Abstract: We describe the linear and nonlinear stability and instability of certain symmetric con... more Abstract: We describe the linear and nonlinear stability and instability of certain symmetric configurations of point vortices on the sphere forming relative equilibria. These configurations consist of one or two rings, and a ring with one or two polar vortices.
Abstract We consider Hamiltonian systems with symmetry, and relative equilibria with isotropy sub... more Abstract We consider Hamiltonian systems with symmetry, and relative equilibria with isotropy subgroup of positive dimension. The stability of such relative equilibria has been studied by Ortega and Ratiu (1999 Nonlinearity 12 693–720) and by Lerman and Singer (1998 Nonlinearity 11 1637–49). In both papers the authors give sufficient conditions for stability which require first determining a splitting of a subalgebra of g, with different splittings giving different criteria.
How many cusps does a swallowtail have, After it becomes a stable map, And how many swallowtails ... more How many cusps does a swallowtail have, After it becomes a stable map, And how many swallowtails does a butter y have, After it . .. (with apologies to B. Dylan)
Résumé–I. Melbourne [9] a énoncé récemment un théoreme d'existence sur la bifurcation générique d... more Résumé–I. Melbourne [9] a énoncé récemment un théoreme d'existence sur la bifurcation générique d'ondes rotatives d'isotropie maximale pour les champs de vecteurs équivariants par l'action absolument irréductible d'un groupe de Lie compact. La démonstration de Melbourne se base sur des résultats récents de Field [4, 5].
Summary. We describe a method for finding the families of relative equilibria of molecules that b... more Summary. We describe a method for finding the families of relative equilibria of molecules that bifurcate from an equilibrium point as the angular momentum is increased from 0. Relative equilibria are steady rotations about a stationary axis during which the shape of the molecule remains constant. We show that the bifurcating families correspond bijectively to the critical points of a function h on the two-sphere which is invariant under an action of the symmetry group of the equilibrium point.
Abstract We describe the linear and nonlinear stability and instability of certain symmetric conf... more Abstract We describe the linear and nonlinear stability and instability of certain symmetric configurations of point vortices on the sphere forming relative equilibria. These configurations consist of one or two rings, and a ring with one or two polar vortices.
where m(r) is an n-periodic function: m(r + n) = m(r) defined on the integers. For example M = ... more where m(r) is an n-periodic function: m(r + n) = m(r) defined on the integers. For example M =
In this introduction, we first recall the basic phase space structures involved in Hamiltonian sy... more In this introduction, we first recall the basic phase space structures involved in Hamiltonian systems, the symplectic form, the Poisson brackets and the Hamiltonian function and vector fields, and the relationship between them. Afterwards we describe a few examples of Hamiltonian systems, both of the classical'kinetic+ potential'type as well as others using the symplectic/Poisson structure more explicitly.
In earlier work [DS Broomhead, JP Huke, MR Muldoon, and J. Stark, Iterated function system models... more In earlier work [DS Broomhead, JP Huke, MR Muldoon, and J. Stark, Iterated function system models of digital channels, Proc. R. Soc. Lond. A 460 (2004), pp. 3123–3142], aimed at developing an approach to signal processing that can be applied as well to nonlinear systems as linear ones, we produced mathematical models of digital communications channels that took the form of iterated function systems (IFS).
Abstract In the visualization of the topology of second rank symmetric tensor fields in the plane ... more Abstract In the visualization of the topology of second rank symmetric tensor fields in the plane one can extract some key points (degenerate points), and curves (separatrices) that characterize the qualitative behaviour of the whole tensor field. This can provide a global structure of the whole tensor field, and effectively reduce the complexity of the original data. To construct this global structure it is important to classify those degenerate points accurately.
Abstract We announce two topological results that may be used to estimate the number of relative ... more Abstract We announce two topological results that may be used to estimate the number of relative periodic orbits of different homotopy classes that are possessed by a symmetric Lagrangian system. The results are illustrated by applications to systems on tori and to strong force N-centre problems.
Abstract. We present a framework for the study of the local qualitative dynamics of equivariant H... more Abstract. We present a framework for the study of the local qualitative dynamics of equivariant Hamiltonian flows specially designed for points in phase space with nontrivial isotropy. This is based on the classical construction of structure-preserving tubular neighborhoods for Hamiltonian Lie group actions on symplectic manifolds. This framework is applied to the obtention of concrete and testable conditions guaranteeing the existence of bifurcations from symmetric branches of Hamiltonian relative equilibria.
