Piero Negrini
P.le Aldo Moro, 2, 00185, Roma, Italy
Dipartimento di Matematica, Universita di Roma "La Sapienza", Roma
Publications:
Bolotin S. V., Negrini P.
Shilnikov Lemma for a Nondegenerate Critical Manifold of a Hamiltonian System
2013, vol. 18, no. 6, pp. 774800
Abstract
Let $M$ be a normally hyperbolic symplectic critical manifold of a Hamiltonian system. Suppose $M$ consists of equilibria with real eigenvalues. We prove an analog of the Shilnikov lemma (strong version of the $\lambda$lemma) describing the behavior of trajectories near $M$. Using this result, trajectories shadowing chains of homoclinic orbits to $M$ are represented as extremals of a discrete variational problem. Then the existence of shadowing periodic orbits is proved. This paper is motivated by applications to the Poincaré’s second species solutions of the 3 body problem with 2 masses small of order $\mu$. As $\mu \to 0$, double collisions of small bodies correspond to a symplectic critical manifold $M$ of the regularized Hamiltonian system. Thus our results imply the existence of Poincaré’s second species (nearly collision) periodic solutions for the unrestricted 3 body problem.

Fusco G., Negrini P., Oliva W. M.
Stationary Motion of a Selfgravitating Toroidal Incompressible Liquid Layer
2012, vol. 17, no. 5, pp. 397416
Abstract
We consider an incompressible fluid contained in a toroidal stratum which is only subjected to Newtonian selfattraction. Under the assumption of infinitesimal thickness of the stratum we show the existence of stationary motions during which the stratum is approximately a round torus (with radii $r$, $R$ and $R \gg r$) that rotates around its axis and at the same time rolls on itself. Therefore each particle of the stratum describes an helixlike trajectory around the circumference of radius $R$ that connects the centers of the cross sections of the torus.

Butta P., Negrini P.
On the stability problem of stationary solutions for the Euler equation on a 2dimensional torus
2010, vol. 15, no. 6, pp. 637645
Abstract
We study the linear stability problem of the stationary solution $\psi^* = −\cos y$ for the Euler equation on a 2dimensional flat torus of sides $2\pi L$ and $2\pi$. We show that $\psi^*$ is stable if $L \in (0, 1)$ 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 $L$ large enough and an unbounded growth of the number of unstable modes as $L$ diverges.

Butta P., Negrini P.
Resonances and $O$curves in Hamiltonian systems
2007, vol. 12, no. 5, pp. 521530
Abstract
We investigate the problem of the existence of trajectories asymptotic to elliptic equilibria of Hamiltonian systems in the presence of resonances.

Negrini P.
Integrability, nonintegrability and chaotic motions for a system motivated by the Riemann ellipsoids problem
2003, vol. 8, no. 4, pp. 349374
Abstract
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.
