arXiv subjects

J. S. W. Lamb

Publications and source records attributed to J. S. W. Lamb.

3 recordsLinked to original sources

arXiv2025

Smooth Persistence of Attractors for Set-Valued Dynamical Systems: A Boundary Map Approach

We study the problem of persistence of attractors with smooth boundary for a class of set-valued dynamical systems that naturally arise in the context of random and control dynamical systems, as well as in systems modeling the dynamical propagation of uncertainty. In order to tackle the inherent difficulties associated to the multi-valued structure of such dynamical systems, we introduce a single-valued map, the so-called boundary map, which is a contactomorphism of the unit-tangent bundle of the state space, with the following characteristic property: boundaries of attractors of the set-valued dynamical system correspond in a unique way to invariant Legendrian manifolds of this map. We show how the underlying contact geometry guarantees the smooth persistence of such attractors under perturbations of the set-valued dynamical system, provided that the associated boundary map is normally hyperbolic at the unit normal bundle of the boundary.

math.DS↗

arXiv2008

On the zero set of G-equivariant maps

Let $G$ be a finite group acting on vector spaces $V$ and $W$ and consider a smooth $G$-equivariant mapping $f:V\to W$. This paper addresses the question of the zero set near a zero $x$ of $f$ with isotropy subgroup $G$. It is known from results of Bierstone and Field on $G$-transversality theory that the zero set in a neighborhood of $x$ is a stratified set. The purpose of this paper is to partially determine the structure of the stratified set near $x$ using only information from the representations $V$ and $W$. We define an index $s(Σ)$ for isotropy subgroups $Σ$ of $G$ which is the difference of the dimension of the fixed point subspace of $Σ$ in $V$ and $W$. Our main result states that if $V$ contains a subspace $G$-isomorphic to $W$, then for every maximal isotropy subgroup $Σ$ satisfying $s(Σ)>s(G)$, the zero set of $f$ near $x$ contains a smooth manifold of zeros with isotropy subgroup $Σ$ of dimension $s(Σ)$. We also present a systematic method to study the zero sets for group representations $V$ and $W$ which do not satisfy the conditions of our main theorem. The paper contains many examples and raises several questions concerning the computation of zero sets of equivariant maps. These results have application to the bifurcation theory of $G$-reversible equivariant vector fields.

math.DS↗

arXiv2002

Normal Forms and Unfoldings of Linear Systems in Eigenspaces of (Anti)-Automorphisms of Order Two

In this article we classify normal forms and unfoldings of linear maps in eigenspaces of (anti)-automorphisms of order two. Our main motivation is provided by applications to linear systems of ordinary differential equations, general and Hamiltonian, which have both time-preserving and time-reversing symmetries. However the theory gives a uniform method to obtain normal forms and unfoldings for a wide variety of linear differential equations with additional structure. We give several examples and include a discussion of the phenomenon of orbit splitting. As a consequence of orbit splitting we observe passing and splitting of eigenvalues in unfoldings.

math.DS↗