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Massimo VILLARINI

Professore Associato
Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede ex-Matematica

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Pubblicazioni

2020 - A Rigidity Property of Perturbations of n Identical Harmonic Oscillators [Articolo su rivista]
Villarini, M.
abstract

Let Xε: S2n-1→ TS2n-1 be a smooth perturbation of X, the vector field associated to the dynamical system defined by n identical uncoupled harmonic oscillators constrained to their 1-energy level. We are dealing with the case when any orbit of every Xε is closed: while in general is false that the vector fields of the perturbation are orbitally equivalent to the unperturbed X (Villarini in Ergod Theory Dyn Syst 39:1–32, 2019), we prove that this rigidity behaviour is indeed true if each Xε restricted to a codimension 2 sphere in S2n-1 is orbitally conjugated to a subsystem of X made by n- 1 harmonic oscillators. In other words: to have a non-rigid, or truly non-linear, perturbation of X at least two harmonic oscillators must be destroyed by the perturbation. We use this rigidity result to prove a linearization theorem for real analytic multicentres. Finally we give an example of a real analytic perturbation of X showing discontinuous changing of integer invariants of the vector fields of the perturbation.

2019 - Smooth foliations by circles of S^7 with unbounded periods and nonlinearizable multicentres [Articolo su rivista]
Villarini, Massimo
abstract

We give an example of a (Formula presented.) vector field (Formula presented.), defined in a neighbourhood (Formula presented.) of (Formula presented.), such that (Formula presented.) is foliated by closed integral curves of (Formula presented.), the differential (Formula presented.) at (Formula presented.) defines a one-parameter group of non-degenerate rotations and (Formula presented.) is not orbitally equivalent to its linearization. Such a vector field (Formula presented.) has the first integral (Formula presented.), and its main feature is that its period function is locally unbounded near the stationary point. This proves in the (Formula presented.) category that the classical Poincaré centre theorem, true for planar non-degenerate centres, is not generalizable to multicentres. Such an example is obtained through a careful study and a suitable modification of a celebrated example by Sullivan [A counterexample to the periodic orbit conjecture. Publ. Math. Inst. Hautes Études Sci. 46 (1976), 5–14], by blowing up the stationary point at the origin and through the construction of a smooth one-parameter family of foliations by circles of (Formula presented.) whose orbits have unbounded lengths (equivalently, unbounded periods) for each value of the parameter and which smoothly converges to the Hopf fibration (Formula presented.).

2011 - Geometric bounds on the linearization domain and analytic dependence on parameters for families of analytic vector fields in a neighborhood ofa singular point [Articolo su rivista]
Villarini, Massimo
abstract

We study families of holomorphic vector fields, holomorphically depending on parameters,in a neighborhood of an isolated singular point. When the singular point is in the Poincarédomain for every vector field of the family we prove, through a modification of classicalSternberg’s linearization argument, cf. Nelson (1969) [7] too, analytic dependence onparameters of the linearizing maps and geometric bounds on the linearization domain:each vector field of the family is linearizable inside the smallest Euclidean sphere which isnot transverse to the vector field, cf. Brushlinskaya (1971) [2], Ilyashenko and Yakovenko(2008) [5] for related results. We also prove, developing ideas in Martinet (1980) [6],a version of Brjuno’s Theorem in the case of linearization of families of vector fields neara singular point of Siegel type, and apply it to study some 1-parameter families of vectorfields in two dimensions.

2010 - Detectability of critical points of smooth functionals from theirfinite-dimensional approximations [Articolo su rivista]
Sani, F.; Villarini, Massimo
abstract

Given a critical point of a C2-functional on a separable Hilbert space, we obtain sufficient conditions for it to be detectable (i.e. `visible') from finite-dimensional Rayleigh-Ritz-Galerkin (RRG) approximations. While examples show that even nondegenerate critical points are, without any further restriction, not visible, we single out relevant classes of smooth functionals, e.g. the Hamiltonian action on the loop space or the functionals associated with boundary value problems for some semilinear elliptic equations, such that their nondegenerate critical points are visible from their RRG approximations.

2009 - Existence of multiple periodic solutions for a natural Hamiltonian system in a potential well [Articolo su rivista]
Villarini, Massimo
abstract

We prove, under a pinching hypothesis, a theorem of existence ofat least n periodic orbits for a closed regular energy hypersurface$\Sigma$ in $\mathbb{R}^{2n}$ which is the level set of a natural (i.e. of the type "potential + kinetic energy") Hamiltonian function and projects onto a potential well in the configuration space.

2007 - Trends in Differential Equations and Dynamical Systems [Esposizione]
Gavioli, Andrea; Malaguti, Luisa; Villarini, Massimo
abstract

The workshop took place at Modena, from November 29th to 30th.The main speakers were J. Andres, from Palachy University (Olomouc, SK), P. K. Maini, from the University of Oxford (UK), V. Obukhovskii, from Voronezh State University (Russia), and other speakers from Italy (R. Johnson, F. Papalini, M. Tarallo). Furthermore, several talks were given by young researchers. The topics of the meeting covered many different areas in the field of differential equations and related problems. In particular, here are some of the exposed subjects: front-propagation in reaction-diffusion equations, which often arise from biological models, Sturm-Liouville operators, perturbation theory for Hamiltonian systems, impulsive control systems, boundary value problems and the theory of bound sets, delay equations, differential inclusions.

