Table of contents

We review an experimental study of the evolution of coherent structures (vortices) in two-dimensional decaying turbulence. The vortices are formed in a thin layer of density-stratified electrolyte, using an electromagnetic forcing. The number of vortices decreases algebraically.

We introduce the concept of porosity for measures and study relations between dimensions and porosities for two classes of measures: measures on ⁿwhich satisfy the doubling condition and strongly porous measures on .

The theory for resonant three-wave interactions in anisotropic and inhomogeneous MHD plasmas is extended. Using a Hamiltonian formalism, we derive symmetric coupling coefficients, relevant for a class of MHD models with anisotropic pressures. For the case with an isotropic pressure term, we present general results valid for arbitrary static background states.

The existence of multisite stable periodic orbits in infinite networks of weakly coupled bistable oscillators is proved through a continuation process from the uncoupled limit. Using a Lyapunov-Schmidt reduction, it is shown that the stability of periodic orbits is determined by a finite-dimensional eigenvalue problem. As an application, the existence of N -site stable periodic orbits in an infinite chain of coupled bistable oscillators is proved for any finite N . Moreover, the stability of spatially disordered periodic orbits is proved in infinite networks of coupled bistable oscillators with exponentially decaying couplings.

We consider a model, isothermal, autocatalytic chemical reaction scheme based on the n ( 1)th-order autocatalytic step, ABat rate kabⁿ , where aand bare the concentrations of reactant Aand autocatalyst B , respectively, and kis the rate constant. We examine the evolution which occurs when a quantity of the autocatalyst Bis introduced locally into an expanse of the reactant A , which is initially at uniform concentration. In addition, the molecular diffusivity of B , say D_B , is very much smaller than that of reactant A , say D_A , so that D_B /D_A <<1. We concentrate on the case when D_B /D_A 0. Under these conditions, single-point blowup occurs in the concentration of the autocatalyst Bas t , when 1n 2, and in finite t , when n >2. We develop both small- and large-time asymptotic solutions for the cases 1<n <2 and n >2, with the cases of n= 1 and 2 having been studied in detail by Billingham J and Needham D J 1991 Phil. Trans. R. Soc.A 336497-539. Finally, we discuss the situation when D_B /D_Ais small but finite, i.e. 0<D_B /D_A <<1, in the context of the studies of Billingham and Needham 1991 and Herrero M A, Lacey A A and Velázquez J J L 1998 Arch. Ration. Mech. Anal.142219-51.

We consider the existence and stability of the hole, or dark soliton, solution to a Ginzburg-Landau perturbation of the defocusing nonlinear Schrödinger equation (NLS), and to the nearly real complex Ginzburg-Landau equation (CGL). By using dynamical systems techniques, it is shown that the dark soliton can persist as either a regular perturbation or a singular perturbation of that which exists for the NLS. When considering the stability of the soliton, a major difficulty which must be overcome is that eigenvalues may bifurcate out of the continuous spectrum, i.e. an edge bifurcationmay occur. Since the continuous spectrum for the NLS covers the imaginary axis, and since for the CGL it touches the origin, such a bifurcation may lead to an unstable wave. An additional important consideration is that an edge bifurcation can happen even if there are no eigenvalues embedded in the continuous spectrum. Building on and refining ideas first presented by Kapitula and Sandstede (1998 PhysicaD 12458-103) and Kapitula (1999 SIAM J. Math. Anal.30273-97), we use the Evans function to show that when the wave persists as a regular perturbation, at most three eigenvalues will bifurcate out of the continuous spectrum. Furthermore, we precisely track these bifurcating eigenvalues, and thus are able to give conditions for which the perturbed wave will be stable. For the NLS the results are an improvement and refinement of previous work, while the results for the CGL are new. The techniques presented are very general and are therefore applicable to a much larger class of problems than those considered here.

In nonlinear dynamics an important distinction exists between uniform bounds on growth rates, as in the definition of hyperbolic sets, and non-uniform bounds as in the theory of Liapunov exponents. In rare cases, for instance in uniquely ergodic systems, it is possible to derive uniform estimates from non-uniform hypotheses. This allowed one of us to show in a previous paper that a strange non-chaotic attractor for a quasiperiodically forced system could not be the graph of a continuous function. This had been a conjecture for some time. In this paper we generalize the uniform convergence of time averages for uniquely ergodic systems to a broader range of systems. In particular, we show how conditions on growth rates with respect to all the invariant measures of a system can be used to derive one-sided uniform convergence in both the Birkhoff and the sub-additive ergodic theorems. We apply the latter to show that any strange compact invariant set for a quasiperiodically forced system must support an invariant measure with a non-negative maximal normal Liapunov exponent; in other words, it must contain some `non-attracting' orbits. This was already known for the few examples of strange non-chaotic attractors that have rigorously been proved to exist. Finally, we generalize our semi-uniform ergodic theorems to arbitrary skew product systems and discuss the application of such extensions to the existence of attracting invariant graphs.

