Searcharxiv⌕ Search

arXiv subjects

Alessandro Portaluri

Publications and source records attributed to Alessandro Portaluri.

At least 37 records · Page 2Linked to original sources
arXiv2019

Morse index and stability of the planar N-vortex problem

This paper concerns the investigation of the stability properties of relative equilibria which are rigidly rotating vortex configurations sometimes called vortex crystals, in the N-vortex problem. Such a configurations can be characterized as critical point of the Hamiltonian function restricted on the constant angular impulse hypersurface in the phase space (topologically a pseudo-sphere whose coefficients are the circulation strengths of the vortices). Relative equilibria are generated by the circle action on the so-called shape pseudo-sphere (which generalize the standard shape sphere appearing in the study of the N-body problem). Inspired by the planar gravitational $N$-body problem, and after a geometrical and dynamical discussion, we investigate the relation intertwining the stability of relative equilibria and the inertia indices of the central configurations generating such equilibria. In the last section we apply our main results to some symmetric three and four vortices relative equilibria.

math.DS↗
arXiv2018

An Index theory for asymptotic motions under singular potentials

We develop an index theory for parabolic and collision solutions to the classical n-body problem and we prove sufficient conditions for the finiteness of the spectral index valid in a large class of trajectories ending with a total collapse or expanding with vanishing limiting velocities. Both problems suffer from a lack of compactness and can be brought in a similar form of a Lagrangian System on the half time line by a regularising change of coordinates which preserve the Lagrangian structure. We then introduce a Maslov-type index which is suitable to capture the asymptotic nature of these trajectories as half-clinic orbits: by taking into account the underlying Hamiltonian structure we define the appropriate notion of geometric index for this class of solutions and we develop the relative index theory.

math.DS↗
arXiv2017

A dihedral Bott-type iteration formula and stability of symmetric periodic orbits

In 1956, Bott in his celebrated paper on closed geodesics and Sturm intersection theory, proved an Index Iteration Formula for closed geodesics on Riemannian manifolds. Some years later, Ekeland improved this formula in the case of convex Hamiltonians and, in 1999, Long generalized the Bott iteration formula by putting in its natural symplectic context and constructing a very effective Index Theory. The literature about this formula is quite broad and the dynamical implications in the Hamiltonian world (e.g. existence, multiplicity, linear stability etc.) are enormous. Motivated by the recent discoveries on the stability properties of symmetric periodic solutions of singular Lagrangian systems, we establish a Bott-type iteration formula for dihedrally equivariant Lagrangian and Hamiltonian systems. We finally apply our theory for computing the Morse indices of the celebrated Chenciner and Montgomery figure-eight orbit for the planar three body problem in different equivariant spaces. Our last dynamical consequence is an hyperbolicity criterion for reversible Lagrangian systems.

math.DS↗
arXiv2017

Bifurcation of heteroclinic orbits via an index theory

Heteroclinic orbits for one-parameter families of nonautonomous vectorfields appear in a very natural way in many physical applications. Inspired by some recent bifurcation results for homoclinic trajectories of nonautonomous vectorfield, we define a new $\mathbf Z_2$-index and we construct a index theory for heteroclinic orbits of nonautonomous vectorfield. We prove an index theorem, by showing that, under some standard transversality assumptions, the $\mathbf Z_2$-index is equal to the parity, a homotopy invariant for paths of Fredholm operators of index 0. As a direct consequence of the index theory developed in this paper, we get a new bifurcation result for heteroclinic orbits.

math.DS↗
arXiv2017

Not every conjugate point of a semi-Riemannian geodesic is a bifurcation point

We revisit an example of a semi-Riemannian geodesic that was discussed by Musso, Pejsachowicz and Portaluri in 2007 to show that not every conjugate point is a bifurcation point. We point out a mistake in their argument, showing that on this geodesic actually every conjugate point is a bifurcation point. Finally, we provide an improved example which yields that the claim in our title is nevertheless true.

