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2013
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# 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.  774-800 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. Keywords: Hamiltonian system, symplectic map, generating function, heteroclinic orbit Citation: Bolotin S. V., Negrini P.,  Shilnikov Lemma for a Nondegenerate Critical Manifold of a Hamiltonian System, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 774-800 DOI:10.1134/S1560354713060142
 Fusco G., Negrini P., Oliva W. M. Stationary Motion of a Self-gravitating Toroidal Incompressible Liquid Layer 2012, vol. 17, no. 5, pp.  397-416 Abstract We consider an incompressible fluid contained in a toroidal stratum which is only subjected to Newtonian self-attraction. 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 helix-like trajectory around the circumference of radius $R$ that connects the centers of the cross sections of the torus. Keywords: self-gravitating liquid figures Citation: Fusco G., Negrini P., Oliva W. M.,  Stationary Motion of a Self-gravitating Toroidal Incompressible Liquid Layer, Regular and Chaotic Dynamics, 2012, vol. 17, no. 5, pp. 397-416 DOI:10.1134/S1560354712050036
 Butta P., Negrini P. On the stability problem of stationary solutions for the Euler equation on a 2-dimensional torus 2010, vol. 15, no. 6, pp.  637-645 Abstract We study the linear stability problem of the stationary solution $\psi^* = −\cos y$ for the Euler equation on a 2-dimensional 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. Keywords: Euler equation, shear flows, linear stability Citation: Butta P., Negrini P.,  On the stability problem of stationary solutions for the Euler equation on a 2-dimensional torus, Regular and Chaotic Dynamics, 2010, vol. 15, no. 6, pp. 637-645 DOI:10.1134/S1560354710510143
 Butta P., Negrini P. Resonances and $O$-curves in Hamiltonian systems 2007, vol. 12, no. 5, pp.  521-530 Abstract We investigate the problem of the existence of trajectories asymptotic to elliptic equilibria of Hamiltonian systems in the presence of resonances. Keywords: Hamiltonian systems, resonances Citation: Butta P., Negrini P.,  Resonances and $O$-curves in Hamiltonian systems, Regular and Chaotic Dynamics, 2007, vol. 12, no. 5, pp. 521-530 DOI:10.1134/S156035470705005X
 Negrini P. Integrability, nonintegrability and chaotic motions for a system motivated by the Riemann ellipsoids problem 2003, vol. 8, no. 4, pp.  349-374 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. Citation: Negrini P.,  Integrability, nonintegrability and chaotic motions for a system motivated by the Riemann ellipsoids problem , Regular and Chaotic Dynamics, 2003, vol. 8, no. 4, pp. 349-374 DOI:10.1070/RD2003v008n04ABEH000251