Table of contents

Let F : ²→² be a homeomorphism. An open F-invariant subset U of ² is a pruning region for F if it is possible to deform F continuously to a homeomorphism F_U for which every point of U is wandering, but which has the same dynamics as F outside of U. This concept is motivated by the Pruning Front Conjecture (PFC) introduced by Cvitanovic, which claims that every Hénon map can be understood as a pruned horseshoe.

This paper contains recent results in pruning theory, concentrating on prunings of the horseshoe. We describe conditions on a disk D which ensure that the orbit of its interior is a pruning region; explain how prunings of the horseshoe can be understood in terms of underlying tree maps; discuss the connection between pruning and Thurston's classification theorem for surface homeomorphisms; motivate a conjecture describing the forcing relation on horseshoe braid types; and use this theory to give a precise statement of the PFC.

This paper studies the destabilizing effects of dissipation on families of relative equilibria in Hamiltonian systems which are non-extremal constraint critical points in the energy-Casimir or the energy-momentum methods. The dissipation is allowed to destroy the conservation law associated with the symmetry group or Casimirs, as long as the family of relative equilibria stays on an invariant manifold. This approach complements earlier work in the literature, in which the dissipation did not affect the conservation law.

Firstly, Chetaev's instability theorem is extended to invariant manifolds and this extended theorem is used to prove the instability of families of relative equilibria for several examples. Secondly, it is shown that families of non-extremal stationary solutions of the two-dimensional incompressible homogeneous Euler equations are unstable for the corresponding viscous perturbations of this system, i.e. for the two-dimensional Navier-Stokes equations. Also, the instability of the sleeping top relative equilibria under friction can easily be proved in this way, even before the Hamiltonian sleeping top becomes linearly unstable. Finally, sufficient conditions are given for which friction destabilizes families of non-extremal relative equilibria in simple mechanical systems with Abelian symmetry.

We study a system of semilinear hyperbolic equations passively advected by smooth white noise in time random velocity fields. Such a system arises in modelling non-premixed isothermal turbulent flames under single-step kinetics of fuel and oxidizer. We derive closed equations for one-point and multi-point probability distribution functions (PDFs) and closed-form analytical formulae for the one-point PDF function, as well as the two-point PDF function under homogeneity and isotropy. Exact solution formulae allow us to analyse the ensemble-averaged fuel/oxidizer concentrations and the motion of their level curves. We recover the empirical formulae of combustion in the thin reaction zone limit and show that these approximate formulae can either underestimate or overestimate average concentrations when the reaction zone is not tending to zero. We show that the averaged reaction rate slows down locally in space due to random advection-induced diffusion, and that the level curves of ensemble-averaged concentration undergo diffusion about mean locations.

In this paper we investigate the large-time asymptotic of linearized very fast diffusion equations with and without potential confinements. These equations do not satisfy, in general, logarithmic Sobolev inequalities, but, as we show by using the `Bakry-Emery reverse approach', in the confined case they have a positive spectral gap at the eigenvalue zero. We present estimates for this spectral gap and draw conclusions on the time decay of the solution, which we show to be exponential for the problem with confinement and algebraic for the pure diffusive case. These results hold for arbitrary algebraically large diffusion speeds, if the solutions have the mass-conservation property.

In this paper, we discuss a stochastic analogue of Aubry-Mather theory in which a deterministic control problem is replaced by a controlled diffusion. We prove the existence of a minimizing measure (Mather measure) and discuss its main properties using viscosity solutions of Hamilton-Jacobi equations. Then we prove regularity estimates on viscosity solutions of the Hamilton-Jacobi equation using the Mather measure. Finally, we apply these results to prove asymptotic estimates on the trajectories of controlled diffusions and study the convergence of Mather measures as the rate of diffusion vanishes.

We investigate the properties of hysteresis cycles produced by a one-dimensional, periodically forced Langevin equation. We show that depending on amplitude and frequency of the forcing and on noise intensity, there are three qualitatively different types of hysteresis cycles. Below a critical noise intensity, the random area enclosed by hysteresis cycles is concentrated near the deterministic area, which is different for small and large driving amplitude. Above this threshold, the area of typical hysteresis cycles depends, to leading order, only on the noise intensity. In all three regimes, we derive mathematically rigorous estimates for expectation, variance and the probability of deviations of the hysteresis area from its typical value.

In this paper we use the mixture of topological and measure-theoretic dynamical approaches to consider riddling of invariant sets for some discontinuous maps of compact regions of the plane that preserve two-dimensional Lebesgue measure. We consider maps that are piecewise continuous and with invertible except on a closed zero measure set. We show that riddling is an invariant property that can be used to characterize invariant sets, and prove results that give a non-trivial decomposion of what we call partially riddled invariant sets into smaller invariant sets. For a particular example, a piecewise isometry that arises in signal processing (the overflow oscillation map), we present evidence that the closure of the set of trajectories that accumulate on the discontinuity is fully riddled. This supports a conjecture that there are typically an infinite number of periodic orbits for this system.

