IRIS Università degli Studi di Padovahttps://www.research.unipd.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Thu, 09 Apr 2020 07:38:59 GMT2020-04-09T07:38:59Z10711A numerical study of the topology of normally hyperbolic invariant manifolds supporting Arnold diffusion in quasi-integrable systemshttp://hdl.handle.net/11577/2378262Titolo: A numerical study of the topology of normally hyperbolic invariant manifolds supporting Arnold diffusion in quasi-integrable systems
Abstract: We investigate numerically the stable and unstable manifolds of the hyperbolic manifolds of the phase
space related to the resonances of quasi-integrable systems in the regime of validity of the Nekhoroshev
and KAM theorems. Using a model of weakly interacting resonances we explain the qualitative features
of these manifolds characterized by peculiar flower-like structures. We detect different transitions in
the topology of these manifolds related to the local rational approximations of the frequencies. We find
numerically a correlation among these transitions and the speed of Arnold diffusion.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/11577/23782622009-01-01T00:00:00ZCantori of the dissipative sawtooth maphttp://hdl.handle.net/11577/2378261Titolo: Cantori of the dissipative sawtooth map
Abstract: We investigate the existence of cantori for a dissipative version of the sawtooth map. Making use of an explicit parametric representation of the solution, we prove that cantori exist for any irrational value of the frequency. We also perform a numerical study aimed to determine some dynamical properties of the dissipative sawtooth map.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/11577/23782612009-01-01T00:00:00ZNekhoroshev-stability of L4 and L5 in the spatial restricted three--body problemhttp://hdl.handle.net/11577/2467625Titolo: Nekhoroshev-stability of L4 and L5 in the spatial restricted three--body problem
Abstract: We show that L4 and L5 in the spatial restricted circular three-body problem are Nekhoroshev-stable for all but a few values of the reduced mass up to the Routh critical value. This result is based on two extensions of previous results on Nekhoroshev-stability of elliptic equilibria, namely to the case of "directional quasi-convexity", a notion introduced here, and to a (non-convex) steep case. We verify that the hypotheses are satisfied for L4 and L5 by means of numerically constructed Birkhoff normal forms.
Thu, 01 Jan 1998 00:00:00 GMThttp://hdl.handle.net/11577/24676251998-01-01T00:00:00ZDiffusion and stability in perturbed non-convex integrable systemshttp://hdl.handle.net/11577/1562601Titolo: Diffusion and stability in perturbed non-convex integrable systems
Abstract: The Nekhoroshev theorem has become an important tool for explaining the
long-term stability of many quasi-integrable systems of interest in physics.
The action variables of systems that satisfy the hypotheses of the Nekhoroshev theorem remain close to their initial value up to very long times, which grow exponentially as an inverse power of the perturbation's norm. In this paper
we study some of the simplest systems that do not satisfy the hypotheses of the Nekhoroshev theorem. These systems can be represented by a perturbed Hamiltonian whose integrable part is a quadratic non-convex function of the action variables. We study numerically the possibility of action diffusion over short times for these systems (continuous or maps) and we compare it with the so-called Arnold diffusion. More precisely we find that, except for very special non-convex functions, for which the effect of non-convexity concerns low-order resonances, the diffusion coefficient decreases faster than a power law (and possibly exponentially) of the perturbation's norm. According to
theory, we find that the diffusion coefficient as a function of the perturbation's
norm decreases more slowly than in the convex case.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/11577/15626012006-01-01T00:00:00ZThe web of three-planet resonances in the outer Solar System II. A source of orbital instability for Uranus and Neptunehttp://hdl.handle.net/11577/1562603Titolo: The web of three-planet resonances in the outer Solar System II. A source of orbital instability for Uranus and Neptune
Abstract: The motion of the giant planets from Jupiter to Neptune is chaotic with Lyapunov time of approximately 10 Myr. A recent theory explains the
presence of this chaos with three-planet mean-motion resonances, i.e. resonances among the orbital periods of at least three planets. We find that
the distribution of these resonances with respect to the semi-major axes of all the planets is compatible with orbital instability. In particular, they
overlap in a region of 10^{-3} AU with respect to the variation of the semi-major axes of Uranus and Neptune. Fictitious planetary systems with
initial conditions in this region can undergo systematic variations of semi-major axes. The true Solar System is marginally in this region, and
Uranus and Neptune undergo very slow systematic variations of semi-major axes with speed of order 10^{-4} AU/Gyr.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/11577/15626032006-01-01T00:00:00ZAnalysis of the chaotic behaviour of orbits diffusing along the Arnold webhttp://hdl.handle.net/11577/1562600Titolo: Analysis of the chaotic behaviour of orbits diffusing along the Arnold web
Abstract: In a previous work [Guzzo et al. DCDS B 5, (2005)] we have provided numerical evidence of global diffusion occurring in slightly perturbed integrable Hamiltonian systems and symplectic maps. We have shown that even if a system is sufficiently close to be
integrable, global diffusion occurs on a set with peculiar topology, the so-called Arnold web,
and is qualitatively different from Chirikov diffusion, occurring in more perturbed systems.
