Table of contents

We analyse the signature type of a cascade of periodic orbits associated with period-doubling renormalizable maps of the two-dimensional disc. The signature is a sequence of rational numbers invariant with respect to orientation-preserving topological conjugacies, which describes how periodic orbits are linked around each other. We prove that in the class of area-contracting maps the signature cannot be a monotone sequence. This explains why classical examples of infinitely renormalizable maps due to Bowen, Franks and Young cannot be achieved by smooth dissipative maps, which shows that there are topological obstructions to realizing infinitely renormalizable maps in the area-contracting case.

The invariant manifold structures of the collinear libration points for the restricted three-body problem provide the framework for understanding transport phenomena from a geometrical point of view. In particular, the stable and unstable invariant manifold tubes associated with libration point orbits are the phase space conduits transporting material between primary bodies for separate three-body systems. These tubes can be used to construct new spacecraft trajectories, such as a 'Petit Grand Tour' of the moons of Jupiter. Previous work focused on the planar circular restricted three-body problem. This work extends the results to the three-dimensional case.

Besides providing a full description of different kinds of libration motions in a large vicinity of these points, this paper numerically demonstrates the existence of heteroclinic connections between pairs of libration orbits, one around the libration point L₁ and the other around L₂. Since these connections are asymptotic orbits, no manoeuvre is needed to perform the transfer from one libration point orbit to the other. A knowledge of these orbits can be very useful in the design of missions such as the Genesis Discovery Mission, and may provide the backbone for other interesting orbits in the future.

We show that the zeta function for the dynamics generated by the map z ↦ z² + c, c < − 2, can be estimated in terms of the dimension of the corresponding Julia set. That implies a geometric upper bound on the number of its zeros, which are interpreted as resonances for this dynamical systems. The method of proof of the upper bound is used to construct a code for counting the number of zeros of the zeta function. The numerical results support the conjecture that the upper bound in terms of the dimension of the Julia set is optimal.

The problem of finding absolutely continuous invariant measures (ACIMs) for a dynamical system can be formulated as a fixed point problem for a Markov operator (the Perron–Frobenius operator). This is an infinite-dimensional problem. Ulam's method replaces the Perron–Frobenius operator by a sequence of finite rank approximations whose fixed points are relatively easy to compute numerically. This paper concerns the optimal choice of Ulam approximations for one-dimensional maps; an adaptive partition selection is used to tailor the approximations to the structure of the invariant measure. The main idea is to select a partition which equally distributes the square root of the derivative of the invariant density amongst the bins of the partition. The results are illustrated for the logistic map where the ACIMs may have inverse square root singularities in their density functions. O(log n/n) convergence rates can be expected, whereas a non-adaptive algorithm yields O(n^−1/2) at best. Studying the convergence of the adaptive algorithm allows an estimate to be made of the measure of the Jakobson parameter set (those logistic maps which admit an ACIM).

A greedy algorithm for scheduling and digital printing with inputs in a polytope lying in an affine space, and vertices of this polytope as successive outputs, has recently been proven to be bounded for any polytope in any dimension in the case when the norm on errors is the Euclidean norm. This boundedness property follows readily from the existence of some invariant sets (both in the affine space and in the associated vector space), for the dynamics associated to the algorithm. We prove several general properties of such invariant sets under the assumption that the greed of the algorithm is driven by the Euclidean norm.

We prove here, by geometric, or rather dynamical, methods, the following theorem. Let G be a non-compact connected Lie subgroup of the isometry group of the real hyperbolic space , which does not fix any point at infinity, i.e. on . Then G preserves a certain hyperbolic subspace and 'contains' all the identity components of its isometry group. We provide an 'algebra-free' proof and present the dynamical tools used, so that the exposition is 'self-contained'.

We study the dynamics of an overcompensatory Leslie population model where the fertility rates decay exponentially with population size. We find a plethora of complicated dynamical behaviour, some of which has not been previously observed in population models and which may give rise to new paradigms in population biology and demography.