CATASTROPHE THEORY is the study of families of functions, and in particular of their critical poi... more CATASTROPHE THEORY is the study of families of functions, and in particular of their critical points. For example the function fa (x)= x3− 3ax has a single (degenerate) critical point when a= 0, it has two critical points when a> 0 and none at all when a< 0 (or two complex ones if one prefers). On the other hand if the family has only nondegenerate critical points when a= 0, then for nearby values of a it will still have nondegnerate critical points and they will be nearby the original ones.
We consider a compact, oriented, smooth Riemannian manifold M (with or without boundary) and we s... more We consider a compact, oriented, smooth Riemannian manifold M (with or without boundary) and we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of G and corresponding vector field XM on M, one defines Wittenʼs inhomogeneous coboundary operator [Formula: see text](even/odd invariant forms on M) and its adjoint [Formula: see text].
In recent work, Belishev and Sharafutdinov show that the generalized Dirichlet to Neumann (DN) op... more In recent work, Belishev and Sharafutdinov show that the generalized Dirichlet to Neumann (DN) operator Λ on a compact Riemannian manifold M with boundary∂ M determines de Rham cohomology groups of M. In this paper, we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of G and the corresponding vector field XM on M, Witten defines an inhomogeneous coboundary operator [Formula: see text] on invariant forms on M.
Abstract: In this note we clarify the relationship between the local and global definitions of du... more Abstract: In this note we clarify the relationship between the local and global definitions of dual pairs in Poisson geometry. It turns out that these are not equivalent. For the passage from local to global one needs a connected fiber hypothesis (this is well known), while the converse requires a dimension condition (which appears not to be known). We also provide examples illustrating the necessity of the extra conditions.
Abstract: We construct a smooth family of Hamiltonian systems, together with a family of group sy... more Abstract: We construct a smooth family of Hamiltonian systems, together with a family of group symmetries and momentum maps, for the dynamics of point vortices on surfaces parametrized by the curvature of the surface. Equivariant bifurcations in this family are characterized, whence the stability of the Thomson heptagon is deduced without recourse to the Birkhoff normal form, which has hitherto been a necessary tool.
Abstract: We describe the linear and nonlinear stability and instability of certain symmetric con... more Abstract: We describe the linear and nonlinear stability and instability of certain symmetric configurations of point vortices on the sphere forming relative equilibria. These configurations consist of one or two rings, and a ring with one or two polar vortices.
Abstract We consider Hamiltonian systems with symmetry, and relative equilibria with isotropy sub... more Abstract We consider Hamiltonian systems with symmetry, and relative equilibria with isotropy subgroup of positive dimension. The stability of such relative equilibria has been studied by Ortega and Ratiu (1999 Nonlinearity 12 693–720) and by Lerman and Singer (1998 Nonlinearity 11 1637–49). In both papers the authors give sufficient conditions for stability which require first determining a splitting of a subalgebra of g, with different splittings giving different criteria.
How many cusps does a swallowtail have, After it becomes a stable map, And how many swallowtails ... more How many cusps does a swallowtail have, After it becomes a stable map, And how many swallowtails does a butter y have, After it . .. (with apologies to B. Dylan)
Résumé–I. Melbourne [9] a énoncé récemment un théoreme d'existence sur la bifurcation générique d... more Résumé–I. Melbourne [9] a énoncé récemment un théoreme d'existence sur la bifurcation générique d'ondes rotatives d'isotropie maximale pour les champs de vecteurs équivariants par l'action absolument irréductible d'un groupe de Lie compact. La démonstration de Melbourne se base sur des résultats récents de Field [4, 5].
Summary. We describe a method for finding the families of relative equilibria of molecules that b... more Summary. We describe a method for finding the families of relative equilibria of molecules that bifurcate from an equilibrium point as the angular momentum is increased from 0. Relative equilibria are steady rotations about a stationary axis during which the shape of the molecule remains constant. We show that the bifurcating families correspond bijectively to the critical points of a function h on the two-sphere which is invariant under an action of the symmetry group of the equilibrium point.