We use a renormalization operator acting on a space of vector fields on ^d, d⩾2, to prove the existence of a submanifold of vector fields equivalent to constant. The result comes from the existence of a fixed point ω of which is hyperbolic. This is done for a certain class KT_d of frequency vectors ω∊^d, called Koch type. The transformation is constructed using a time rescaling, a linear change of basis plus a periodic non-linear map isotopic to the identity, which we derive by a `homotopy trick'.

We construct a rigorous renormalization scheme for analytic vector fields on ² of Poincaré type. We show that, iterating this procedure, there is convergence to a limit set with a `Gauss map' dynamics on it, related to the continued fraction expansion of the slope of the frequencies. This is valid for diophantine frequency vectors.

We develop a semiclassical approximation for the spectral Wigner and Husimi functions in the neighbourhood of a classically unstable periodic orbit of chaotic 2-D maps. The prediction of hyperbolic fringes for the Wigner function, asymptotic to the stable and unstable manifolds, is verified computationally for a (linear) cat map, after the theory is adapted to a discrete phase space appropriate to a quantized torus. The characteristic fringe patterns can be distinguished even for quasi-energies where the fixed point is not Bohr-quantized. The corresponding Husimi function dampens these fringes with a Gaussian envelope centred on the periodic point. Even though the hyperbolic structure is then barely perceptible, more periodic points stand out due to the weakened interference.

We analyse the infinitesimal effect of holomorphic perturbations of the dynamics of a structurally stable rational map on a neighbourhood of its Julia set. This implies some restrictions on the behaviour of critical points.

We prove a general existence theorem for the `viscous profile problem' for singular shocks associated with the weak solutions of some pairs of conservation laws. The structure obtained approximately satisfies the regularized system (identity viscosity matrix) in the space of measures. Several features of the work of Keyfitz and Kranzer for a model problem of this type are recovered.

We show that in order to estimate the Hausdorff dimension of conformal infinite iterated function systems (IFSs), it is completely sufficient to consider finite subsystems with N<∞ generators. We give estimates on the accuracy of this approximation and present a simple straightforward algorithm which allows us to reduce this setting further to finite iteration time n. Our method allows us to calculate the Hausdorff dimension of an IFS with the same speed of convergence as the algorithm proposed by McMullen (McMullen C 1998 Am. J. Math.120 691-721) (i.e. the distance of the approximation to the actual value is ~1/n, where n is the iteration depth, hence corresponding to ~Nⁿ calculation steps). This is much slower than Jenkinson/Pollicott's method for bounded continued fractions (which gives superexponential accuracy ~exp (-n^3/2) for the same depth) but does not require extensive knowledge about the periodic points of the involved transformations and is applicable in a more general setting. The method is applied in order to perform numerical calculations for certain classes of examples.

We consider a pair of coupled Painlevé equations arising as a scaling similarity reduction of the Hirota-Satsuma system of partial differential equations. Bäcklund transformations constructed in a previous work are presented explicitly as discrete shifts in a two-dimensional parameter space. τ-functions derived from a Hamiltonian description are also presented, which satisfy multilinear lattice equations built from Hirota bilinear operators, and these are used to calculate polynomial τ-functions for rational solutions.

We study the Cauchy problem for the (2+1)-dimensional relativistic Abelian Chern-Simons-Higgs model. We prove the global existence and uniqueness of the solutions for the finite energy data.

For a class of generalized Korteweg-de Vries equations of the form

u_t + (u^p)_x-D^βu_x = 0            (*)

posed in and for the focusing nonlinear Schrödinger equations

iu_t + Δu + |u|^pu = 0            (**)

posed on ⁿ, it is well known that the initial-value problem is globally in time well posed provided the exponent p is less than a critical power p_crit. For p⩾p_crit, it is known for equation (**) and suspected for equation (*) (known for p = 5 and β = 2) that large initial data need not lead to globally defined solutions. It is our purpose here to investigate the critical case p = p_crit in more detail than heretofore. Building on previous work of Weinstein, Laedke, Spatschek and their collaborators, earlier work of the present authors and others, a stability result is formulated for small perturbations of ground-state solutions of (**) and solitary-wave solutions of (*). This theorem features a scaling that is natural in the critical case. When interpreted in the contexts in view, our general result provides information about singularity formation in the case the solution blows up in finite time and about large-time asymptotics in the case the solution is globally defined.

The analogy between the expansion of a real number as a continued fraction and the summation of a formal power series by means of Padé approximants is studied. The elementary dynamics of the map analogous to the well-known Gauss map of the remainder for the expression of a real number as a continued fraction is thereby developed.

