Table of contents

The authors show that the way in which finite differences are applied to the nonlinear term in certain partial differential equations (PDES) can mean the difference between dissipation and blow up. For fixed parameter values and arbitrarily fine discretizations they construct solutions which blow up in finite time for two semi-discrete schemes. They also show the existence of spurious steady states whose unstable manifolds, in some cases, contain solutions which explode. This connection between the blow-up phenomenon and spurious steady states is also explored for Galerkin and nonlinear Galerkin semi-discrete approximations. Two fully discrete finite difference schemes derived from a third semi-discrete scheme, reported to be dissipative, are analysed. Both latter schemes are shown to have a stability condition which is independent of the initial data.

The universal description of orbits in the domain swept by a slowly varying separatrix is provided through a symplectic map derived by means of an extension of classical adiabatic theory. This map connects action-angle-like variables of an orbit when far from the instantaneous separatrix to time-energy variables at a reference point of the orbit very close to the corresponding separatrix. When the separatrix pulsates periodically with a small frequency epsilon , the authors combine this map with WKB theory to obtain a description of the structure underlying chaos: the homoclinic tangle related to the hyperbolic fixed point whose separatrix is pulsating. For each extremum of the area within the pulsating separatrix, an initial branch of length O(1/ epsilon ) of the stable manifold is explicitly constructed, and makes O(1/ epsilon ) transverse homoclinic intersections with a similar branch of the unstable manifold.

The authors consider an energy-dependent version of the third-order scalar Lax operator, thus extending previous results. Unlike the second-order case it is no longer possible to expand the potentials as arbitrary polynomials in lambda . They prove that there are exactly four cases. They present Hamiltonian operators, Hamiltonian Mura maps and modifications for two new examples.

The author proves the equivalence of degenerate Hopf bifurcations which have all their closed orbits at the bifurcation point. Although these Hopf bifurcations have infinity codimension, they can nevertheless occur generically in dynamical systems under constraint such as in the Hamiltonian systems or in the replicator equations; and so in these contexts a treatment of their equivalence is required. The analysis is rather delicate. The Poincare return maps of the flows give rise to a one-parameter family of one-dimensional maps and the authors start by determining the conjugacy classes of such families: There are surprisingly only two classes depending upon the finiteness or infiniteness of an integral modulus. The conjugacy class of the return maps is then used to show the equivalence of the Hopf bifurcations.

The authors study a set of three complex ordinary differential equations describing modal interactions in a system equivariant under the group O(2) of planar rotations and reflections and appropriate to the interaction among Fourier modes with spatial wavenumbers in the ratio 1:2:4. Such systems are known to possess structurally stable heteroclinic cycles and they focus on the bifurcations occurring near a degenerate point at which these cycles simultaneously change their stability type and become unstable to travelling waves. They find a subtle interaction between local and global dynamics and they show that multiple branches of modulated travelling waves emerge. Their methods include centre manifolds and normal forms coupled with estimates on the global return of solutions obtained with the aid of numerical simulations.

The author considers a generalization of the one-sided shift, suitable for describing a certain class of maps in the interval that preserve a Cantor set. The author shows that if such a map is single-valued, it has a finite Markov partition (i.e. that its symbolic dynamics is regular), but if it is multiple-valued, its symbolic dynamics can be an arbitrary context-free language. The scaling properties of sets corresponding to such languages are discussed, and an example is given where the semigroup of scaling operations is infinite-dimensional but finitely describable. The author also discusses the problems of embedding a computationally complex process in one dimension.

A model is considered representing an elastically jointed pair of articulated pipes conveying fluid. The motion is described by a four-component system of autonomous ordinary differential equations. Numerical techniques are used to investigate changes in the dynamics as two parameters are varied. These parameters represent the fluid flow-rate and a form of symmetry-breaking. Evidence is found that the global bifurcation picture is surprisingly complicated, involving chaos and two types of homoclinic behaviour: namely, Sil'nikov homoclinic orbits to a saddle-focus stationary point, and homoclinic tangencies to periodic orbits. Local theory respective to each type of homoclinicity is reviewed and compared with the numerical results.

The authors generalize notions of transport in phase space associated with the classical Poincare map reduction of a periodically forced two-dimensional system to apply to a sequence of nonautonomous maps derived from a quasiperiodically forced two-dimensional system. They obtain a global picture of the dynamics in homoclinic and heteroclinic tangles using a sequence of time-dependent two-dimensional lobe structures derived from the invariant global stable and unstable manifolds of one or more normally hyperbolic invariant sets in a Poincare section of an associated autonomous system phase space. The invariant manifold geometry is studied via a generalized Melnikov function. Transport in phase space is specified in terms of two-dimensional lobes mapping from one to another within the sequence of lobe structures, which provides the framework for studying several features of the dynamics associated with chaotic tangles.

The authors consider the variational problem describing the static deformation of a thin elastic superconducting shell in a magnetic field; the shell is supposed to be clamped along the edge. This problem is essentially nonlinear because the functional in the problem depends on the unknown deformed shell middle surface. For sufficiently weak fields and under some additional simplifications they prove that the solution of this problem exists and is unique.

