Related papers: Normal stability of slow manifolds in nearly-perio…
We prove homological stability for both general linear groups of modules over a ring with finite stable rank and unitary groups of quadratic modules over a ring with finite unitary stable rank. In particular, we do not assume the modules…
We study Hamiltonian systems near a compact symplectic Morse-Bott minimum. Our first result shows that if the flow is Zoll (that is, it induces a free circle action) along a sequence of energy levels converging to the minimum, then the…
We analyse the effect of intrinsic fluctuations on the properties of bistable stochastic systems with time scale separation operating under1 quasi-steady state conditions. We first formulate a stochastic generalisation of the quasi-steady…
The stability of heteroclinic cycles may be obtained from the value of the local stability index along each connection of the cycle. We establish a way of calculating the local stability index for quasi-simple cycles: cycles whose…
We study two resonant Hamiltonian systems on the phase space $L^2(\mathbb{R} \rightarrow \mathbb{C})$: the quintic one-dimensional continuous resonant equation, and a cubic resonant system that has appeared in the literature as a modified…
We apply the Nyquist method to the Hamiltonian Mean Field (HMF) model in order to settle the linear dynamical stability of a spatially homogeneous distribution function with respect to the Vlasov equation. We consider the case of Maxwell…
The limiting slow dynamics of slow-fast, piecewise-linear, continuous systems of ODEs occurs on critical manifolds that are piecewise-linear. At points of non-differentiability, such manifolds are not normally hyperbolic and so the…
In this paper we prove the existence of an almost invariant symplectic slow manifold for analytic Hamiltonian slow-fast systems with finitely many slow degrees of freedom for which the error field is exponentially small. We allow for…
Let $M$ be a $10$-dimensional closed oriented smooth manifold. Set $$\mathcal{D}_{M} := \{ x \in H^{2}(M; \Z/2) \mid x^{2} + w_{2}(M) x \in \rho_{2} ( TH^{4}(M;\Z) ) \}.$$ Suppose that $H_{1}(M;\Z)=0$ and $\mathcal{D}_{M} \subset \rho_{2}(…
We apply round-off to planar rotations, obtaining a one-parameter family of invertible maps of a two-dimensional lattice. As the angle of rotation approaches pi/2, the fourth iterate of the map produces piecewise-rectilinear motion, which…
In this paper we study long time stability of a class of nontrivial, quasi-periodic solutions depending on one spacial variable of the cubic defocusing non-linear Schr\"odinger equation on the two dimensional torus. We prove that these…
We introduce a class of convex, higher-dimensional billiard models which generalise stadium billiards. These models correspond to the free motion of a point-particle in a region bounded by cylinders cut by planes. They are motivated by…
It is shown that in perfectly quasi-isodynamic stellarators, trapped particles with a bounce frequency much higher than the frequency of the instability are stabilizing in the electrostatic and collisionless limit. The collisionless…
We study the properties of the asymptotic Maslov index of invariant measures for time-periodic Hamiltonian systems on the cotangent bundle of a compact manifold M. We show that if M has finite fundamental group and the Hamiltonian satisfies…
Concepts like `typicality' and the `eigenstate thermalization hypothesis' aim at explaining the apparent equilibration of quantum systems, possibly after a very long time. However, these concepts are not concerned with the specific way in…
Localized patterns in singularly perturbed reaction-diffusion equations typically consist of slow parts -- in which the associated solution follows an orbit on a slow manifold in a reduced spatial dynamical system -- alternated by fast…
We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and…
This paper studies a class of $1\frac12$-degree-of-freedom Hamiltonian systems with a slowly varying phase that unfolds a Hamiltonian pitchfork bifurcation. The main result of the paper is that there exists an order of…
This article provides an example of fast-slow system such that most orbits remain as close as possible to the unstable manifold of the fast dynamics for an arbitrarily long time.
We investigate the stability properties of an abstract class of semi-linear systems. Our main result establishes rational rates of decay for classical solutions assuming a certain non-uniform observability estimate for the linear part and…