Abstract We describe the linear and nonlinear stability and instability of certain symmetric conf... more Abstract We describe the linear and nonlinear stability and instability of certain symmetric configurations of point vortices on the sphere forming relative equilibria. These configurations consist of one or two rings, and a ring with one or two polar vortices.
where m(r) is an n-periodic function: m(r + n) = m(r) defined on the integers. For example M = ... more where m(r) is an n-periodic function: m(r + n) = m(r) defined on the integers. For example M =
In this introduction, we first recall the basic phase space structures involved in Hamiltonian sy... more In this introduction, we first recall the basic phase space structures involved in Hamiltonian systems, the symplectic form, the Poisson brackets and the Hamiltonian function and vector fields, and the relationship between them. Afterwards we describe a few examples of Hamiltonian systems, both of the classical'kinetic+ potential'type as well as others using the symplectic/Poisson structure more explicitly.
In earlier work [DS Broomhead, JP Huke, MR Muldoon, and J. Stark, Iterated function system models... more In earlier work [DS Broomhead, JP Huke, MR Muldoon, and J. Stark, Iterated function system models of digital channels, Proc. R. Soc. Lond. A 460 (2004), pp. 3123–3142], aimed at developing an approach to signal processing that can be applied as well to nonlinear systems as linear ones, we produced mathematical models of digital communications channels that took the form of iterated function systems (IFS).
Abstract In the visualization of the topology of second rank symmetric tensor fields in the plane ... more Abstract In the visualization of the topology of second rank symmetric tensor fields in the plane one can extract some key points (degenerate points), and curves (separatrices) that characterize the qualitative behaviour of the whole tensor field. This can provide a global structure of the whole tensor field, and effectively reduce the complexity of the original data. To construct this global structure it is important to classify those degenerate points accurately.
Abstract We announce two topological results that may be used to estimate the number of relative ... more Abstract We announce two topological results that may be used to estimate the number of relative periodic orbits of different homotopy classes that are possessed by a symmetric Lagrangian system. The results are illustrated by applications to systems on tori and to strong force N-centre problems.
Abstract. We present a framework for the study of the local qualitative dynamics of equivariant H... more Abstract. We present a framework for the study of the local qualitative dynamics of equivariant Hamiltonian flows specially designed for points in phase space with nontrivial isotropy. This is based on the classical construction of structure-preserving tubular neighborhoods for Hamiltonian Lie group actions on symplectic manifolds. This framework is applied to the obtention of concrete and testable conditions guaranteeing the existence of bifurcations from symmetric branches of Hamiltonian relative equilibria.
CATASTROPHE THEORY is the study of families of functions, and in particular of their critical poi... more CATASTROPHE THEORY is the study of families of functions, and in particular of their critical points. For example the function fa (x)= x3− 3ax has a single (degenerate) critical point when a= 0, it has two critical points when a> 0 and none at all when a< 0 (or two complex ones if one prefers). On the other hand if the family has only nondegenerate critical points when a= 0, then for nearby values of a it will still have nondegnerate critical points and they will be nearby the original ones.
We consider a compact, oriented, smooth Riemannian manifold M (with or without boundary) and we s... more We consider a compact, oriented, smooth Riemannian manifold M (with or without boundary) and we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of G and corresponding vector field XM on M, one defines Wittenʼs inhomogeneous coboundary operator [Formula: see text](even/odd invariant forms on M) and its adjoint [Formula: see text].
In recent work, Belishev and Sharafutdinov show that the generalized Dirichlet to Neumann (DN) op... more In recent work, Belishev and Sharafutdinov show that the generalized Dirichlet to Neumann (DN) operator Λ on a compact Riemannian manifold M with boundary∂ M determines de Rham cohomology groups of M. In this paper, we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of G and the corresponding vector field XM on M, Witten defines an inhomogeneous coboundary operator [Formula: see text] on invariant forms on M.
Abstract: In this note we clarify the relationship between the local and global definitions of du... more Abstract: In this note we clarify the relationship between the local and global definitions of dual pairs in Poisson geometry. It turns out that these are not equivalent. For the passage from local to global one needs a connected fiber hypothesis (this is well known), while the converse requires a dimension condition (which appears not to be known). We also provide examples illustrating the necessity of the extra conditions.
Abstract: We construct a smooth family of Hamiltonian systems, together with a family of group sy... more Abstract: We construct a smooth family of Hamiltonian systems, together with a family of group symmetries and momentum maps, for the dynamics of point vortices on surfaces parametrized by the curvature of the surface. Equivariant bifurcations in this family are characterized, whence the stability of the Thomson heptagon is deduced without recourse to the Birkhoff normal form, which has hitherto been a necessary tool.
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Papers by James Montaldi