P. Negrini

• Full Profesor
• Professor (Full) at Sapienza University of Rome

We prove an analog of Shilnikov Lemma for a normally hyperbolic symplectic critical manifold $M\subset H^{-1}(0)$ of a Hamiltonian system. Using this result, trajectories with small energy $H=\mu>0$ shadowing chains of homoclinic orbits to $M$ are represented as extremals of a discrete variational problem, and their existence is proved. This paper...

We consider the plane 3 body problem with 2 of the masses small. Periodic solutions with near collisions of small bodies were named by Poincar�e second species periodic solutions. Such solutions shadow chains of collision orbits of 2 uncoupled Kepler problems. Poincar�e only sketched the proof of the existence of second species solutions. Rigorous...

We consider an incompressible fluid contained in a toroidal stratum which is only subjected to Newtonian self-attraction. Under the assumption of infinitesimal tickness of the stratum we show the existence of stationary motions during which the stratum is approximatly a round torus (with radii r, R and R>>r) that rotates around its axis and at the...

We study the linear stability problem of the stationary solution ψ* = −cos y for the Euler equation on a 2-dimensional flat torus of sides 2πL and 2π. We show that ψ* is stable if L ∈ (0, 1) and that exponentially unstable modes occur in a right neighborhood of L = n for any integer n. As a corollary, we gain exponentially instability for any L lar...

The subject of the present lecture is part of a work [1] devoted to some questions related to the problem of turbulence as proposed by the Landau model [2]. We give a very brief survey of the problems which are dealt with. Let μ, be a real parameter, Fμ a vector field family de-fined on a suitable vectorial space E and consider the corresponding ev...

We prove the existence of a number of smooth periodic motions u ∗ of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group R\mathcal{R} of one of the five Platonic polyhedra. The number N coincides with the order |R||\mathcal{R}| of R\mathcal{R} and the particles have all the sam...

We prove the existence of a number of smooth periodic motions $u_*$ of the classical Newtonian $N$-body problem which, up to a relabeling of the $N$ particles, are invariant under the rotation group ${\cal R}$ of one of the five Platonic polyhedra. The number $N$ coincides with the order of ${\cal R}$ and the particles have all the same mass. Our a...

We investigate the problem of the existence of trajectories asymptotic to elliptic equilibria of Hamiltonian systems in the presence of resonances.

We investigate the problem of the existence of trajectories asymptotic to elliptic equilibria of Hamiltonian systems in the presence of resonances.

We consider the motion of a particle in a plane under the gravitational action of 3 fixed centers (the 3-center planar problem). As it is well known (Vestnik Moskov. Univ. Ser. 1 Matem. Mekh 6 (1984) 65; Prikl. Matem. i Mekhan. 48 (1984) 356; Classical Planar Scattering by Coulombic Potentials, Lecture Notes in Physics, Springer, Berlin, 1992) on t...

We consider a system obtained by coupling two Euler-Poinsot systems. The motivation to consider such a system can be traced back to the Riemann Ellipsoids problem. We deal with the problems of integrability and existence of region of chaotic motions.

We describe a global version of the KS regularization of the n-center problem on a closed 3-dimensional manifold. The regularized configuration manifold turns out to be 4 or 5 dimensional closed manifold depending on whether n is even or odd. As an application, we show that the n center problem in S3 has positive topological entropy for n ≥ 5 and e...

We show that the n center problem in R 3 has positive topological entropy for n 3. The proof is based on global regularization of singularities and the results of Gromov and Paternain on the topological entropy of geodesic flows. The n-center problem in S 3 is also studied. 1 Introduction Let P = fp 1 ; : : : ; p n g be a finite set in R 3 . The n-...

The non-integrability and existence of chaotic trajectories in the high-energy zone are proved for a double mathematical pendulum with certain constraints on the ratio of the masses.

We prove a variational criterion for the nonintegrability of an analytic Lagrangian system, with two degrees of freedom, whose configuration space is a torus or a cylinder. As an application, we prove the nonintegrability of the double pendulum in a certain domain of the parameters.

This paper deals with the problem of the instability of an equilibrium, say (q = 0, q̇ = 0), of a lagrangian differential system, in the presence of "gyroscopic forces." More precisely, we examine the case in which the gyroscopic forces start with linear terms A(0) q̇, A(0) being an invertible antisymmetric matrix, while the conservative forces ari...

We consider a natural Lagrangian system. We assume that the potential U(q) has a critical point q = 0, which is not a local maximum; furthermore this property depends only on a suitable k-jet of U (k ≥ 3). Then, if the Lagrangian function is Ch, h ≥ k + 3, and if a "weak coupling" condition is satisfied, we prove that q = 0 is an unstable equilibri...

The existence of 2 -dimensional invariant tori and their bifurcation in 3-dimensional invariant tori are investigated for a family of (non- hamiltonian) differential sistems in R 4.Techniques inspired to the K.A.M. theory are used to identify paths of bifurcation in the parameters space.

The inversion of the Lagrange-Dirichlet theorem is proved under the hypothesis that the potential function U of the acting force is h-differentiable, h > 3, and the lack of a local maximum of U at the equilibrium position is recognizable by means of the nonvanishing terms with lowest degree in the expansion of U. This result extends a previous one...

Sunto Viene compiuta un'analisi completa del problema della biforcazione di Hopf relativa ad arbitrarie piccole perturbazioni del secondo membro di un'equazione differenziale in Rn, p=f0(p). Gli autovalori di f'0(O) soddisfano una condizione di non risonanza. I risultati sono forniti in termini delle proprietá di stabilità di un sistema dinamico pi...

This chapter discusses the analysis of the bifurcating periodic orbits arising in the generalized Hopf bifurcation problem. The existence of these periodic orbits has often been obtained by using techniques such as the Liapunov–Schmidt method or topological degree arguments. The approach, on the other hand, is based upon stability properties of the...

Explicit criteria for the asymptotic stability (or instability) of bifurcating closed orbits are given for a class of abstract evolution equations.

This chapter discusses the stability problems for Hopf bifurcation. It explores the general problem of asymptotic stability of the periodic orbits arising in the Hopf bifurcation. The bifurcating periodic orbits are found to be attracting under the general assumption that 0 is asymptotically stable for μ = 0, which is the critical value of the para...

An analysis of the relationship between asymptotic stability and total stability for abstract dynamic systems is presented. The measure of perturbations on initial data as well as the measure of the distance at any given time between two motions are evaluated in terms of a measure h of stability, introduced as a function defined on the domain of th...

A concept of total stability for continuous or discrete dynamical systems and a generalized definition of bifurcation are given: it is possible to show the link between an abrupt change of the asymptotic behaviour of a family of flows and the arising of new invariant sets, with determined asymptotic properties. The theoretical results are a contrib...

The dispersion properties of drift waves in a low β weakly collisional plasma, in the presence of ion acoustic or Langmuir waves parallel to magnetic lines, are investigated. It is shown that the non-linear interaction with an ion acoustic wave mainly leads to a shift of the drift wave spectrum towards lower frequencies (and only under very particu...