James Montaldi

Periodic solutions near equilibria of symmetric Hamiltonian systems

We consider the effects of symmetry on the dynamics of a nonlinear hamiltonian system invariant under the action of a compact Lie group T, in the vicinity of an isolated equilibrium: in particular, the local existence and stability of periodic trajectories. The main existence result, an equivariant version of the Weinstein—Moser theorem, asserts the existence of periodic trajectories with certain prescribed symmetries Z c T x S1, independently of the precise nonlinearities. We then describe the constraints put on the Floquet operators of these periodic trajectories by the action of T. This description has three ingredients: an analysis of the linear symplectic maps that commute with a symplectic representation, a study of the momentum mapping and its relation to Floquet multipliers, and Krein Theory. We find that for some 2, which we call cylospetral, all eigenvalues of the Floquet operator are forced by the group action to lie on the unit circle; that is, the periodic trajectory is spectrally stable. Similar results for equilibria are described briefly. The results are applied to a number of simple examples such as T = SO(2), 0(2 ), Zn, Dn, SU (2) ; and also to the irreducible symplectic actions of O(3) on spaces of complex spherical harmonics, modelling oscillations of a liquid drop.

Relative equilibria of point vortices on the sphere

We describe the linear and nonlinear stability and instability of certain configurations of point vortices on the sphere forming relative equilibria. These configurations consist of up to two rings, with and without polar vortices. Such configurations have dihedral symmetry, and the symmetry is used both to block diagonalize the relevant matrices and to distinguish the subspaces on which their eigenvalues need to be calculated.

Existence of nonlinear normal modes of symmetric Hamiltonian systems

The authors analyse the existence of nonlinear normal modes of a (nonlinear) Hamiltonian system, i.e. periodic solutions that approximate periodic solutions of the system linearised around an elliptic (and semisimple) equilibrium point. In particular they consider systems with symmetry, including time-reversible symmetry which involves an antisymplectic operator. The general form for such a system contains free parameters (Taylor series coefficients), and their aim is to calculate how the number of nonlinear normal modes varies with these parameters.

Persistence and stability of relative equilibria

We consider relative equilibria in symmetric Hamiltonian systems, and their persistence or bifurcation as the momentum is varied. In particular, we extend a classical result about persistence of relative equilibria from values of the momentum map that are regular for the coadjoint action, to arbitrary values, provided that either (i) the relative equilibrium is at a local extremum of the reduced Hamiltonian or (ii) the action on the phase space is (locally) free. The first case uses just point-set topology, while in the second we rely on the local normal form for (free) symplectic group actions, and then apply the splitting lemma. We also consider the Lyapunov stability of extremal relative equilibria. The group of symmetries is assumed to be compact.

Geometric Mechanics and Symmetry:The Peyresq Lectures

This consists of lecture notes from 6 courses held at 2 summer schools in Peyresq, France in 2000 and 2001. The notes were written up by the lecturers together with some participants.

Vortex Dynamics on a Cylinder

Point vortices on a cylinder (periodic strip) are studied geometrically. The Hamiltonian formalism is developed, a nonexistence theorem for relative equilibria is proved, equilibria are classified when all vorticities have the same sign, and several results on relative periodic orbits are established, including as corollaries classical results on vortex streets and leapfrogging.

Golden gaskets: variations on the Sierpinski sieve

We consider the iterated function systems (IFSs) that consist of three general similitudes in the plane with centres at three non-collinear points, with a common contraction factor λ in (0, 1). As is well known, for λ = 1/2 the attractor, S_λ, is a fractal called the Sierpinski sieve and for λ < 1/2 it is also a fractal. Our goal is to study S_λ for this IFS for 1/2 < λ < 2/3 , i.e. when there are ‘overlaps’ in S_λ as well as ‘holes’. In this introductory paper we show that despite the overlaps (i.e. the breaking down of the open set condition (OSC)), the attractor can still be a totally self-similar fractal, although this happens only for a very special family of algebraic λ (so-called multinacci numbers). We evaluate the ausdorff dimension of S_λ for these special values by showing that S_λ is essentially the attractor for an infinite IFS that does satisfy the OSC. We also show that the set of points in the attractor with a unique ‘address’ is self-similar and compute its dimension. For non-multinacci values of λ we show that if λ is close to 2/3 , then S_λ has a non-empty interior. Finally we discuss higher-dimensional analogues of the model in question.

Real continuation from the complex quadratic family: Fixed-point bifurcation sets

This paper is primarily a numerical study of the fixed-point bifurcation loci — saddle-node, period-doubling and Hopf bifurcations — present in the family: where z is a complex dynamic (phase) variable, its complex conjugate, and C and A are complex parameters. We treat the parameter C as a primary parameter and A as a secondary parameter, asking how the bifurcation loci projected to the C plane change as the auxiliary parameter A is varied. For A=0, the resulting two-real-parameter family is a familiar one-complex-parameter quadratic family, and the local fixed-point bifurcation locus is the main cardioid of the Mandlebrot set. For A ≠ 0, the resulting two-real-parameter families are not complex analytic, but are still analytic (quadratic) when viewed as a map of ℛ2. Saddle-node and period-doubling loci evolve from points on the main cardioid for A=0 into closed curves for A ≠ 0. As A is varied further from 0 in the complex plane, the three sets interact in a variety of interesting ways. More generally, we discuss bifurcations of families of maps with some parameters designated as primary and the rest as auxiliary. The auxiliary parameter space is then divided into equivalence classes with respect to a specified set of bifurcation loci. This equivalence is defined by the existence of a diffeomorphism of corresponding primary parameter spaces which preserves the specified set of specified bifurcation loci. In our study there is a huge amount of complexity added by specifying the three fixed-point bifurcation loci together, rather than one at a time. We also provide a preliminary classification of the types of codimension-one bifurcations one should expect in general studies of families of two-parameter families of maps of the plane. Comments on numerical continuation techniques are provided as well.

Openness of momentum maps and persistence of extremal relative equilibria

We prove that for every proper Hamiltonian action of a Lie group G in finite dimensions the momentum map is locally G-open relative to its image (i.e. images of G-invariant open sets are open). As an application we deduce that in a Hamiltonian system with continuous Hamiltonian symmetries, extremal relative equilibria persist for every perturbation of the value of the momentum map, provided the isotropy subgroup of this value is compact. We also demonstrate how this persistence result applies to an example of ellipsoidal figures of rotating fluid. We also provide an example with plane point vortices which shows how the compactness assumption is related to persistence.

Group theoretic conditions for existence of robust relative homoclinic trajectories

We consider robust relative homoclinic trajectories ( RHT s) for G -equivariant vector fields. We give some conditions on the group and representation that imply existence of equivariant vector fields with such trajectories. Using these results we show very simply that abelian groups cannot exhibit relative homoclinic trajectories. Examining a set of group theoretic conditions that imply existence of RHT s, we construct some new examples of robust relative homoclinic trajectories. We also classify RHT s of the dihedral and low order symmetric groups by means of their symmetries.