math.DG↗
arXiv2017

Index theory for heteroclinic orbits of Hamiltonian systems

Index theory revealed its outstanding role in the study of periodic orbits of Hamiltonian systems and the dynamical consequences of this theory are enormous. Although the index theory in the periodic case is well-established, very few results are known in the case of homoclinic orbits of Hamiltonian systems. Moreover, to the authors' knowledge, no results have been yet proved in the case of heteroclinic and halfclinic (i.e. parametrised by a half-line) orbits. Motivated by the importance played by these motions in understanding several challenging problems in Classical Mechanics, we develop a new index theory and we prove at once a general spectral flow formula for heteroclinic, homoclinic and halfclinic trajectories. Finally we show how this index theory can be used to recover all the (classical) existing results on orbits parametrised by bounded intervals.

math.DS↗
arXiv2016

A K-theoretical Invariant and Bifurcation for Homoclinics of Hamiltonian Systems

We revisit a K-theoretical invariant that was invented by the first author some years ago for studying multiparameter bifurcation of branches of critical points of functionals. Our main aim is to apply this invariant to investigate bifurcation of homoclinic solutions of families of Hamiltonian systems which are parametrised by tori.

math.FA↗
arXiv2015

Biological control of the chestnut gall wasp with \emph{T. sinensis}: a mathematical model

The Asian chestnut gall wasp \emph{Dryocosmus kuriphilus}, native of China, has become a pest when it appeared in Japan, Korea, and the United States. In Europe it was first found in Italy, in 2002. In 1982 the host-specific parasitoid \emph{Torymus sinensis} was introduced in Japan, in an attempt to achieve a biological control of the pest. After an apparent initial success, the two species seem to have locked in predator-prey cycles of decadal length. We have developed a spatially explicit mathematical model that describes the seasonal time evolution of the adult insect populations, and the competition for finding egg deposition sites. In a spatially homogeneous situation the model reduces to an iterated map for the egg density of the two species. While the map would suggest, for realistic parameters, that both species should become locally extinct (somewhat corroborating the hypothesis of biological control), the full model, for the same parameters, shows that the introduction of \emph{T. sinensis} sparks a traveling wave of the parasitoid population that destroys the pest on its passage. Depending on the value of the diffusion coefficients of the two species, the pest can later be able to re-colonize the empty area left behind the wave. When this occurs the two populations do not seem to attain a state of spatial homogeneity, but produce an ever-changing pattern of traveling waves.

q-bio.PE↗
arXiv2015

A Morse-Smale index theorem for indefinite elliptic systems and bifurcation

We generalise the semi-Riemannian Morse index theorem to elliptic systems of partial differential equations on star-shaped domains. Moreover, we apply our theorem to bifurcation from a branch of trivial solutions of semilinear systems, where the bifurcation parameter is introduced by shrinking the domain to a point. This extends recent results of the authors for scalar equations.

math.AP↗
arXiv2015

A mathematical model for voice leading and its complexity

We give a formal definition of the musical concept of voice leading in mathematical terms, based on the idea of partial permutations of certain ordered multisets. Then we associate a partial permutation matrix with a voice leading in a unique way and write an algorithm to easily transform any musical composition into a sequence of such matrices; we then generalise it in order to include in the model also rhythmic independence and rests. From that we extract a vector whose components return information about the movements of the voices in the piece and hence about the complexity of the voice leading. We provide some examples by analysing three compositions, also visualising complexity as a point cloud for each piece. Finally, we interpret the sequence of complexity vectors associated with each composition, thus considering the position of each observation with respect to time. The Dynamic Time Warping allows us to compute the distance between two pieces and to show that our approach distinguishes well the examples that we took into account, exhibiting a strong indication that the notion of complexity we propose is a good tool to identify and classify musical pieces.

