamical meaning. Furthermore, we give sharp quantitative estimates on the 'density' of surviving tori, when the distance to the resonant periodic orbit goes to zero, and show that the four-dimensional invariant Cantor manifold holding them admits a Whitney-C^∞ extension. Due to the strong degeneracy of the problem, some standard KAM methods for elliptic low-dimensional tori of Hamiltonian systems do not apply directly, so one needs to properly suit these techniques to the context.

The Landau equation with an external force describes the time evolution of charged particles in an external field. In this paper we construct the global existence of classical solutions to the Landau equation with the external force in the whole space when the initial data are a perturbation of the given local Maxwellian. Moreover, the time decay rate of the solution is also obtained.

The Stokes geometry for the propagator of the quantum Hénon map is studied. Virtual turning points and new Stokes curves, which have been proposed as entirely new notions appearing only in higher-order differential equations, are defined through deriving a set of differential equations acting on the propagator. A concrete recipe to construct Stokes geometry is presented and its validity is examined using three independent methods. The tree-pruning rule, a phenomenological scheme to cope with the complex WKB description of chaotic systems, is also examined in the light of the exact WKB treatment of the Stokes phenomenon.

We study the asymptotic behaviour of a general class of discrete energies defined on functions of the form , as the mesh size ε goes to 0. We prove that under general assumptions that cover the case of bounded and unbounded spin systems in the thermodynamic limit, the variational limit of E_ε has the form . The cases of homogenization and of non-pairwise interacting systems (e.g. multiple-exchange spin systems) are also discussed.

In this paper, we propose a geometric integrator for nonholonomic mechanical systems. It can be applied to discrete Lagrangian systems specified through a discrete Lagrangian , where Q is the configuration manifold, and a (generally nonintegrable) distribution . In the proposed method, a discretization of the constraints is not required. We show that the method preserves the discrete nonholonomic momentum map, and also that the nonholonomic constraints are preserved in average. We study, in particular, the case where Q has a Lie group structure and the discrete Lagrangian and/or nonholonomic constraints have various invariance properties, and show that the method is also energy-preserving in some important cases.

We study a Markov random process describing muscle molecular motor behaviour. Every motor is either bound up with a thin filament or unbound. In the bound state the motor creates a force proportional to its displacement from the neutral position. In both states the motor spends an exponential time depending on the state. The thin filament moves at a velocity proportional to the average of all displacements of all motors.

We assume that the time which a motor stays in the bound state does not depend on its displacement. Then one can find an exact solution of a nonlinear equation appearing in the limit of an infinite number of motors.

We study non-topological solitons, so-called Q-balls, which carry a non-vanishing Noether charge and arise as lump solutions of self-interacting complex scalar field models. Explicit examples of new axially symmetric non-spinning Q-ball solutions that have not been studied so far are constructed numerically. These solutions can be interpreted as angular excitations of the fundamental Q-balls and are related to the spherical harmonics. Correspondingly, they have higher energy and their energy densities possess two local maxima on the positive z-axis.

We also study two Q-balls interacting via a potential term in 3 + 1 dimensions and construct examples of stationary, solitonic-like objects in (3 + 1)-dimensional flat space–time that consist of two interacting global scalar fields. We concentrate on configurations composed of one spinning and one non-spinning Q-ball and study the parameter-dependence of the energy and charges of the configuration.

In addition, we present numerical evidence that for fixed values of the coupling constants two different types of 2-Q-ball solutions exist: solutions with defined parity, but also solutions that are asymmetric with respect to reflection through the x–y plane.