A new integrable nonlinear equation is derived from a previously known integrable equation by means of an asymptotically exact nonlinear reduction method based on Fourier expansion and spatio-temporal rescaling. The new equation is likely to be of applicative relevance due to the particular features of the reduction method. The integrability by the weak Lax pair formulation and the inverse scattering method is explicitly demonstrated, by applying the reduction technique to the Lax pair of the starting equation, and the corresponding Lax pair of the new equation is found.

We show that if f is a C¹ (continuously differentiable) map from the unit interval I into itself, of type 2^∞ in the Sarkovskii order, then there are C¹ maps g,h:I→I, of respective types greater than and less than 2^∞ as close to f in the C¹-topology as required. This proves a conjecture by Block and Hart (1982 Ergodic Theory Dynam. Systems2 125-9) (restated in MacKay and Tresser (1988 J. Lond. Math. Soc. 37 164-81) and Hu and Tresser (1998 Fund. Math.155 237-49)).

Let M be a two-dimensional closed Riemannian manifold and denote by Diff¹(M) the set of C¹ diffeomorphisms on M. Then, the C¹ interior of {f∊Diff¹(M):h(f) = 0} is equal to the C¹ interior of the closure of the Morse-Smale systems and equal to the C¹ interior of the set of diffeomorphisms having no horseshoe.

By a dynamical system we mean a pair (X,T), where X is a compact metric space and T:X→X is surjective and continuous. We study weak disjointness in topological dynamics. (X,T) is scattering iff it is weakly disjoint from all minimal systems and (X,T) is strongly scattering iff it is weakly disjoint from all E-systems, i.e. transitive systems having invariant measures with full support. It is clear that a weakly mixing system is strongly scattering and the latter is scattering. An existential proof of scattering and a non-weakly mixing example is obtained by Akin and Glasner (2001 J. Anal. Math.84 243-86). In this paper, we will give an explicit example which is strongly scattering and not weakly mixing. We also define extreme scattering, weak scattering and study the relationships of the various definitions.

For a dynamical property P stronger than transitivity, let P^⋏ be the property such that a system has P^⋏ iff it is weakly disjoint from any system having P. We show that P^⋏ = P^⋏⋏⋏. Moreover, we prove that (thickly syndetic-transitive)^⋏ = piecewise-syndetic-transitive and (piecewise-syndetic-transitive)^⋏ = thickly syndetic-transitive.

In this paper, we give an upper bound on the number of zeros of Abelian integrals for the quadratic centres having almost all their orbits formed by quartics, under polynomial perturbations of arbitrary degree n. The bound is linearly dependent on n.

There is a vast body of literature devoted to the study of bifurcation phenomena in autonomous systems of differential equations. However, there is currently no well-developed theory that treats similar questions for the non-autonomous case. Inspired in part by the theory of pullback attractors, we discuss generalizations of various autonomous concepts of stability, instability, and invariance. Then, by means of relatively simple examples, we illustrate how the idea of a bifurcation as a change in the structure and stability of invariant sets remains a fruitful concept in the non-autonomous case.

The quantum mechanical propagators of the linear automorphisms of the two-torus (cat maps) determine a projective unitary representation of the theta group Γ_θ⊂SL(2,). We prove that there exists an appropriate choice of phases in the propagators that defines a proper representation of Γ_θ. We also give explicit formulae for the propagators in this representation.

In this paper, we explore some properties of a Markov finite element approximation on a shape-regular triangulation over a polygonal region Ω⊂R^N. In order to find a fixed density of a Markov operator P:L¹(Ω)→L¹(Ω) efficiently, we propose and analyse a finite element scheme for approximating P, which preserves the Markov structure of the operator.

It has recently been emphasized that all known exact evaluations of gap probabilities for classical unitary matrix ensembles are in fact τ-functions for certain Painlevé systems. We show that all exact evaluations of gap probabilities for classical orthogonal matrix ensembles, either known or derivable from the existing literature, are likewise τ-functions for certain Painlevé systems. In the case of symplectic matrix ensembles, all exact evaluations, either known or derivable from the existing literature, are identified as the mean of two τ-functions, both of which correspond to Hamiltonians satisfying the same differential equation, differing only in the boundary condition. Furthermore the product of these two τ-functions gives the gap probability in the corresponding unitary symmetry case, while one of these τ-functions is the gap probability in the corresponding orthogonal symmetry case.

Unfortunately, due to an unforeseen software conversion error, all occurrences of the word `translate' were rendered incorrectly in printed issues. This affected the following articles:

R S Ellis, K Haven and B Turkington 2002 Nonequivalent statistical equilibrium ensembles and refined stability theorems for most probable flows Nonlinearity15 239-255

A Iatrou and J A G Roberts 2002 Integrable mappings of the plane preserving biquadratic invariant curves II Nonlinearity15 459-489

O Runborg, C Theodoropoulos and I G Kevrekidis 2002 Effective bifurcation analysis: a time-stepper-based approach Nonlinearity15 491-511

We apologise for this error. The Electronic Journals version is unaffected.