Table of contents

We consider dynamical systems that are equivariant under a noncompact Lie group of symmetries and the drift of relative equilibria in such systems. In particular, we investigate how the drift for a parametrized family of normally hyperbolic relative equilibria can change character at what we call a `drift bifurcation'. To do this, we use results of Arnold to analyse parametrized families of elements in the Lie algebra of the symmetry group.

We examine effects in physical space of such drift bifurcations for planar reaction-diffusion systems and note that these effects can explain certain aspects of the transition from rigidly rotating spirals to rigidly propagating `retracting waves'. This is a bifurcation observed in numerical simulations of excitable media where the rotation rate of a family of spirals slows down and gives way to a semi-infinite translating wavefront.

An inequality is derived which must be satisfied for a relative equilibrium of the n-body problem to be spectrally stable. This inequality is studied in the equal mass case and using simple geometric estimates, it is shown that any relative equilibrium of n equal masses is not spectrally stable for n 24 306.

We consider travelling wave solutions of a class of differential-difference equations. Our interest is in understanding propagation failure, the directional dependence due to the discrete Laplacian, and the relationship between travelling wave solutions of the spatially continuous and spatially discrete limits of this equation. The differential-difference equations that we study include damped and undamped nonlinear wave and reaction-diffusion equations as well as their spatially discrete counterparts. Both analytical and numerical results are given.

We develop a Melnikov-type global perturbation technique for detecting the existence of transverse homoclinic orbits and occurrence of homoclinic bifurcations in periodic perturbations of multi-degree-of-freedom Hamiltonian systems. The unperturbed system is assumed to have a saddle-centre whose stable and unstable manifolds do not coincide but intersect in a lower-dimensional manifold, and does not have to be completely integrable. Other Melnikov-type methods do not apply in this situation. We also apply our technique to a four-degree-of-freedom model of nonplanar vibrations of a forced, buckled beam.

We consider small Hamiltonian perturbations of a system of infinitely many harmonic oscillators. We assume that the frequencies _j, j 1, fulfil _j ~ j^d with d 1, and a suitable non-resonance condition of Diophantine-type. We prove a Nekhoroshev-type theorem for solutions corresponding to initial data such that the energy on the jth linear oscillator is bounded by Kexp(-j^a) with a given a > 1 and positive , K. We show, precisely that up to times of order exp|ln|^1+b, where is the size of the perturbation and b a positive parameter, the solution remains close to a quasi-periodic motion. Closedness is measured in a weighted ² norm with an exponential weight. For our Diophantine-type condition we show that if the frequencies depend on a real parameter, and a suitable non-degeneracy condition is fulfilled, it is satisfied for almost all values of the parameter. Finally, we apply the general result to some nonlinear Klein-Gordon equations.

The conservation laws for linear and angular momenta in a Schrödinger-Chern-Simons field theory modelling vortex dynamics in planar superconductors are studied. In analogy with fluid vortices it is possible to express the linear and angular momenta as low moments of vorticity. The conservation laws are shown to be consistent with those obtained in the moduli space approximation for vortex dynamics, valid close to the Bogomol'nyi limit. For Bogomol'nyi vortices, the relevant moments of vorticity can be evaluated fairly explicitly, as can the integral of log||², where is the scalar field. Conservation of angular momentum prevents a single vortex from escaping to infinity.

We study in a systematic way all static solutions of the Goldstone model in 1 + 1 dimensions with a periodicity condition imposed on the spatial coordinate. The solutions are presented in terms of standard trigonometric functions and of Jacobi elliptic functions. Their stability analysis is carried out, and the complete list of classically stable quasi-topological solitons is given.

AMS classification scheme number: 12B90

We obtain bounds for the generalized q-dimensions of measures supported by self-affine sets that are valid for `almost all' families of affine mappings. Exact values are established for self-affine measures and for Gibbs measures when 1<q2. These q-dimensions may exhibit phase transitions as q varies.

