Papers by Roberto Castelli
arXiv (Cornell University), Apr 12, 2017
In this paper, we present and apply a computer-assisted method to study steady states of a triang... more In this paper, we present and apply a computer-assisted method to study steady states of a triangular cross-diffusion system. Our approach consist in an a posteriori validation procedure, that is based on using a fixed point argument around a numerically computed solution, in the spirit of the Newton-Kantorovich theorem. It allows us to prove the existence of various non homogeneous steady states for different parameter values. In some situations, we get as many as 13 coexisting steady states. We also apply the a posteriori validation procedure to study the linear stability of the obtained steady states, proving that many of them are in fact unstable.
; −t) Jacobi Integral: C = 2Ω(x, y , z) − (ẋ 2 +ẏ 2 +ż 2) = −2E Equilibrium points: Lagrangian Po... more ; −t) Jacobi Integral: C = 2Ω(x, y , z) − (ẋ 2 +ẏ 2 +ż 2) = −2E Equilibrium points: Lagrangian Points L j , j = 1, ..., 5. Hill's Region: H(C) = {(x, y , z) : 2Ω(x, y , z) − C ≥ 0} 9th March 2011 Dynamical system theory for mission design Roberto Castelli
Journal of Mathematical Analysis and Applications, 2015
In a central force system the apsidal angle is the angle at the centre of force between two conse... more In a central force system the apsidal angle is the angle at the centre of force between two consecutive apsis and measures the precession rate of the line of apsis. The apsidal angle has applications in different fields and the Newton's apsidal precession theorem has been extensively studied by astronomers, physicist and mathematicians. The perihelion precession of Mercury, the dynamics of galaxies, the vortex dynamics, the JWKB quantisation condition are some examples where the apsidal angle is of interest. In case of eccentric orbits and forces far from inverse square, numerical investigations provide evidence of the monotonicity of the apsidal angle with respect to the orbit parameters, such as the orbit eccentricity. However, no proof of this statement is available. In this paper central force systems with f (r) ∼ µr −(α+1) are considered. We prove that for any −2 < α < 1 the apsidal angle is a monotonic function of the orbital eccentricity, or equivalently of the angular momentum. As a corollary, the conjecture stating the absence of isolated cases of zero precession is proved.
SIAM Journal on Applied Dynamical Systems, 2015
We present an efficient numerical method for computing Fourier-Taylor expansions of (un)stable ma... more We present an efficient numerical method for computing Fourier-Taylor expansions of (un)stable manifolds associated with hyperbolic periodic orbits. Three features of the method are that (1) we obtain accurate representation of the invariant manifold as well as the dynamics on the manifold, (2) it admits natural a posteriori error analysis, and (3) it does not require numerically integrating the vector field. Our approach is based on the parameterization method for invariant manifolds, and studies a certain partial differential equation which characterizes a chart map of the manifold. The method requires only that some mild nonresonance conditions hold. The novelty of the present work is that we exploit the Floquet normal form in order to efficiently compute the Fourier-Taylor expansion. A number of example computations are given including manifolds in phase space dimension as high as ten and manifolds which are two and three dimensional. We also discuss computations of cycle-to-cycle connecting orbits which exploit these manifolds.
Trajectories connecting LEOs with halos around libration points of the Earth-Moon CRTBP are prese... more Trajectories connecting LEOs with halos around libration points of the Earth-Moon CRTBP are presented. Exploiting the coupled circular restricted three-body problem approximation suitable first guess trajectories are derived detecting intersections between stable manifolds related to halo orbits of EM spatial CRTBP and Earth-escaping trajectories integrated in planar SE CRTBP. The accuracy of the intersections in configuration space and the discontinuities in terms of ∆v are controlled through the box covering structure implemented in the software GAIO. Finally first guess solutions are optimized in the bicircular four-body problem and single-impulse and two-impulse transfers are presented.
