arXiv subjects

Andrea Giacobbe

Publications and source records attributed to Andrea Giacobbe.

6 recordsLinked to original sources

arXiv2020

An algebro-geometric classification of spectral types of equilibria

We give three algebraic equations which allow a geometric classification of all spectral types of equilibria of a given $m$-dimensional dynamical system, and we analyse them thoroughly in dimension 3 and 4. The loci defined by these equations correspond to definite types of bifurcations. The complement of such loci give a geometric decomposition of the space of invariants in open domains in which the equilibrium has a given spectral types. The usefulness of this approach lays in the fact that, when dealing with a parameter-dependent dynamical system, the pull-back of the loci from the space of invariants to the parameter space gives the bifurcation-decomposition of parameter space for the dynamical system at hand. We also give effective methods to explicitly compute the spectral indices.

math.DS↗

arXiv2020

A theoretical model for realistic local climates

We write a nonlinear model that predicts the climate (temperature and humidity) on the surface of a small region on Earth, perform numerical investigations using the model, and compare the results to real climate on a variety of regions on Earth. It the parameters are chosen keeping into consideration the climatic Köppen zone to which the region belongs, the numerical model accurately reproduces the real climate. The model takes into account the doubly-periodic forcing of the solar radiation (annual and daily), the laws of irradiance, the fact that the Earth has land and oceans with different thermic inertia, and the humidity of the air due to evaporation. This enables us to reproduce remarkable features of Earth's climate such as lag of seasons, lag of noons, and asymmetric evolution of daily temperatures. The model can easily be adapted to planets with non-terrestrial astronomic parameters. We conclude this article with an investigation of an Earth with eccentricity higher than real.

physics.ao-ph↗

arXiv2018

Nonlinear stability results for plane Couette and Poiseuille flows

In this article we prove, choosing an appropriately weighted $L_2$-energy equivalent to the classical energy, that the plane Couette and Poiseuille flows are nonlinearly stable with respect to streamwise perturbations for any Reynolds number. In this case the coefficient of time-decay of the energy is $π^2/(2 {\rm Re})$, and it is a bound from above of the time-decay of streamwise perturbations of linearized equations. We also prove that the plane Couette and Poiseuille flows are nonlinearly stable if the Reynolds number is less then ${\rm Re}_{Orr}/\sin φ$ when the perturbation is a tilted perturbation, i.e. 2D perturbations with wave vector which forms an angle $φ\in [0, π/2]$ with the direction ${\bf i}$ of the motion. ${\rm Re}_{Orr}$ is the Orr (1907) critical Reynolds number for spanwise perturbations which, for the Couette flow is ${\rm Re}_{Orr}=177.22$ and for the Poiseuille flow is ${\rm Re}_{Orr}=175.31$. In particular these results improve those obtained by Joseph (1966), who found for streamwise perturbations a critical nonlinear value of $82.6$ in the Couette case, and those obtained by Joseph and Carmi (1969), who found the value $99.1$ for plane Poiseuille flow for streamwise perturbations. The results we obtain here are, for any angle, in a good agreement with the experiments (see Prigent et al. 2003) and the numerical simulations (see Barckley and Tuckerman 2005, 2007).

physics.flu-dyn↗

arXiv2015

Quasi-periodicity in relative quasi-periodic tori

At variance from the cases of relative equilibria and relative periodic orbits of dynamical systems with symmetry, the dynamics in relative quasi-periodic tori (namely, subsets of the phase space that project to an invariant torus of the reduced system on which the flow is quasi-periodic) is not yet completely understood. Even in the simplest situation of a free action of a compact and abelian connected group, the dynamics in a relative quasi-periodic torus is not necessarily quasi-periodic. It is known that quasi-periodicity of the unreduced dynamics is related to the reducibility of the reconstruction equation, and sufficient conditions for it are virtually known only in a perturbation context. We provide a different, though equivalent, approach to this subject, based on the hypothesis of the existence of commuting, group-invariant lifts of a set of generators of the reduced torus. Under this hypothesis, which is shown to be equivalent to the reducibility of the reconstruction equation, we give a complete description of the structure of the relative quasi-periodic torus, which is a principal torus bundle whose fibers are tori of a dimension which exceeds that of the reduced torus by at most the rank of the group. The construction can always be done in such a way that these tori have minimal dimension and carry ergodic flow.

math.DS↗

arXiv2007

Geometry of Invariant Tori of Certain Integrable Systems with Symmetry and an Application to a Nonholonomic System

Bifibrations, in symplectic geometry called also dual pairs, play a relevant role in the theory of superintegrable Hamiltonian systems. We prove the existence of an analogous bifibrated geometry in dynamical systems with a symmetry group such that the reduced dynamics is periodic. The integrability of such systems has been proven by M. Field and J. Hermans with a reconstruction technique. We apply the result to the nonholonomic system of a ball rolling on a surface of revolution.

math.SG↗

arXiv2000

Convexity of multi-valued momentum map

We extend the famous convexity theorem of Atiyah, Guillemin and Sternberg to the case of non-Hamiltonian actions. We show that the image of a generalized momentum map is a bounded polytope times a vector space. We prove that this picture is stable for small perturbations of the symplectic form. We also observe that an n-dimensional torus symplectic action on a 2n-dimensional symplectic manifold, with fixed point, is Hamiltonian. We finally prove that an extension of Kirwan's convexity theorem (for compact group actions) is also true.

math.SG↗