In the present work we study in more detail the chaotic behaviour of a set of 90 orbits which
diffuse on the Arnold web. We find that the largest Lyapunov exponent does not seem to
converge for the individual orbits while the mean Lyapunov exponent on the set of 90 orbits
does converge. In other words, a kind of average mixing characterizes the diffusion. Moreover, the Local Lyapunov Characteristic Numbers (LLCNs), on individual orbits appear to reflect the different zones of the Arnold web revealed by the Fast Lyapunov Indicator. Finally,
using the LLCNs we study the ergodicity of the chaotic part of the Arnold web.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/11577/15626002006-01-01T00:00:00ZLong--term stability of proper rotations of the Euler perturbed rigid bodyhttp://hdl.handle.net/11577/2435116Titolo: Long--term stability of proper rotations of the Euler perturbed rigid body
Abstract: We study the long term stability of the proper rotations of the Euler perturbed rigid body, in the framework of Nekhoroshev theory. For simplicity we treat here in detail only the kinetically symmetric case (the potential needs not to be symmetric), but we indicate how to extend the results to the triaxial case. We show that the proper rotations around the symmetry axis are Nekhoroshev stable: more precisely, if the initial datum is suÆciently close to a proper rotation, then for a very long time it remains such, and the tip of the unit vector \mu parallel to the angular momentum precesses, up to a small noise, along the level curves of a regular function on the unit sphere. If the proper rotations are resonant, chaotic motions with positive Lyapunov exponents are possible, but chaos (unlike the case of ordinary motions, that is motions not close to proper rotations) is always localized, and does not aect in an essential way the motion of the angular momentum in space. Preliminary numerical result indicate that the theory is, in many aspects, optimal, although in some points it can still be improved.
Thu, 01 Jan 2004 00:00:00 GMThttp://hdl.handle.net/11577/24351162004-01-01T00:00:00ZNumerical studies of hyperbolic manifolds supporting diffusion in symplectic mappingshttp://hdl.handle.net/11577/2424972Titolo: Numerical studies of hyperbolic manifolds supporting diffusion in symplectic mappings
Abstract: Diﬀusion in generic quasi integrable systems at small values of the perturbing parameters has been a very studied subject since the pioneering work of Arnold. For moderate values of the perturbing parameter a diﬀerent kind of diﬀusion occurs, the so called Chirikov diﬀusion, since the Chirikov’s papers [....]. The two underlying mechanisms are diﬀerent, the ﬁrst has an analytic demonstration only on speciﬁc models, the second is based on an heuristic argument. Even if the relation between chaos and diﬀusion is far to be completely understood, a key role is played by the topology of hyperbolic manifolds related to the resonances. Diﬀerent methods can be found in the literature for the detection of hyperbolic manifolds, at least for two dimensional systems. For higher dimensional ones some sophisticated methods have been recently developed (for a review see [....]). In this paper we review some of these methods and an easy tool of detection of invariant manifolds that we have developed based on the Fast Lyapunov Indicator. The relation between the topology of hyperbolic manifolds and diﬀusion is discussed in the framework of Arnold diﬀusion.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/11577/24249722010-01-01T00:00:00ZNekhoroshev stability of quasi--integrable degenerate Hamiltonian systemshttp://hdl.handle.net/11577/145938Titolo: Nekhoroshev stability of quasi--integrable degenerate Hamiltonian systems
Abstract: A perturbation of a degenerate integrable Hamiltonian system has the form H=h(I) + ε f(I, φ, p, q) with (I, φ)\in R^n × T^n, (p, q)\in R^{2m} and the two-form is dI^dφ + dp^dq. In the case h is convex, Nekhoroshev theorem provides the usual bound to the motion of the actions I, but only for a time which is the smaller between the usual exponentially-long time and the escape time of p, q from the domain. Furthermore, the theorem does not provide any estimate for the "degenerate variables" p, q better than the a priori one , and in the literature there are examples of systems with degenerate variables that perform large chaotic motions in short times. The problem of the motion of the degenerate variables is relevant to understand the long time stability of several systems, like the three body problem, the asteroid belt dynamical system and the fast rotations of the rigid body. In this paper we show that if the "secular" Hamiltonian of H, i.e. the average of H with respect to the fast angles φ, is integrable (or quasi-integrable) and if it satisfies a convexity condition, then a Nekhoroshev-like bound holds for the degenerate variables (actually for the actions of the secular integrable system) for all initial data with initial action I(0) outside a small neighbourhood of the resonant manifolds of order lower than ln(1 /ε). This paper generalizes a result proved in connection with the problem of the long-time stability in the Asteroid Main Belt.
Fri, 01 Jan 1999 00:00:00 GMThttp://hdl.handle.net/11577/1459381999-01-01T00:00:00ZCOMPUTATION OF TRANSIT ORBITS IN THE THREE-BODY-PROBLEM WITH FAST LYAPUNOV INDICATORShttp://hdl.handle.net/11577/3256621Titolo: COMPUTATION OF TRANSIT ORBITS IN THE THREE-BODY-PROBLEM WITH FAST LYAPUNOV INDICATORS
Abstract: We describe in this paper how to compute special orbits of the three-body-problem which transit from a region which is internal to the secondary mass to the region which is external to the binary system, by using a recent variant of the Fast Lyapunov Indicator method. The orbits are obtained by slightly changing the initial conditions of orbits which are heteroclinic to Lyapunov orbits of the Lagrangian equilibrium points L1 and L2 of the restricted three-body-problem.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/11577/32566212017-01-01T00:00:00Z