The author gives an example of an invariant set that is not asymptotically stable but which has the following strong attracting properties. 'Almost all' trajectories that start close to the invariant set behave as if the set were asymptotically stable, that is, these trajectories remain close and converge to the invariant set. The term 'almost all' means that the only trajectories that escape lie in a cuspoidal region abutting the invariant set. The example is a heteroclinic cycle forced by symmetry. The surprising feature is that nodes on the cycle may have unstable eigenvalues in directions 'normal' to the cycle, and yet the cycle is stable in the above sense. This type of stability appears to explain some numerical experiments.

The author considers Abelian integrals associated with generic polynomials of a given degree n+1 and with polynomial 1-forms of degree <or=n. He gives an explicit bound C(n) for the multiplicity of zeros of the Abelian integrals considered. A consequence is that C(n) is a bound for the cyclicity of regular cycles of a generic polynomial Hamiltonian vector field of degree n deformed within a nonconservative polynomial vector field family of degree n. He also gives explicit bounds C₀(n) and C_l(n) for the cyclicity of centres and homoclinic loops of generic Hamiltonian vector fields within the considered family.

There are two simple discrete versions of the nonlinear Schrodinger equation: (i) the discrete self-trapping system which preserves the standard norm, and (ii) the Ablowitz-Ladik system which is integrable using the inverse scattering method. In this paper the quantum theories for the two systems are sketched and compared.

Recently models describing the dynamics of large arrays of Josephson junctions coupled through a variety of loads have been studied. Since, in applications, these systems are to be operated in a state of stable synchronous oscillation, these studies have emphasized how the synchronous periodic state can lose stability. A common feature of the models equations is that they are invariant under permutation of the individual junctions. In the authors' study they focus on the effects that these symmetries have on the resulting bifurcations when the synchronous solution loses stability. In these systems the causes for loss of stability are: fixed-point bifurcations and period-doubling bifurcations. Moreover, these two bifurcations can coalesce in a new codimension-two bifurcation which they call a homoclinic twist bifurcation. Due to the S_N symmetry, it can be shown that the fixed-point bifurcations must lead to families of unstable periodic orbits. The period-doubling bifurcations, however, can lead to stable period-doubled oscillations, and the possible states and their stabilities are classified. In particular, generically, all of the period-doubled oscillations are described by dividing the junctions into two or three groups within which the junctions oscillate synchronously. The existence of these states in the model equations have been confirmed by numerical simulation. In addition to these period-doubled states, the existence of the homoclinic twist bifurcation and periodic solutions where the junctions oscillate with the same waveform but (1/N)th of a period out of phase with each other is observed in the numerical simulation. These last types of solution are called ponies on a merry-go-round (POMS). In these equations POMS do not arise from a local bifurcation. This issue is discussed in the companion paper.

Numerical simulation of periodic solutions in large arrays of Josephson junctions indicates the existence of periodic solutions where each junction oscillates with the same waveform, but with equal phase lags. These solutions are called ponies on a merry-go-round or POMs for short. The authors prove the existence of POMs in the equations modelling large arrays of Josephson junctions by using global bifurcation techniques. The basic idea is to view the period of the solution and the phase lag as independent parameters and to prove, using a priori estimates, that the synchronous solution (with phase lag set to zero) can be continued to a solution with phase lag equal to (1/N)th of the period, a POM.

The author introduces the concept of a map with memory associated to a given map f and to a normalized weight sequence P. The stabilization properties of maps with memory are established. It is shown that fixed points and periodic orbits, in the neighbourhood of which a map f has an oscillating behaviour, can be 'coherentized' by the use of a memory. Maps with exponential decaying fading memory are characterized as displaying a phase transition near the critical value beta _c of the relaxation constant beta with universal exponent nu =1.

Using the uniform Lyapunov exponents for a compact hyperbolic set, the author gives an upper bound for the Hausdorff dimension of the compact hyperbolic set, provided the diffeomorphism satisfies pinching condition. This improves previous results.

The author considers one-parameter families f(t, z) of orientation-preserving diffeomorphisms of the circle which satisfy some natural assumptions. A prototype mapping of this type is the sine map y=x+ Omega +(A/2 pi )sin(2 pi x) (mod 1). His goal is to give a mathematical proof of the universality of harmonic scaling in the case of families of diffeomorphisms. As a corollary he obtains that the rotation number depends Holder continuously on the parameter value with an exponent alpha >or=1/2. He also discusses the asymptotic behaviour of the considered scaling. As a conclusion he obtains that the typical Holder exponent is equal to 1/2.

The authors present a new technique for constructing a computer-assisted proof of the reliability of a long computer-generated trajectory of a dynamical system. Auxiliary calculations made along the noise-corrupted computer trajectory determine whether there exists a true trajectory which follows the computed trajectory closely for long times. A major application is to verify trajectories of chaotic differential equations and discrete systems. They apply the main results to computer simulations of the Henon map and the forced damped pendulum.

A critical function K( nu ) of a two degrees of freedom Hamiltonian system represents a fractal boundary between regular and chaotic motion as a function of frequency. The method of modular smoothing uses the transformation properties of K( nu ) for unit translation and inversion of nu to provide a rapid method of computing K( nu ). The authors demonstrate two unusual properties of the method: it is simpler for continuous time systems than for maps, and for at least one system exact results can be obtained.