An exact two loop soliton solution to the Vakhnenko equation is found. The key step in finding this solution is to transform the independent variables in the equation. This leads to a transformed equation for which it is straightforward to find an exact explicit 2-soliton solution by use of Hirota's method. The exact two loop soliton solution to the Vakhnenko equation is then found in implicit form by means of a transformation back to the original independent variables. The nature of the interaction between the two loop solitons depends on the ratio of their amplitudes.

The vanishing viscosity limit is considered for the incompressible 2D Navier-Stokes equations in a bounded domain. Motivated by studies of turbulent flow we suppose Navier's friction condition in the tangential direction, i.e. the creation of a vorticity proportional to the tangential velocity. We prove the existence of the regular solutions for the Navier-Stokes equations with smooth compatible data and of the solutions with bounded vorticity for initial vorticity being only bounded. Finally, we establish a uniform L-infinity-bound for the vorticity and convergence to the incompressible 2D Euler equations in the inviscid limit.

This paper provides an analytical insight into the observed nonlinear behaviour of a simple widely used power electronic circuit (the buck converter) and draws parallels with a wider class of piecewise-smooth systems. After introducing the buck converter model and background, the most fascinating features of its dynamical behaviour are reviewed. So-called grazing and sliding solutions are discussed and their role in determining many of the buck converter's dynamical oddities is demonstrated. In particular, a local map is studied which explains how grazing bifurcations cause sharp turning points in the bifurcation diagram of periodic orbits. Moreover, these orbits are shown to accumulate onto a sliding trajectory through a 'spiralling' impact adding scenario. The structure of such a diagram is derived analytically and is shown to be closely related to the analysis of homoclinic bifurcations, The results are shown to match perfectly with numerical simulations. The sudden jump to large-scale chaos and the fingered structure of the resulting attractor are also explained.

The energy-equation approach used to prove the existence of the global attractor by establishing the so-called asymptotic compactness property of the semigroup is considered, and a general formulation that can handle a number of weakly damped hyperbolic equations and parabolic equations on either bounded or unbounded spatial domains is presented. As examples, three specific and physically relevant problems are considered, namely the flows of a second-grade fluid, the flows of a Newtonian fluid in an infinite channel past an obstacle, and a weakly damped, forced Korteweg-de Vries equation on the whole line.

A representative model of a return map near homoclinic bifurcation is studied. This model is the so-called fattened Arnold map, a diffeomorphism of the annulus. The dynamics is extremely rich, involving periodicity, quasiperiodicity and chaos. The method of study is a mixture of analytic perturbation theory, numerical continuation, iteration to an attractor and experiments, in which the guesses are inspired by the theory. In rum the results lead to fine-tuning of the theory. This approach is a natural paradigm for the study of complicated dynamical systems. By following generic bifurcations, both local and homoclinic, various routes to chaos and strange attractors are detected. Here, particularly, the 'large' strange attractors which wind around the annulus are of interest. Furthermore, a global phenomenon regarding Arnold tongues is important. This concerns the accumulation of tongues on lines of homoclinic bifurcation. This phenomenon sheds some new light on the occurrence of infinitely many sinks in certain cases, as predicted by the theory.

We analyse the structure of minimal-energy solutions of the baby Skyrme model for any topological charge n; the baby multi-skyrmions. Unlike in the (3+1)D nuclear Skyrme model, a potential term must be present in the (2+1)D baby Skyrme model to ensure stability. The form of this potential term has a crucial effect on the existence and structure of baby multi-skyrmions. The simplest holomorphic baby Skyrme model has no known stable minimal-energy solution for n greater than one. The other baby Skyrme model studied in the literature possesses non-radially-symmetric minimal-energy configurations that look like 'skyrmion lattices' formed by skyrmions with n = 2. We discuss a baby Skyrme model with a potential that has two vacua. Surprisingly, the minimal-energy solution for every n is radially symmetric and the energy grows linearly for large n. Further, these multi-skyrmions are lighter bound, have less energy and the same large r behaviour than in the model with one vaccum. We rely on numerical studies and approximations to test and verify this observation.

We study the dynamics of Lyapunov vectors in various models of one-dimensional distributed systems with spacetime chaos. We demonstrate that the vector corresponding to the maximum exponent is always localized and the localization region wanders irregularly. This localization is explained by interpreting the logarithm of the Lyapunov vector as a roughening interface. We show that for many systems, the 'interface' belongs to the Kardar-Parisi-Zhang universality class. Accordingly, we discuss the scaling behaviour of finite-size effects and self-averaging properties of the Lyapunov exponents.

An explicit upper bound Z(3, n) less than or equal to 5n + 15 is derived for the number of the zeros of the integral h > I (h) = integral(H=h) g(x, y) dx - f(x, y) dy of degree n polynomials f, g, on the open interval C for which the cubic curve {H = h} contains an oval. The proof exploits the properties of the Picard-Fuchs system satisfied by the four basic integrals integral integral(H

We consider the behaviour of attractors near invariant subspaces on varying a parameter that changes the dynamics in the invariant subspace of a dynamical system. We refer to such a parameter as 'non-normal'. In the presence of chaos that is fragile, we find blowout bifurcations that are blurred over a range of parameter values. We demonstrate that this can occur on a set of positive measure in the parameter space. Under an assumption that the dynamics is not of skew product form, these blowout bifurcations can create attractors displaying 'in-out intermittency', a generalized form of on-off intermittency. We characterize in-out intermittency both in terms of its structure in phase space and statistically by means of a Markov model. We discuss some other dynamical and bifurcation effects associated with non-normal parameters, in particular non-normal bifurcation to riddled basins and transition between on-off and in-out intermittency.