Communications in Nonlinear Science and Numerical Simulation, 2012
This work deals with the design of transfers connecting LEOs with halo orbits around libration po... more This work deals with the design of transfers connecting LEOs with halo orbits around libration points of the Earth-Moon CRTBP using impulsive maneuvers. Exploiting the coupled circular restricted three-body problem approximation, suitable first guess trajectories are derived detecting intersections between stable manifolds related to halo orbits of EM spatial CRTBP and Earth-escaping trajectories integrated in planar Sun-Earth CRTBP. The accuracy of the intersections in configuration space and the discontinuities in terms of ∆v are controlled through the box covering structure implemented in the software GAIO. Finally first guess solutions are optimized in the bicircular four-body problem and single-impulse and two-impulse transfers are presented.
Applications of Mathematics, 2015
Journal of Mathematical Analysis and Applications, 2014
This paper concerns the behaviour of the apsidal angle for orbits of central force system with ho... more This paper concerns the behaviour of the apsidal angle for orbits of central force system with homogenous potential of degree -2 ≤ α ≤ 1 and logarithmic potential. We derive a formula for the apsidal angle as a fixed-end points integral and we study the derivative of the apsidal angle with respect to the angular momentum . The monotonicity of the apsidal angle as function of is discussed and it is proved in the logarithmic potential case.
In this paper it is demonstrated how rigorous numerics may be applied to the one-dimensional nonl... more In this paper it is demonstrated how rigorous numerics may be applied to the one-dimensional nonlinear Schrödinger equation (NLS); specifically, to determining bound–state solutions and establishing certain spectral properties of the linearization. Since the results are rigorous, they can be used to complete a recent analytical proof [6] of the local exact controllability of NLS. Key words: rigorous numerics, radii polynomials, controllability of PDEs, spectral analysis, BEC
arXiv: Dynamical Systems, 2011
In this paper, a rigorous computational method to enclose eigendecompositions of complex interval... more In this paper, a rigorous computational method to enclose eigendecompositions of complex interval matrices is proposed. Each eigenpair x = (; v) is found by solving a nonlinear equation of the form f(x) = 0 via a contraction argument. The set-up of the method relies on the notion of radii polynomials, which provide an ecient mean
In this paper, a rigorous computational method to enclose eigenpairs of complex interval matrices... more In this paper, a rigorous computational method to enclose eigenpairs of complex interval matrices is proposed. Each eigenpair x = (; v) is found by solving a nonlinear equation of the form f(x) = 0 via a contraction argument. The set-up of the method relies on the notion of radii polynomials, which provide an ecient mean of determining a domain on which the contraction mapping theorem is applicable.
Journal of Differential Equations
Abstract In this paper, we find some new patterns regarding the periodic solvability of the Brill... more Abstract In this paper, we find some new patterns regarding the periodic solvability of the Brillouin electron beam focusing equation x ¨ + β ( 1 + cos ( t ) ) x = 1 x . In particular, we prove that there exists β ⁎ ≈ 0.248 for which a 2π-periodic solution exists for every β ∈ ( 0 , β ⁎ ] , and the bifurcation diagram with respect to β displays a fold for β = β ⁎ . This result significantly contributes to the discussion about the well-known conjecture asserting that the Brillouin equation admits a periodic solution for every β ∈ ( 0 , 1 / 4 ) , leading to doubt about its truthfulness. For the first time, moreover, we prove multiplicity of periodic solutions for a range of values of β near β ⁎ . The technique used relies on rigorous computation and can be extended to some generalizations of the Brillouin equation, with right-hand side equal to 1 / x p ; we briefly discuss the cases p = 2 and p = 3 .
Journal of Differential Equations
In this paper, we present and apply a computer-assisted method to study steady states of a triang... more In this paper, we present and apply a computer-assisted method to study steady states of a triangular cross-diffusion system. Our approach consist in an a posteriori validation procedure, that is based on using a fixed point argument around a numerically computed solution, in the spirit of the Newton-Kantorovich theorem. It allows us to prove the existence of various non homogeneous steady states for different parameter values. In some situations, we get as many as 13 coexisting steady states. We also apply the a posteriori validation procedure to study the linear stability of the obtained steady states, proving that many of them are in fact unstable.
Archive for Rational Mechanics and Analysis
In this paper, we develop computer-assisted techniques for the analysis of periodic orbits of ill... more In this paper, we develop computer-assisted techniques for the analysis of periodic orbits of ill-posed partial differential equations. As a case study, our proposed method is applied to the Boussinesq equation, which has been investigated extensively because of its role in the theory of shallow water waves. The idea is to use the symmetry of the solutions and a Newton-Kantorovich type argument (the radii polynomial approach), to obtain rigorous proofs of existence of the periodic orbits in a weighted 1 Banach space of space-time Fourier coefficients with geometric decay. We present several computer-assisted proofs of existence of periodic orbits at different parameter values.