We study the two- and three-dimensional models and find a large variety of complicated behaviour: all codimension 1 local bifurcations, period doubling cascades, attracting closed curves that bifurcate into strange attractors, multiple coexisting strange attractors with large basins (which cause an intrinsic lack of 'ergodicity'), crises that can cause a discontinuous large population swing, merging of attractors, phase locking and transient chaos. We find (and explain) two different bifurcation cascades transforming an attracting invariant closed curve into a strange attractor. We also find one-parameter families that exhibit most of these phenomena. We show that some of the more exotic phenomena arise from homoclinic tangencies.

We present some numerical evidence for universality associated with the breakup of shearless invariant tori, by studying a renormalization group transformation acting on an appropriate space of Hamiltonians.

We construct a real-analytic circle map for which the corresponding Perron–Frobenius operator has a real-analytic eigenfunction with an eigenvalue outside the essential spectral radius when acting upon C¹-functions.

We consider a system of two nonhyperbolic conservation laws modelling incompressible two-phase flow in one space dimension. The purpose of this paper is to justify the use of singular shocks in the solution of Riemann problems. We prove that both strictly and weakly overcompressive singular shocks are limits of viscous structures. Using Riemann solutions we solve Cauchy problems with piecewise constant data for the nonhyperbolic two-fluid model.

In this paper, we study a three-dimensional thermocline planetary geostrophic 'horizontal' hyper-diffusion model of the gyre-scale midlatitude ocean. We show the global existence and uniqueness of the weak and strong solutions to this model. Moreover, we establish the existence of a finite-dimensional global attractor to this dissipative evolution system. Preliminary computational tests indicate that our hyper-diffusion model does not exhibit any of the non-physical instabilities near the lateral boundary which are observed numerically in other models.

We show that near a simple focus–focus value in a Liouville integrable Hamiltonian system with two degrees of freedom lines of locally constant rotation number in the image of the energy–momentum map are spirals determined by the eigenvalue of the equilibrium. From this representation of the rotation number we derive that the twist condition for the isoenergetic KAM condition vanishes on a curve in the image of the energy–momentum map that is transversal to the line of constant energy. In contrast to this, we also show that the frequency map is non-degenerate for every point in a neighbourhood of a simple focus–focus point.

Piecewise rotations are natural generalizations of interval exchange maps. They appear naturally in the theory of digital filters, Hamiltonian systems and polygonal dual billiards. We construct a rational piecewise rotation system with three atoms for which the return time to one of the atoms is unbounded. We show that the return map gives rise to a self-similar structure of induced atoms. The constructions are based on the angle of rotation π/7. Moreover, we construct a continuous class of examples with an infinite number of periodic cells. These periodic cells alternate between two atoms and they form a self-similar structure. Our investigation here may be viewed as generalizations of results obtained by Boshernitzan and Caroll, as well as Adler, Kitchens and Tresser, Kahng, Lowenstein and others. The main tools in the investigation are algebraic computations in a cyclotomic field determined by fourteenth roots of unity.

We study diffusion phenomena in a priori unstable (initially hyperbolic) Hamiltonian systems. These systems are perturbations of integrable ones, which have a family of hyperbolic tori. We prove that in the case of two and a half degrees of freedom the action variable generically drifts (i.e. changes on a trajectory by a quantity of order one). Moreover, there exists a trajectory such that the velocity of this drift is ε/logε, where ε is the parameter of the perturbation.

We consider the Cauchy problem for the Kadomstev–Petviashvili II (KPII) equation with generic data that do not decay along a line. The linearization is in terms of spectral properties of the heat operator with a decaying 'time independent' potential. A bounded Green's function of this operator is constructed and its main properties are determined. The solution of the KPII equation is obtained via linear integral equations.