A class of planar dynamical systems is considered which models a wide variety of physical problems; this class is of a form to which Melnikov's method may be applied. It is shown that a family of excitations exists which is optimal for inducing a homoclinic tangle, and hence horseshoe dynamics, in such dynamical systems (in the sense that the optimal excitations are functions of smallest L^p (0,T ) norm, 1p , which will produce a simple zero in the Melnikov function). These optimal functions are particularly effective at inducing complex dynamics in the nonlinear system of interest; they are of both theoretical and practical interest. An illustration is given of this approach for two well known problems in nonlinear dynamics: the pendulum, and Duffing's equation. Numerical results are presented which confirm the accuracy and practical significance of this approach.

We discuss the existence of a rational map whose Julia set is homeomorphic to a given topological self-similar set. We prove that a self-similar set Kis homeomorphic to the Julia set of a postcritically finite rational map if and only if Kis embedded in S²such that the dynamics of Kcan be extended to a postcritically finite branched covering on S² .

In this paper, sufficiently smooth Hamiltonian systems with perturbations are considered. By combining a smooth version of the Kolmogorov-Arnold-Moser theorem and the theory of normally hyperbolic invariant manifolds, we show that under the conditions of nonresonance and nondegeneracy, most hyperbolic invariant tori and their stable and unstable manifolds survive smoothly under sufficiently smooth autonomous perturbation. This result can be generalized directly to the case of time-dependent quasi-periodic perturbations. Finally, an example from geometrical optics is used to illustrate our method.

We show that in the neighbourhood of the tripling bifurcation of a periodic orbit of a Hamiltonian flow or of a fixed point of an area-preserving map, there is generically a bifurcation that creates a `twistless' torus. At this bifurcation, the twist, which is the derivative of the rotation number with respect to the action, vanishes. The twistless torus moves outward after it is created and eventually collides with the saddle-centre bifurcation that creates the period-three orbits. The existence of the twistless bifurcation is responsible for the breakdown of the non-degeneracy condition required in the proof of the KAM theorem for flows or the Moser twist theorem for maps. When the twistless torus has a rational rotation number, there are typically reconnection bifurcations of periodic orbits with that rotation number.

The slow passage through a Hopf bifurcation leads to the delayed appearance of large-amplitude oscillations. We construct a smooth scalar feedback control which suppresses the delay and causes the system to follow a stable equilibrium branch. This feature can be used to detect in time the loss of stability of an ageing device. As a by-product, we obtain results on the slow passage through a bifurcation with double-zero eigenvalue, described by a singularly perturbed cubic Liénard equation.

We study the local equation of energy for weak solutions of three-dimensional incompressible Navier-Stokes and Euler equations. We define a dissipation term D (u ) which stems from an eventual lack of smoothness in the solution u . We give in passing a simple proof of Onsager's conjecture on energy conservation for the three-dimensional Euler equation, slightly weakening the assumption of Constantin et al . We suggest calling weak solutions with non-negative D (u ) `dissipative'.

We consider the problem of scheduling a sequence of actions when the benefit obtained from an action depends on only the current time modulo a period and the time since the previous action. A simple example is a model of a shuttle bus service where the profit from running a bus depends on the time of day and the time since the previous bus. Mathematically, such problems can be formulated as maximizing V (t_{n -1} ,t_n ) over schedules (t_n )_{n 0}as t_Ngoes to infinity, for functions V (t ,t ´) satisfying the periodicity condition V (t +T ,t ´+T ) = V (t ,t ´), for some T >0. The problem is related to Aubry-Mather theory in the dynamics of area-preserving maps. We extend this theory in order to prove the existence of optimizing schedules for each initial condition t₀ , to characterize the properties of such schedules and to analyse their dependence on t₀and on the parameters of the model.

In this paper we investigate the structure of the set of step sizes with which symplectic algorithms can simulate invariant tori of a given non-resonant frequency of general integrable Hamiltonian systems. It turns out that in the general nonlinear systems case, the set has a Cantor structure, very similar to, but more complex than, the Cantor structure of the set of Diophantine frequencies in the KAM theory. The Cantor set is of density one at the origin of the real line. Some remarks about the Cantor set and about the invariant tori are given.

We explore some connections between round-off errors in linear planar rotations and algebraic number theory. We discretize a map on a lattice in such a way as to retain invertibility, restricting the system parameter (the trace) to rational values with power-prime denominator pⁿ . We show that this system can be embedded into a smooth expansive dynamical system over the p -adic integers, consisting of multiplication by a unit composed with a Bernoulli shift. In this representation, the original round-off system corresponds to restriction to a dense subset of the p -adic integers. These constructs are based on symbolic dynamics and on the representation of the discrete phase space as a ring of integers in a quadratic number field.