On the Arnold Diffusion Mechanism in Medium Earth Orbit - Journal of Nonlinear Science

Space debris mitigation guidelines represent the most effective method to preserve the circumterrestrial environment. Among them, end-of-life disposal solutions play a key role. In this regard, effective strategies should be conceived not only on the basis of novel technologies, but also following an advanced theoretical understanding. A growing effort is devoted to exploit natural perturbations to lead the satellites toward an atmospheric reentry, reducing the disposal cost, also if departing from high-altitude regions. In the case of the Medium Earth Orbit region, home of the navigation satellites (like GPS and Galileo), the main driver is the gravitational perturbation due to the Moon, that can increase the eccentricity in the long term. In this way, the pericenter altitude can get into the atmospheric drag domain and the satellite can eventually reenter. In this work, we show how an Arnold diffusion mechanism can trigger the eccentricity growth. Focusing on the case of Galileo, we consider a hierarchy of Hamiltonian models, assuming that the main perturbations on the motion of the spacecraft are the oblateness of the Earth and the gravitational attraction of the Moon. First, the Moon is assumed to lay on the ecliptic plane and periodic orbits and associated stable and unstable invariant manifolds are computed for various energy levels, in the neighborhood of a given resonance. Along each invariant manifold, the eccentricity increases naturally, achieving its maximum at the first intersection between them. This growth is, however, not sufficient to achieve reentry. By moving to a more realistic model, where the inclination of the Moon is taken into account, the problem becomes non-autonomous and the satellite is able to move along different energy levels. Under the ansatz of transversality of the stable and unstable manifolds in the autonomous case, checked numerically, Poincaré–Melnikov techniques are applied to show how the Arnold diffusion can be attained, by constructing a sequence of homoclinic orbits that connect invariant tori at different energy levels on the normally hyperbolic invariant manifold.

On Nested Central Configurations of the 3n Body Problem - Journal of Nonlinear Science

In this work, we consider the existence of (3, n)–crowns in the classical Newtonian 3n–body problem, which are central configurations formed by three groups of n bodies with the same mass within each group, located at the vertices of three concentric regular polygons. We consider the case with dihedral symmetry, called nested (3, n)–crowns, where the vertices of the polygons are aligned. We characterize the set of admissible radii for the polygons for which nested (3, n)–crowns exist. We conclude with numerical evidences that suggest uniqueness for each set of three masses.

Oscillatory Motions, Parabolic Orbits and Collision Orbits in the Planar Circular Restricted Three-Body Problem - Communications in Mathematical Physics

In this paper we consider the planar circular restricted three body problem (PCRTBP), which models the motion of a massless body under the attraction of other two bodies, the primaries, which describe circular orbits around their common center of mass. In a suitable system of coordinates, this is a two degrees of freedom Hamiltonian system. The orbits of this system are either defined for all (future or past) time or eventually go to collision with one of the primaries. For orbits defined for all time, Chazy provided a classification of all possible asymptotic behaviors, usually called final motions. By considering a sufficiently small mass ratio between the primaries, we analyze the interplay between collision orbits and various final motions and construct several types of dynamics. In particular, we show that orbits corresponding to any combination of past and future final motions can be created to pass arbitrarily close to the massive primary. Additionally, we construct arbitrarily large ejection-collision orbits (orbits which experience collision in both past and future times) and periodic orbits that are arbitrarily large and get arbitrarily close to the massive primary. Furthermore, we also establish oscillatory motions in both position and velocity, meaning that as time tends to infinity, the superior limit of the position or velocity is infinity while the inferior limit remains a real number.

On small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting - Inventiones mathematicae

Breathers are nontrivial time-periodic and spatially localized solutions of nonlinear dispersive partial differential equations (PDEs). Families of breathers have been found for certain integrable PDEs but are believed to be rare in non-integrable ones such as nonlinear Klein-Gordon equations. In this paper we show that small amplitude breathers of any temporal frequency do not exist for semilinear Klein-Gordon equations with generic analytic odd nonlinearities. A breather with small amplitude exists only when its temporal frequency is close to be resonant with the linear Klein-Gordon dispersion relation. Our main result is that, for such frequencies, we rigorously identify the leading order term in the exponentially small (with respect to the small amplitude) obstruction to the existence of small breathers in terms of the so-called Stokes constant, which depends on the nonlinearity analytically, but is independent of the frequency. This gives a rigorous justification of a formal asymptotic argument by Kruskal and Segur (Phys. Rev. Lett. 58(8):747, 1987) in the analysis of small breathers. We rely on the spatial dynamics approach where breathers can be seen as homoclinic orbits. The birth of such small homoclinics is analyzed via a singular perturbation setting where a Bogdanov-Takens type bifurcation is coupled to infinitely many rapidly oscillatory directions. The leading order term of the exponentially small splitting between the stable/unstable invariant manifolds is obtained through a careful analysis of the analytic continuation of their parameterizations. This requires the study of another limit equation in the complexified evolution variable, the so-called inner equation.

Don Estep (@DonEstep@mathstodon.xyz)

We have posted lectures of some of the proofs and examples in our book "A Ramble Through Probability: How I Learned to Stop Worrying and Love Measure Theory" on our vimeo page. https://vimeo.com/showcase/ramble?share=copy

4th Nonlinear Processes in Oceanic and Atmospheric Flows - Centre de Recerca Matemàtica

4th Nonlinear Processes in Oceanic and Atmospheric Flows Sign in Conference From January 22, 2025 to January 24, 2025 Dates: January 22-24, 2025 Location: Institut de Ciències del Mar (CSIC) Registration deadline 08 / 01 / 2025 REGISTRATION FEE 250€ Early Registration* 300€ Late Registration Students: 190€ Early Registration* 220€ Late Registration * Deadline for

Session NP6.1