2005 - Normalization of Poincare' singularities via variation of constants [Articolo su rivista]
T., Carletti; A., Margheri; Villarini, Massimo
abstract

We present a geometric proof of the Poincare-Dulac Normalization Theorem for analytic vector fields with singularities of Poincare type. Our approach allows us to relate the size of the convergence domain of the linearizing transformation to the geometry of the complex foliation associated to the vector field. A similar construction is considered in the case of linearization of maps in a neighborhood of a hyperbolic fixed point.

2005 - Trends in Differential Equations and Dynamical Systems [Esposizione]
Gavioli, Andrea; Malaguti, Luisa; Villarini, Massimo
abstract

The workshop took place at Reggio Emilia, from September 29th to 30th. The main speakers were S. Kamin from Tel-Aviv University, Tel-Aviv (Israel) and L. Sanchez from Lisbon University (Lisbon, Portugal) and other speakers from Italy (A. Agrachev and S. Terracini). Furthermore, several talks were given by young researchers. The topics of the meeting covered many different areas in the field of differential equations and control problems.

2004 - Existence of periodic orbits for vector fields via Fuller index and the averaging method [Articolo su rivista]
P., Benevieri; Gavioli, Andrea; Villarini, Massimo
abstract

We prove a generalization of a theorem proved by Seifert and Fuller concerning the existence of periodic orbits of vector fields via the averaging method. Also we show applications of these results to Kepler motion and to geodesic flows on spheres.

2002 - A geometric approach to the existence of sets of periodic orbits [Articolo su rivista]
Margheri, A; Villarini, Massimo
abstract

We study the problem of the existence and of the geometric structure of the set of periodic orbits of a vector field in presence of a first integral. We give a unified treatment and a geometric proof of existence results of periodic orbits by Moser (local case) and Bottkol (global case) under a suitable nonresonance condition.The local resonance case is considered, too. For analytic vector fields admitting an analytic first integral, we give a geometric description of the set of periodic orbits, proving that it is an analytic set, hence extending a theorem by Siegel.

2001 - Periodic orbits for vector fields with nondegenerate first integrals [Articolo su rivista]
Villarini, Massimo
abstract

We give a geometric proof in the particular case of nondegenerate first integrals. of a theorem by Moser about the existence of periodic orbits on each level set of the integral, in a neighbourhood of a singular point of a vector field satisfying a nonresonance hypothesis. We use the same geometric approach to deal with the resonance case, obtaining a generalization of previous results by Sweet.

2000 - Smooth linearization of centres [Articolo su rivista]
Villarini, Massimo
abstract

We extend the Poincarè-Lyapunov Centre Theorem to the smooth case: namely we prove that a $C^k$ vector field, $k \geq 3$, having a nondegenerate centre at the origin is always $C^{k-2}$-orbitally equivalent to its linear part at the origin.

1999 - On the Poincare'-Lyapunov Centre theorem [Articolo su rivista]
Brunella, M.; Villarini, Massimo
abstract

We give a geometric proof and a slight generalization of a result of Sibuya and Urabe concerning a higher dimensional version of the classical Poincaré-Lyapunov Centre Theorem. This is done from a complexified point of view.

1998 - Jets of differential equations having integrable extensions [Articolo su rivista]
Villarini, Massimo
abstract

We characterize the set of $n$-jets admitting an extension which is a germ of a differential equation with an analytic first integral, and compute its codimension in the $n$-jet space. Some applications in the case of the centre-focus problem are given.

1997 - Algebraic criteria for the existence of analytic first integrals [Articolo su rivista]
Villarini, Massimo
abstract

We discuss the problem of existence of holomorphic first integrals of analytic differential equations from the point of view of algebraic solvability.

1997 - Limit cycles and bifurcation from a separatrix for a polynomial Lienard system in the plane. [Articolo su rivista]
G., Villari; Villarini, Massimo
abstract

We study bifurcations of limit cycles from a separatrix in a polynomial Lienard equation.

1995 - Algebraic nonsolvability of the problem of existence of holomorphic first integrals [Articolo su rivista]
Villarini, Massimo
abstract

We prove that the problem of existence of holomorphic first integrals of analytic differential equations in a neighbourhood of a singular point is not algebraically solvable

1995 - On the problem of existence of holomorphic first integrals [Articolo su rivista]
Villarini, Massimo
abstract

We discuss the Mattei-Moussu topological criterion for existence of a holomorphic first integral in a neighbourhood of a singular point of an analytic differential equation, and we prove that it cannot be used to get algebraic solvability of such an existence problem.

1992 - Regularity properties of the period function near a centre of a planar vector field [Articolo su rivista]
Villarini, Massimo
abstract

We study the problem of analytic extension up to the singular point of the period function of a centre of a planar analytic vector field, and discuss condition of isochronicity.

1992 - Some properties of planar polynomial systems of even degree [Articolo su rivista]
Galeotti, M; Villarini, Massimo
abstract

We prove that a planar polynomial vector field of even degree always has an unbounded solution

1990 - Error bounds for solutions of systems of quasi linear hyperbolic partial differential equations in diagonal form [Articolo su rivista]
R., Ceppitelli; Villarini, Massimo
abstract

We study a quasi linear hyperbolic system of partial differential equations and find bounds for the difference of its solutions with respect to the solutions of the linear system.

1986 - Error bounds for solutions of systems of quasilinear hyperbolic partial differential equations in bicharacteristic form [Articolo su rivista]
R., Ceppitelli; Villarini, Massimo
abstract

We find bounds for the difference between the solutions of a nonlinear systems of hyperbolic differential equations in bicharacteristic form and its linear counterpart.