arXiv subjects

Roberto Castelli

Publications and source records attributed to Roberto Castelli.

6 recordsLinked to original sources

arXiv2017

Existence and instability of steady states for a triangular cross-diffusion system: a computer-assisted proof

In this paper, we present and apply a computer-assisted method to study steady states of a triangular cross-diffusion system. Our approach consist in an a posteriori validation procedure, that is based on using a fxed point argument around a numerically computed solution, in the spirit of the Newton-Kantorovich theorem. It allows us to prove the existence of various non homogeneous steady states for different parameter values. In some situations, we get as many as 13 coexisting steady states. We also apply the a posteriori validation procedure to study the linear stability of the obtained steady states, proving that many of them are in fact unstable.

math.AP↗

arXiv2013

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

In this paper it is demonstrated how rigorous numerics may be applied to the one-dimensional nonlinear Schrödinger equation (NLS); specifically, to determining bound--state solutions and establishing certain spectral properties of the linearization. Since the results are rigorous, they can be used to complete a recent analytical proof [6] of the local exact controllability of NLS.

math.DS↗

arXiv2013

A study of the apsidal angle and a proof of monotonicity in the logarithmic potential case

This paper concerns the behaviour of the apsidal angle for orbits of central force system with homogenous potential of degree $-2\leq α\leq 1$ and logarithmic potential. We derive a formula for the apsidal angle as a fixed-end points integral and we study the derivative of the apsidal angle with respect to the angular momentum $\ell$. The monotonicity of the apsidal angle as function of $\ell$ is discussed and it is proved in the logarithmic potential case.

math.DS↗

arXiv2011

A method to rigorously enclose eigendecompositions of interval matrices

In this paper, a rigorous computational method to enclose eigendecompositions of complex interval matrices is proposed. Each eigenpair $x=(\lambda,v)$ is found by solving a nonlinear equation of the form $f(x)=0$ via a contraction argument. The set-up of the method relies on the notion of radii polynomials, which provide an efficient mean of determining a domain on which the contraction mapping theorem is applicable.

math.DS↗

arXiv2011

Rigorous numerics in Floquet theory: computing stable and unstable bundles of periodic orbits

In this paper, a new rigorous numerical method to compute fundamental matrix solutions of non-autonomous linear differential equations with periodic coefficients is introduced. Decomposing the fundamental matrix solutions $\Phi(t)$ by their Floquet normal forms, that is as product of real periodic and exponential matrices $\Phi(t)=Q(t)e^{Rt}$, one solves simultaneously for $R$ and for the Fourier coefficients of $Q$ via a fixed point argument in a suitable Banach space of rapidly decaying coefficients. As an application, the method is used to compute rigorously stable and unstable bundles of periodic orbits of vector fields. Examples are given in the context of the Lorenz equations and the $\zeta^3$-model.

math.DS↗

arXiv2011

Global dynamics under a weak potential on a sphere

We give a detailed analytical description of the global dynamics of a point mass moving on a sphere under the action of a logarithmic potential. After performing a McGehee-type blow-up in order to cope with the singularity of the potential, we investigate the rest-points of the flow, the invariant (stable and unstable) manifolds and we give a complete dynamical description of the motion.

math.DS↗