Table of contents

We describe a renormalization group transformation that is related to the break-up of golden invariant tori in Hamiltonian systems with two degrees of freedom. This transformation applies to a large class of Hamiltonians, is conceptually simple, and allows for accurate numerical computations. In a numerical implementation, we find a non-trivial fixed point and determine the corresponding critical index and scaling. Our computed values for various universal constants are in good agreement with existing data for area-preserving maps. We also discuss the flow associated with the non-trivial fixed point.

We show that there exists a rational change of coordinates of Painlevé's P1 equation and of the elliptic equation after which these two equations become analytically equivalent in a region in the complex phase space where y and are unbounded. The region of equivalence comprises all singularities of solutions of P1 (i.e. outside the region of equivalence, solutions are analytic). The Painlevé property of P1 (that the only movable singularities are poles) follows as a corollary. Conversely, we argue that the Painlevé property is crucial in reducing P1, in a singular regime, to an equation integrable by quadratures.

The global constraints on chaotic dynamics induced by the analyticity of smooth flows are used to dispense with individual periodic orbits and derive infinite families of exact sum rules for several simple dynamical systems. The associated Fredholm determinants are of particularly simple polynomial form. The theory developed suggests an alternative to the conventional periodic orbit theory approach for determining eigenspectra of transfer operators.

AMS classification scheme numbers: 58F20

We construct a series of n-unimodal approximations to maps of the Hénon type and utilize the associated symbolic dynamics to describe the possible bifurcation structures for such maps. We construct the bifurcation surfaces of the short periodic orbits in the topological parameter space and check numerically that the Hénon map parameter plane (a,b) is topologically equivalent to a two-dimensional section through the infinite-dimensional parameter space characterizing a generic map of the Hénon type.

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.

In this paper we study the time-dependent Ginzburg-Landau equation of the Schrödinger type in two dimensions. The initial conditions are chosen to describe several well-separated vortices. Our task is to understand the vortex structure of the corresponding solutions as well as corrections due to radiation. To this end we develop the nonlinear adiabatic theory. Using the methods of effective action and of geometric solvability we derive equations for the vortex dynamics and radiation. As an example we consider the special case of radiation by two 1-vortices.

In this paper we continue our study of the long-time behaviour of solutions of the non-stationary Ginzburg Landau equation of the Schrödinger type. Here we consider initial conditions corresponding to either two vortices of charges +1 each or two vortices of opposite charges +1 and -1. We show that in the first case the vortices radiate while rotating around each other. As a result of this radiation they move apart at the rate of asymptotically. In the second case we show that the radiation is absent for large vortex separations which conforms with previous results. For separations of orders O (1) and less we argue that the vortex pair produces a shock wave. As a result it loses energy and eventually collapses.

The model, with and without a generalized Hopf term, is studied using the collective coordinate approximation. In the spirit of this approximation, an ansatz is given for the head-on collision of two solitons. The equations of motion for the collective coordinates are then solved analytically, for solitons close together and for solitons far apart. The solutions show how the generalized Hopf term changes the scattering angle, which in its absence, is .

We study the propagation of round-off error near the periodic orbits of a linear area-preserving map - a planar rotation by a rational angle - which is discretized on a lattice in such a way as to retain invertibility. We consider the round-off error probability distribution as a function of time t, and we show that for each t this is an algebraic number, which can be calculated exactly. We prove that its kth moment increases asymptotically as , where is the fractional dimension of a self-similar set related to periodic orbits of long-period, while G is a bounded function, periodic in the logarithm of t. This implies the diffusion coefficient displays bounded variations, while all higher order transport coefficients diverge, resulting in anomalous transport. This result contrasts with the case of irrational rotations, where the existence of a central limit theorem has been recently established (Vladimirov I 1996 Preprint Deakin University).

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-dimendional 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.

In this short note we uncover the geometry behind the classical result on averaging high-frequency vibrations . It is shown that the classical effective potential of Kapitsa is produced by a centrifugal force of a point mass constrained to a certain curve determined by V.

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.

Assuming that the properties of nonequilibrium stationary states of systems of particles can be expressed in terms of weighted orbital measures, we derive the Onsager reciprocal relations for systems of N particles subject to a Gaussian thermostat and prove that the relevant transport coefficients are positive.

PACS Numbers: 0545, 0560, 0570L

We study the time evolution of the interfaces in one-dimensional bistable extended dynamical systems with discrete time. The dynamics are governed by the competition between a local piecewise affine bistable mapping and any couplings given by the convolution with a function of bounded variation. We prove the existence of travelling wave interfaces, namely fronts, and the uniqueness of the corresponding selected velocity and shape. This selected velocity is shown to be the propagating velocity for any interface, to depend continuously on the couplings and to increase with the symmetry parameter of the local nonlinearity. We apply the results to several examples including discrete and continuous couplings, and the dynamics of planar fronts in multidimensional coupled map lattices. We use our technique to study the existence of other kinds of fronts. Finally we consider a more general class of bistable extended mappings for which the couplings are allowed to be nonlinear and the local map to be smooth.

Motivated by recent analytical and numerical work on two- and three-dimensional convection with imposed spatial periodicity, we analyse three examples of bifurcations from a continuous group orbit of spatio-temporally symmetric periodic solutions of partial differential equations. Our approach is based on centre manifold reduction for maps, and is in the spirit of earlier work by Iooss (1986) on bifurcations of group orbits of spatially symmetric equilibria. Two examples, two-dimensional PWs and three-dimensional alternating pulsating waves (APW), have discrete spatio-temporal symmetries characterized by the cyclic groups , n = 2 (PW) and n = 4 (APW). These symmetries force the Poincaré return map to be the nth iterate of a map : . The group orbits of PW and APW are generated by translations in the horizontal directions and correspond to a circle and a two-torus, respectively. An instability of PWs can lead to solutions that drift along the group orbit, while bifurcations with Floquet multiplier (FM) +1 of APWs do not lead to drifting solutions. The third example we consider, alternating rolls, has the spatio-temporal symmetry of APWs as well as being invariant under reflections in two vertical planes. This leads to the possibility of a doubling of the marginal FM and of bifurcation to two distinct types of drifting solutions. We conclude by proposing a systematic way of analysing steady-state bifurcations of periodic orbits with discrete spatio-temporal symmetries, based on applying the equivariant branching lemma to the irreducible representations of the spatio-temporal symmetry group of the periodic orbit, and on the normal form results of Lamb J S W (1996 Local bifurcations in k-symmetric dynamical systems Nonlinearity 9 537-57). This general approach is relevant to other pattern formation problems, and contributes to our understanding of the transition from ordered to disordered behaviour in pattern-forming systems.