In this paper we prove that a diffeomorphism with a homoclinic orbit associated to a generic Hopf bifurcation point can be perturbed to obtain a homoclinic tangency. Let {F_t} be a generic one-parameter family of diffeomorphisms that unfolds a Hopf bifurcation point which has associated a transverse homoclinic point. For this kind of family we prove also that F_t can be perturbed to obtain a homoclinic tangency for that t such that the rotation number of F_t restricted to the invariant circle, produced by the Hopf bifurcation, is irrational.

We consider simultaneous binary collisions in the general planar four-body problem. We prove they are regularizable in the sense of continuity with respect to initial conditions using a blow-up of the singularity. Furthermore, numerical evidence is given that the differentiability of the regularization should be, in general, less than C^8/3 . As a simple example, the double isosceles four-body problem, displays that kind of behaviour.

The equation y´´ = f(x)y² arises in the study of a class of fluid models in relativity and possesses the Painlevéproperty (closely connected with integrability) if and only if f satisfies a certain sixth-order ODE which admits GL(2,) as its symmetry group. Using differential invariants of this non-solvable group, the general solution is obtained. A special case of the sixth-order equation is equivalent to the generalized Chazy equation with parameter n = (6/7). All known explicit choices for f considered in the literature arise in a natural way in this framework. Generalizations of the techniques described here lead to a novel class of integrable equations.

The trace formula for the evolution operator associated with nonlinear stochastic flows with weak additive noise is cast in the path integral formalism. We integrate over the neighbourhood of a given saddlepoint exactly by means of a smooth conjugacy, a locally analytic nonlinear change of field variables. The perturbative corrections are transferred to the corresponding Jacobian, which we expand in terms of the conjugating function, rather than the action used in defining the path integral. The new perturbative expansion which follows by a recursive evaluation of derivatives appears more compact than the standard Feynman diagram perturbation theory. The result is a stochastic analogue of the Gutzwiller trace formula with the `' corrections computed an order higher than what has so far been attainable in stochastic and quantum mechanical applications.

The truncation method is a collective name for techniques that arise from truncating a Laurent series expansion (with leading term) of generic solutions of nonlinear partial differential equations (PDEs). Despite its utility in finding Bäcklund transformations and other remarkable properties of integrable PDEs, it has not been generally extended to ordinary differential equations (ODEs). Here we give a new general method that provides such an extension and show how to apply it to the classical nonlinear ODEs called the Painlevé equations. Our main new idea is to consider mappings that preserve the locations of a natural subset of the movable poles admitted by the equation. In this way we are able to recover all known fundamental Bäcklund transformations for the equations considered. We are also able to derive Bäcklund transformations onto other ODEs in the Painlevéclassification.

We prove that all the invariant measures for the monotone gradient dynamics of the (non-tilted) Frenkel-Kontorova model are supported on the set of stationary configurations, and that the -limit set of each configuration is either a single stationary configuration or contains an invariant circle of stationary configurations. A consequence is a new proof of the existence of Mather's shadowing orbits of twist diffeomorphisms. We compare the results with the dynamics of other dissipative spatially extended systems, such as the real Ginzburg-Landau equation, and suggest application to non-conservative diffusion on 1D lattices.

By explicit construction of the ADHM data, we prove the existence of a charge-seven instanton with icosahedral symmetry. By computing the holonomy of this instanton we obtain a Skyrme field which approximates the minimal energy charge-seven skyrmion. We also present a one-parameter family of tetrahedrally symmetric instantons whose holonomy gives a family of Skyrme fields which models a skyrmion scattering process, where seven well-separated skyrmions collide to form the icosahedrally symmetric skyrmion.

We show that transcendental meromorphic functions of a certain type have infinitely many invariant Baker domains, with dynamics at similar to those in Leau domains at a parabolic fixed point. We deduce, for example, that, for each p , there exists an entire function which has infinitely many p-cycles of Baker domains.

