Poles Distribution of PVI Transcendents close to a Critical Point (summer 2011)

The distribution of the poles of branches of the Painleve' VI transcendents associated to semi-simple Frobenius manifolds is determined close to a critical point. It is shown that the poles accumulate at the critical point, asymptotically along two rays. The example of the Frobenius manifold given by the quantum cohomology of the two-dimensional complex projective space is also considered.Comment: 35 pages, 10 figures; Physica D (2012

Notes on non-generic isomonodromy deformations

Some of the main results of [Cotti G., Dubrovin B., Guzzetti D., Duke Math. J., to appear, arXiv:1706.04808], concerning non-generic isomonodromy deformations of a certain linear differential system with irregular singularity and coalescing eigenvalues, are reviewed from the point of view of Pfaffian systems, making a distinction between weak and strong isomonodromic deformations. Such distinction has a counterpart in the case of Fuchsian systems, which is well known as Schlesinger and non-Schlesinger deformations, reviewed in Appendix A

Isomonodromic Laplace Transform with Coalescing Eigenvalues and Confluence of Fuchsian Singularities

We consider a Pfaffian system expressing isomonodromy of an irregular system of Okubo type, depending on complex deformation parameters u=(u_1,...,u_n), which are eigenvalues of the leading matrix at the irregular singuilarity. At the same time, we consider a Pfaffian system of non-normalized Schlesinger type expressing isomonodromy of a Fuchsian system, whose poles are the deformation parameters u_1,...,u_n. The parameters vary in a polydisc containing a coalescence locus for the eigenvalues of the leading matrix of the irregular system, corresponding to confluence of the Fuchsian singularities. We construct isomonodromic selected and singular vector solutions of the Fuchsian Pfaffian system together with their isomonodromic connection coefficients, so extending a result of references [4] and [20] to the isomonodromic case, including confluence of singularities. Then, we introduce an isomonodromic Laplace transform of the selected and singular vector solutions, allowing to obtain isomonodromic fundamental solutions for the irregular system, and their Stokes matrices expressed in terms of connection coefficients. These facts, in addition to extending [4] and [20] to the isomonodromic case with coalescences/confluences, allow to prove by means of Laplace transform the main result of reference [11], which is the analytic theory of non-generic isomonodromic deformations of the irregular system with coalescing eigenvalues.Comment: 57 pages, 4 figure

Isomonodromic deformations along a stratum of the coalescence locus

We consider deformations of a differential system with Poincaré rank 1 at infinity and Fuch-sian singularity at zero along a stratum of a coalescence locus. We give necessary and sufficient conditions for the deformation to be strongly isomonodromic, both as an explicit Pfaffian system (integrable de- formation) and as a non linear system of PDEs on the residue matrix A at the Fuchsian singularity. This construction is complementary to that of [13]. For the specific system here considered, the results generalize those of [26], by giving up the generic conditions, and those of [3], by giving up the Lidskii generic assumption. The importance of the case here considered originates form its applications in the study of strata of Dubrovin-Frobenius manifolds and F -manifolds

Coupled orbit-attitude mission design in the circular restricted three-body problem

Trajectory design increasingly leverages multi-body dynamical structures that are based on an understanding of various types of orbits in the Circular Restricted Three-Body Problem (CR3BP). Given the more complex dynamical environment, mission applications may also benefit from deeper insight into the attitude motion. In this investigation, the attitude dynamics are coupled with the trajectories in the CR3BP. In a highly sensitive dynamical model, such as the orbit-attitude CR3BP, periodic solutions allow delineation of the fundamental dynamical structures. Periodic solutions are also a subset of motions that are bounded over an infinite time-span (assuming no perturbing factors), without the necessity to integrate over an infinite time interval. Euler equations of motion and quaternion kinematics describe the rotational behavior of the spacecraft, whereas the translation of the center of mass is modeled in the CR3BP equations. A multiple shooting and continuation procedure are employed to target orbit-attitude periodic solutions in this model. Application of Floquet theory, Poincaré mapping, and grid search to identify initial guesses for the targeting algorithm is described. In the Earth-Moon system, representative scenarios are explored for axisymmetric vehicles with various inertia characteristics, assuming that the vehicles move along Lyapunov, halo as well as distant retrograde orbits. A rich structure of possible periodic behaviors appears to pervade the solution space in the coupled problem. The stability analysis of the attitude dynamics for the selected families is included. Among the computed solutions, marginally stable and slowly diverging rotational behaviors exist and may offer interesting mission applications. Additionally, the solar radiation pressure is included and a fully coupled orbit-attitude model is developed. With specific application to solar sails, various guidance algorithms are explored to direct the spacecraft along a desired path, when the mutual interaction between orbit and attitude dynamics is considered. Each strategy implements a different form of control input, ranging from instantaneous reorientation of the sail pointing direction to the application of control torques, and it is demonstrated within a simple station keeping scenario

Asymptotic solutions for linear ODEs with not-necessarily meromorphic coefficients: a Levinson type theorem on complex domains, and applications

In this paper, we consider systems of linear ordinary differential equations, with analytic coefficients on big sectorial domains, which are asymptotically diagonal for large values of $|z|$. Inspired by N. Levinson's work [Lev48], we introduce two conditions on the dominant diagonal term (the $L$-$condition$) and on the perturbation term (the $good\,\,decay\,\,condition$) of the coefficients of the system, respectively. Under these conditions, we show the existence and uniqueness, on big sectorial domains, of an $asymptotic$ fundamental matrix solution, i.e. asymptotically equivalent (for large $|z|$) to a fundamental system of solutions of the unperturbed diagonal system. Moreover, a refinement (in the case of subdominant solutions) and a generalization (in the case of systems depending on parameters) of this result are given. As a first application, we address the study of a class of ODEs with not-necessarily meromorphic coefficients. We provide sufficient conditions on the coefficients ensuring the existence and uniqueness of an asymptotic fundamental system of solutions, and we give an explicit description of the maximal sectors of validity for such an asymptotics. Furthermore, we also focus on distinguished examples in this class of ODEs arising in the context of open conjectures in Mathematical Physics relating Integrable Quantum Field Theories and affine opers ($ODE/IM\,\,correspondence$). Our results fill two significant gaps in the mathematical literature pertaining to these conjectural relations. As a second application, we consider the classical case of ODEs with meromorphic coefficients. Under an $adequateness$ condition on the coefficients, we show that our results reproduce (with a shorter proof) the main asymptotic existence theorems of Y. Sibuya [Sib62, Sib68] and W. Wasow [Was65] in their optimal refinements.Comment: 43 pages, 7 figure