The breathing circle is a two-dimensional generalization of the Fermi accelerator. It is shown that the billiard map associated to this model has invariant curves in phase space, implying that any particle will have bounded gain of energy.

A spatially discrete version of the general kink-bearing nonlinear Klein-Gordon model in (1 + 1) dimensions is constructed which preserves the topological lower bound on kink energy. It is proved that, provided the lattice spacing h is sufficiently small, there exist static kink solutions attaining this lower bound centred anywhere relative to the spatial lattice. Hence there is no Peierls-Nabarro (PN) barrier impeding the propagation of kinks in this discrete system. An upper bound on h is derived and given a physical interpretation in terms of the radiation of the system. The construction, which works most naturally when the nonlinear Klein-Gordon model has a squared polynomial interaction potential, is applied to a recently proposed continuum model of polymer twistons. Numerical simulations are presented which demonstrate that kink pinning is eliminated, and radiative kink deceleration is greatly reduced in comparison with the conventional discrete system. So even on a very coarse lattice, kinks behave much as they do in the continuum. It is argued, therefore, that the construction provides a natural means of numerically simulating kink dynamics in nonlinear Klein-Gordon models of this type. The construction is compared with the inverse method of Flach, Zolotaryuk and Kladko. Using the latter method, alternative spatial discretizations of the twiston and sine-Gordon models are obtained which are also free of the PN barrier.

It is proved that periodic and homoclinic trajectories which are tangent to the boundary of any scattering (ergodic) billiard produce elliptic islands in the 'nearby' Hamiltonian Bows i.e. in a family of two-degrees-of-freedom smooth Hamiltonian Bows which converge to the singular billiard flow smoothly where the billiard flow is smooth and continuously where it is continuous. Such Hamiltonians exist; indeed, sufficient conditions are supplied, and thus it is proved that a large class of smooth Hamiltonians converges to billiard flows in this manner. These results imply that ergodicity may be lost in the physical setting, where smooth Hamiltonians which are arbitrarily close to the ergodic billiards, arise.

We consider piecewise twice differentiable maps T on [0, 1] with indifferent fixed points giving rise to infinite invariant measures. Without assuming the existence of a Markov partition and only requiring that the first image of the fundamental partition is finite, we prove that the interval decomposes into a finite number of ergodic cycles with exact powers plus a dissipative part. T is shown to be exact on components containing indifferent fixed points. We also determine the order of the singularities of the invariant densities.

We derive the global bifurcation diagram of a three-parameter family of cubic Liénard systems. This family seems to have a universal character in that its bifurcation diagram (or parts of it) appears in many models from applications for which a combination of hysteretic and self-oscillatory behaviour is essential. The family emerges as a partial unfolding of a doubly degenerate Bogdanov-Takens point, that is. of the codimension-four singularity with nilpotent linear part and no quadratic terms in the normal form. We give a new presentation of a local four-parameter bifurcation diagram which is a candidate for the universal unfolding of this singularity.

A reduced periodic orbit is one which is periodic module a rigid motion. If such an orbit for the planar N-body problem is collision free then it represents a conjugacy class in the projective coloured braid group. Under a 'strong force' assumption which excludes the original 1/r Newtonian potential we prove that in most conjugacy classes there is a collision-free reduced periodic solution to Newton's N-body equations. These are the classes that are 'tied' in the sense of Gordon. We give explicit homological conditions which ensure that a class is tied. The method of proof is the direct method of the calculus of variations. For the three-body problem we obtain qualitative information regarding the shape of our solutions which leads to a partial symbolic dynamics.

We study classical solutions of the vector O(3) sigma model in (2 + 1) dimensions, spontaneously broken to O(2) x Z(2). The model possesses Skyrmion-type solutions as well as stable domain walls which connect different vacua. We show that different types of waves can propagate on the wall, including waves carrying a topological charge. The domain wall can also absorb Skyrmions and, under appropriate initial conditions, it is possible to emit a Skyrmion from the wall.

We develop a general, coordinate-free theory for the reduction of volume-preserving Bows with a volume-preserving symmetry on three-manifolds. The reduced flow is generated by a one-degree-of-freedom Hamiltonian which is the generalization of the Bernoulli invariant from hydrodynamics. The reduction procedure also provides global coordinates for the study of symmetry-breaking perturbations. Our theory gives a unified geometric treatment of the integrability of three-dimensional, steady Euler flows and two-dimensional, unsteady Euler flows, as well as quasigeostrophic and magnetohydrodynamic flows.

For piecewise expanding one-dimensional maps without periodic turning points we prove that isolated eigenvalues of small (random) perturbations of these maps are close to isolated eigenvalues of the unperturbed system. (Here 'eigenvalue' means eigenvalue of the corresponding Perron-Frobenius operator acting on the space of functions of bounded variation.) This result applies e.g. to the approximation of the system by a finite state Markov chain and generalizes Ulam's conjecture about the approximation of the Sinai-Bowen-Ruelle invariant measure of such a map. We provide several simple examples showing that for maps with periodic turning points and for general multidimensional smooth hyperbolic maps isolated eigenvalues are typically unstable under random perturbations. Our main tool in the one-dimensional case is a special technique for 'interchanging' the map and the perturbation, developed in our previous paper (Blank M L and Keller G 1997 Stochastic stability versus localization in chaotic dynamical systems Nonlinearity 10 81-107), combined with a compactness argument.

Complete proofs are given for some claims of Middleton and of Floria and Maze about the asymptotic behaviour of chains of balls and springs in a tilted periodic potential and generalizations, under gradient dynamics.