We consider an Abel equation (*)y´ = p(x)y²+q(x)y³ with p(x), q(x) - polynomials in x. A centre condition for this equation (closely related to the classical centre condition for polynomial vector fields on the plane) is that y₀ = y(0)y(1) for any solution y(x). This condition is given by the vanishing of all the Taylor coefficients v_k(1) in the development . Following Briskin et al (Centre Conditions, Composition of Polynomials and Moments on Algebraic Curves to appear) we introduce periods of the equation (*) as those , for which y(0)y() for any solution y(x) of (*). The generalized centre conditions are conditions on p, q under which given a₁, ... ,a_k are (exactly all) the periods of (*).

A new basis for the ideals I_k = {v₂, ... ,v_k} has been produced in Briskin et al (1998 The Bautin ideal of the Abel equation Nonlinearity10), defined by a linear recurrence relation. Using this basis and a special representation of polynomials, we extend results of Briskin et al (Centre Conditions, Composition of Polynomials and Moments on Algebraic Curves to appear), proving for small degrees of p and q a composition conjecture, as stated in Alwash and Lloyd (1987 Non-autonomous equations related to polynomial two-dimensional systems Proc. R. Soc. Edinburgh A 105 129-52), Briskin et al (Centre Conditions, Composition of Polynomials and Moments on Algebraic Curves to appear), Briskin et al (Center Conditions II: Parametric and Model Centre Problems to appear). In particular, this provides transparent generalized centre conditions in the cases considered. We also compute maximal possible multiplicity of the zero solution of (*), extending the results of Alwash and Lloyd (1987 Non-autonomous equations related to polynomial two-dimensional systems Proc. R. Soc. Edinburgh A 105 129-52).

We consider the approximation of absolutely continuous invariant measures (ACIMs) of systems defined by random compositions of piecewise monotonic transformations. Convergence of Ulam's finite approximation scheme in the case of a single transformation was dealt with by Li (1976 J. Approx. Theory17 177-86). We extend Ulam's construction to the situation where a family of piecewise monotonic transformations are composed according to either an iid or Markov law, and prove an analogous convergence result. In addition, we obtain a convergence rate for our approximations to the unique ACIM, and provide rigorous bounds for the L¹ error of the Ulam approximation.

We consider the canonical symplectic form evaluated explicitly on the solitons of a general integrable model defined in terms of the inverse scattering procedure. This paper establishes a direct link between the symplectic form for the solitons and the underlying loop groups which occur in the inverse scattering procedure. The integral over space in the form which arises because the canonical method uses the Lagrangian density, is performed explicitly in terms of functions arising in the group doublecross product formulation of the inverse scattering procedure, and we are left with a simple expression given by two boundary terms. The expression is then evaluated explicitly in terms of the changes in the positions and momenta of the solitons, after we restrict our attention to the sine-Gordon case for simplicity. In this case, we find agreement with a result of Babelon and Bernard who have evaluated the form using a different argument, where it is diagonal in terms of `in' or `out' coordinates.

Using the result, we also investigate the higher conserved charges within the inverse scattering framework, obtaining expressions for them, and check that they Poisson commute.

We study a parabolic-elliptic system of partial differential equations that arises in modelling the overdamped gravitational interaction of a cloud of particles or chemotaxis in bacteria. The system has a rich dynamics and the possible behaviours of the solutions include convergence to time-independent solutions and the formation of finite-time singularities. Our goal is to describe the different kinds of solutions that lead to these outcomes. We restrict our attention to radial solutions and find that the behaviour of the system depends strongly on the space dimension d. For 2<d<10 there are two stable blowup modalities (self-similar and Burgers-like) and one stable steady state. On unbounded domains, there exists a one-parameter family of unstable steady solutions and a countable number of unstable blowup behaviours. We document connections between one unstable blowup behaviour and both a stable steady state and a stable blowup, as well as connections between one unstable blowup and two different stable blowups. There is a topological and stability correspondence between the various asymptotic behaviours and this suggests the possibility of constructing a global phase portrait for the system that treats the global in time solutions and the blowing up solutions on an equal footing.