A traditional subject in statistical physics is the linear response of a molecular dynamical system to changes in an external forcing agency, e.g. the Ohmic response of an electrical conductor to an applied electric field. For molecular systems the linear response matrices, such as the electrical conductivity, can be represented by Green–Kubo formulae as improper time-integrals of 2-time correlation functions in the system. Recently, Ruelle has extended the Green–Kubo formalism to describe the statistical, steady-state response of a 'sufficiently chaotic' nonlinear dynamical system to changes in its parameters. This formalism potentially has a number of important applications. For instance, in studies of global warming one wants to calculate the response of climate-mean temperature to a change in the atmospheric concentration of greenhouse gases. In general, a climate sensitivity is defined as the linear response of a long-time average to changes in external forces. We show that Ruelle's linear response formula can be computed by an ensemble adjoint technique and that this algorithm is equivalent to a more standard ensemble adjoint method proposed by Lea, Allen and Haine to calculate climate sensitivities.

In a numerical implementation for the 3-variable, chaotic Lorenz model it is shown that the two methods perform very similarly. However, because of a power-law tail in the histogram of adjoint gradients their sum over ensemble members becomes a Lévy flight, and the central limit theorem breaks down. The law of large numbers still holds and the ensemble-average converges to the desired sensitivity, but only very slowly, as the number of samples is increased. We discuss the implications of this example more generally for ensemble adjoint techniques and for the important practical issue of calculating climate sensitivities.

A new family of exact solutions is analysed, which models two-dimensional circulations of an ideal fluid in a uniformly rotating elliptical tank, under the semi-geostrophic approximation from meteorology and oceanography. The fluid pressure and stream function remain quadratic functions of space at each instant in time, and their fluctuations are described by a single degree of freedom Hamiltonian system depending on two conserved parameters: domain eccentricity and the constant value of potential vorticity. These parameters determine the presence or absence of periodic orbits with arbitrarily long periods, fixed points of the dynamics, and aperiodic homoclinic orbits linking hyperbolic saddle points. The energy relative to these parameters selects the frequency and direction in which isobars nutate or precess, as well as the steady circulation direction of the fluctuating flow. The canonically conjugate variables are the moment of inertia and angle of inclination of an elliptical inverse-potential-vorticity patch evolving in dual coordinates.

Discrete breathers are time-periodic, spatially localized solutions of the equations of motion for a system of classical degrees of freedom interacting on a lattice. We study the existence of energy thresholds for discrete breathers, i.e. the question of whether, in a certain system, discrete breathers of arbitrarily low energy exist, or whether a threshold has to be overcome in order to excite a discrete breather. Breather energies are found to have a positive lower bound if the lattice dimension d is greater than or equal to a certain critical value d_c, whereas no energy threshold is observed for d < d_c. The critical dimension d_c is system dependent and can be computed explicitly, taking on values between zero and infinity. Three classes of Hamiltonian systems are distinguished, being characterized by different mechanisms affecting the existence (or non-existence) of an energy threshold.

A non-linear two-dimensional system is studied by making use of both the Lagrangian and the Hamiltonian formalisms. This model is obtained as a two-dimensional version of a one-dimensional oscillator previously studied at the classical and also at the quantum level. First, it is proved that it is a super-integrable system, and then the non-linear equations are solved and the solutions are explicitly obtained. All the bounded motions are quasiperiodic oscillations and the unbounded (scattering) motions are represented by hyperbolic functions. In the second part the system is generalized to the case of n degrees of freedom. Finally, the relation of this non-linear system to the harmonic oscillator on spaces of constant curvature, the two-dimensional sphere S² and hyperbolic plane H², is discussed.

The aim of this paper is to introduce a technique for describing trajectories of systems of ordinary differential equations (ODEs) passing near saddle-fixed points. In contrast to classical linearization techniques, the methods of this paper allow for perturbations of the underlying vector fields. This robustness is vital when modelling systems containing small uncertainties, and in the development of numerical ODE solvers producing rigorous error bounds.

This corrigendum corrects the erroneous conclusion made in section 3.2 (using the same notation), see the PDF for full details.