Acta Applicandae Mathematicae
In this paper a method to rigorously compute several non trivial solutions of the Gray-Scott reac... more In this paper a method to rigorously compute several non trivial solutions of the Gray-Scott reaction-diffusion system defined on a 2-dimensional bounded domain is presented. It is proved existence, within rigorous bounds, of non uniform patterns significantly far from being a perturbation of the homogenous states. As a result, a non local diagram of families that bifurcate from the homogenous states is depicted, also showing coexistence of multiple solutions at the same parameter values. Combining analytical estimates and rigorous computations, the solutions are sought as fixed points of a operator in a suitable Banach space. To address the curse of dimensionality, a variation of the existing technique is presented, necessary to enable successful computations in reasonable time.
Celestial Mechanics and Dynamical Astronomy, 2016
In this contribution, an efficient technique to design direct low-energy trajectories in multi-mo... more In this contribution, an efficient technique to design direct low-energy trajectories in multi-moon systems is presented. The method relies on analytical two-body approximations of trajectories originating from the stable and unstable invariant manifolds of two coupled circular restricted three-body problems. We provide a means to perform very fast and accurate computations of the minimum-cost trajectories between two moons. Eventually, we validate the methodology by comparison with numerical integrations in the three-body problem. Motivated by the growing interest in the robotic exploration of the Jovian system, which has given rise to numerous studies and mission proposals, we apply the method to the design of minimum-cost low-energy direct trajectories between Galilean moons, and the case study is that of Ganymede and Europa. However, the domain of applicability of the method is much wider. It can be employed, for instance, in geocentric orbit context, whenever a rendezvous or an orbit change is sought, in the optimization of high-energy patched-conics tours of multi-moon systems, and in the design of interplanetary deep space maneuvers.
Astrophysics and Space Science Proceedings, 2016
Over the past two decades, the robotic exploration of the Solar System has reached the moons of t... more Over the past two decades, the robotic exploration of the Solar System has reached the moons of the giant planets. In the case of Jupiter, a strong scientific interest towards its icy moons has motivated important space missions (e.g., ESAs' JUICE and NASA's Europa Mission). A major issue in this context is the design of efficient trajectories enabling satellite tours, i.e., visiting the several moons in succession. Concepts like the Petit Grand Tour and the Multi-Moon Orbiter have been developed to this purpose, and the literature on the subject is quite rich. The models adopted are the two-body problem (with the patched conics approximation and gravity assists) and the three-body problem (giving rise to the so-called lowenergy transfers, LETs). In this contribution, we deal with the connection between two moons, Europa and Ganymede, and we investigate a two-body approximation of trajectories originating from the stable/unstable invariant manifolds of the two circular restricted three body problems, i.e., Jupiter-Ganymede and Jupiter-Europa. We develop ad-hoc algorithms to determine the intersections of the resulting elliptical arcs, and the magnitude of the maneuver at the intersections. We provide a means to perform very fast and accurate evaluations of the minimum-cost trajectories between the two moons. Eventually, we validate the methodology by comparison with numerical integrations in the three-body problem.
Archive for Rational Mechanics and Analysis, 2016
This work concerns the planar N-center problem with homogeneous potential of degree −α (α ∈ [1, 2... more This work concerns the planar N-center problem with homogeneous potential of degree −α (α ∈ [1, 2)). The existence of infinitely many, topologically distinct, non-collision periodic solutions with a prescribed energy is proved. A notion of admissibility in the space of loops on the punctured plane is introduced so that in any admissible class and for any positive h the existence of a classical periodic solution with energy h for the N-center problem with α ∈ (1, 2) is proven. In case α = 1 a slightly different result is shown: it is the case that there is either a non-collision periodic solution or a collision-reflection solution. The results hold for any position of the centres and it is possible to prescribe in advance the shape of the periodic solutions. The proof combines the topological properties of the space of loops in the punctured plane with variational and geometrical arguments.
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Papers by Roberto Castelli