We consider the second-order equation where f and g are polynomials with deg f,deg gn. Our interest is in the maximum number of isolated periodic solutions which can bifurcate from the steady state solution x = 0. Alternatively, this is equivalent to seeking the maximum number of limit cycles which can bifurcate from the origin for the Liénard system, Assuming the origin is not a centre, we show that if either f or g are quadratic, then this number is . If f or g are cubic we show that this number is , for all 1n50. The results also hold for generalized Liénard systems.

We investigate the `semiclassical Fredholm determinant' for strongly chaotic billiards derived from the semiclassical limit of the Fredholm determinant of the boundary element method. We show that it is the same as a cycle-expanded Gutzwiller - Voros zeta function when the bounce number of the periodic orbit with the billiard boundary corresponds to the length of the symbolic sequence of its symbolic dynamical expression. A numerical experiment on a `concave triangle billiard' shows that the series defining the semiclassical Fredholm determinant does not converge absolutely in spite of the absolute convergence of the series defining the Fredholm determinant. However, the series behaves like an asymptotic series, and the finite sum obtained by optimal truncation of the series defining the semiclassical Fredholm determinant gives the semiclassical eigenenergies precisely enough such that the error of the semiclassical approximation is much smaller than the mean spacing of the exact eigenenergies.

It is well established that a formal language generated from a unimodal map is regular if and only if the map's kneading sequence is either periodic or eventually periodic. A previously proposed conjecture said that if a language generated from a unimodal map is context-free, then it must be regular, i.e. there exists no proper context-free language which can be generated from a unimodal map. This paper is a step forward in answering this conjecture showing that under two situations the conjecture is true. The main results of this paper are: (1) if the kneading map of a unimodal map is unbounded, then the map's language is not context-free, (2) all nonregular languages generated from the Fibonacci substitutions are context-sensitive, but not context-free. These results strongly suggest that the conjecture may be indeed true.

A codimension-three unfolding for the ₂-symmetric Hopf-pitchfork bifurcation, in the presence of an additional nonlinear degeneracy, is analysed.

Up to ten distinct topological equivalence classes for the unfolding are found. A rich variety of dynamical and bifurcation behaviours is pointed out. Beyond the bifurcations present in the nondegenerate case, we show that the following bifurcations appear locally: Takens - Bogdanov of periodic orbits, degenerate pitchfork of periodic orbits, and global connections involving equilibria and/or periodic orbits.

The local results achieved, extended by means of numerical continuation methods, are used to understand the dynamics of a modified van der Pol - Duffing electronic oscillator, for a certain range of the parameters.

Given a dynamical system and a function f from the state space to the real numbers, an optimal orbit for f is an orbit over which the time average of f is maximal. In this paper we consider some basic mathematical properties of optimal orbits: existence, sensitivity to perturbations of f, and approximability by periodic orbits with low period. For hyperbolic systems, we conjecture that for (topologically) generic smooth functions, there exists an optimal periodic orbit. In support of this conjecture, we prove that optimal periodic orbits are insensitive to small C¹ perturbations of f, while the optimality of a non-periodic orbit can be destroyed by arbitrarily small C¹ perturbations. In case there is no optimal periodic orbit for a given f, we discuss the question of how fast the maximum average over orbits of period at most p must converge to the optimal average, as p increases.

In this paper we consider the class of stochastic stationary sources induced by one-dimensional Gibbs states, with Hölder continuous potentials. We show that the time elapsed before the source repeats its first n symbols, when suitably renormalized, converges in law either to a log-normal distribution or to a finite mixture of exponential random variables. In the first case we also prove a large deviation result.