math.HO↗
arXiv2015

Morse index and linear stability of the Lagrangian circular orbit in a three-body-type problem via index theory

It is well known that the linear stability of the Lagrangian elliptic solutions in the classical planar three-body problem depends on a mass parameter $β$ and on the eccentricity $e$ of the orbit. We consider only the circular case ($e = 0$) but under the action of a broader family of singular potentials: $α$-homogeneous potentials, for $α\in (0,2)$, and the logarithmic one. It turns out indeed that the Lagrangian circular orbit persists also in this more general setting. We discover a region of linear stability expressed in terms of the homogeneity parameter $α$ and the mass parameter $β$, then we compute the Morse index of this orbit and of its iterates and we find that the boundary of the stability region is the envelope of a family of curves on which the Morse indices of the iterates jump. In order to conduct our analysis we rely on a Maslov-type index theory devised and developed by Y.~Long, X.~Hu and S.~Sun; a key role is played by an appropriate index theorem and by some precise computations of suitable Maslov-type indices.

math.DS↗
arXiv2014

A Mathematical Model of Flavescence Dorée Epidemiology

Flavescence dorée (FD) is a disease of grapevine transmitted by an insect vector, $Scaphoideus$ $titanus$ Ball. At present, no prophylaxis exists, so mandatory control procedures (e.g. removal of infected plants, and insecticidal sprays to avoid transmission) are in place in Italy and other European countries. We propose a model of the epidemiology of FD by taking into account the different aspects involved into the transmission process (acquisition of the disease, latency and expression of symptoms, recovery rate, removal and replacement of infected plants, insecticidal treatments, and the effect of hotbeds). The model was constructed as a system of first order nonlinear ODEs in four compartment variables. We perform a bifurcation analysis of the equilibria of the model using the severity of the hotbeds as the control parameter. Depending on the non-dimensional grapevine density of the vineyard we find either a single family of equilibria in which the health of the vineyard gradually deteriorates for progressively more severe hotbeds, or multiple equilibria that give rise to sudden transitions from a nearly healthy vineyard to a severely deteriorated one when the severity of the hotbeds crosses a critical value. These results suggest some lines of intervention for limiting the spread of the disease.

q-bio.PE↗
arXiv2014

Linear instability of relative equilibria for $n$-body problems in the plane

Following Smale, we study simple symmetric mechanical systems of $n$ point particles in the plane. In particular, we address the question of the linear and spectral stability properties of relative equilibria, which are special solutions of the equations of motion. Our main result is a sufficient condition to detect spectral (hence linear) instability. Namely, we prove that if the Morse index of an equilibrium point with even nullity is odd, then the associated relative equilibrium is spectrally unstable. The proof is based on some refined formulæ for computing the spectral flow. As a notable application of our theorem, we examine two important classes of singular potentials: the $α$-homogeneous one, with $α\in (0, 2)$, which includes the gravitational case, and the logarithmic one. We also establish, for the $α$-homogeneous potential, an inequality which is useful to test the spectral instability of the associated relative equilibrium.

math.DS↗
arXiv2013

Modes in modern music from a topological viewpoint

The aim of this paper is twofold: on one side we review the classical concept of musical mode from the viewpoint of modern music, reading it as a superimposition of a base-chord (seventh chord) and a tension-chord (triad). We associate to each modal scale an oriented plane graph whose homotopy properties give a measure of the complexity of the base-chord associated to a certain mode. Using these graphs we prove the existence of special modes which are not deducible in the standard way. On the other side we give a more deep musical insight by developing a braid theoretical interpretation of some cadential harmonic progressions in modern music and we use braid theory in order to represent them and voice leadings among them. A striking application is provided by the analysis of an harmonic fragment from Peru by Tribal Tech We approximate the octatonic scale used in the improvisation by Scott Henderson, through the special mode myxolydian b2#4 and we finally associate a braid representation to the fragment we analysed.

math.AT↗