Roberto Castelli

Parameterization of Invariant Manifolds for Periodic Orbits I: Efficient Numerics via the Floquet Normal Form

We present an efficient numerical method for computing Fourier--Taylor expansions of (un)stable manifolds associated with hyperbolic periodic orbits. Three features of the method are that (1) we obtain accurate representation of the invariant manifold as well as the dynamics on the manifold, (2) it admits natural a posteriori error analysis, and (3) it does not require numerically integrating the vector field. Our approach is based on the parameterization method for invariant manifolds, and studies a certain partial differential equation which characterizes a chart map of the manifold. The method requires only that some mild nonresonance conditions hold. The novelty of the present work is that we exploit the Floquet normal form in order to efficiently compute the Fourier--Taylor expansion. A number of example computations are given including manifolds in phase space dimension as high as ten and manifolds which are two and three dimensional. We also discuss computations of cycle-to-cycle connecting orbits ...

Rigorous Numerics in Floquet Theory: Computing Stable and Unstable Bundles of Periodic Orbits

In this paper, a rigorous method to compute Floquet normal forms of fundamental matrix solutions of nonautonomous linear differential equations with periodic coefficients is introduced. The Floquet normal form of a fundamental matrix solution

Rigorous Numerics for ill-posed PDEs: Periodic Orbits in the Boussinesq Equation

\Phi(t)

The monotonicity of the apsidal angle in power-law potential systems

is a canonical decomposition of the form

Two-Body Approximations in the Design of Low-Energy Transfers Between Galilean Moons

\Phi(t)=Q(t)e^{Rt}

Rigorous numerics for NLS: Bound states, spectra, and controllability

, where

A METHOD TO RIGOROUSLY ENCLOSE EIGENPAIRS OF COMPLEX INTERVAL MATRICES

Q(t)

Topologically Distinct Collision-Free Periodic Solutions for the \({N}\)-Center Problem

is a real periodic matrix and

Parameterization of Invariant Manifolds for Periodic Orbits (II): A Posteriori Analysis and Computer Assisted Error Bounds

R

Singular dynamics under a weak potential on a sphere

is a constant matrix. To rigorously compute the Floquet normal form, the idea is to use the regularity of