Elena Fantino (Khalifa University of Science Technology, United Arab Emirates)

Gerard Gómez (Universitat de Barcelona)

The course provides fundamentals of spacecraft trajectory design near the  libration points based on dynamical systems theory and associated numerical tools. After illustrating the characteristics of the main dynamical models, i.e., the two-body problem, the circular restricted three-body problem (including solar radiation pressure), Hill's model, the bicircular and quasi-bicircular four-body model and the full n-body Solar System model, the focus will be on the description of the dynamics in the vicinity of the libration points of the circular restricted three-body problem: equilibrium points, periodic orbits, quasi-periodic orbits, and invariant stable and unstable manifolds. The Lindstedt-Poincaré procedure to approximate periodic and quasi-periodic solutions will be illustrated in detail, also with the aid of numerical examples and exercises for the students. The last part of the course is an overview of past and future missions whose trajectories have been designed with this theory: from Earth-Moon transfers to missions to the libration points of the Sun-Earth system and the low-energy tours of the Galilean moons.

Enrique Ponce (Universidad de Sevilla, Spain)

Piece-wise smooth differential systems are needed nowadays both as accurate models of real engineering devices and as global reasonable models of different branches in bio-sciences. In most cases, (possibly discontinuous) piece-wise linear systems (PWL systems, for short) are enough to cope with all the problems of interest. We will review the existing theory of PWL systems in a low-dimensional context (2D/3D), putting the emphasis on the more relevant bifurcations and illustrating their usefulness in the analysis of realistic applications.

Anthony Quas (University of Victoria, Canada)

The well-known Birkhoff ergodic theorem concerns the behaviour of sums of a function along an orbit of a dynamical system. This idea first arose in kinetic theory, but the ideas have proved very fruitful in many parts of mathematics (geometry, information theory, probability, analytic and algebraic number theory to name a few). Multiplicative ergodic theory is concerned with multiplying matrices along the orbit of a dynamical system. One is given a (discrete) dynamical system T:X→X and a matrix-valued function A:X→Mdxd(R). Starting from an initial point, x0, one forms the orbit (xn) by xn=T(xn-1). In multiplicative ergodic theory, we study the behaviour of the matrix products A(xn-1) ··· A(x0). This leads to the critical idea of Lyapunov exponents and underlies the study of differential dynamical systems.
In this course, we will with an outline of ergodic theory, including some key examples. We will then recall the singular value decomposition from linear algebra, and will give a proof of the multiplicative ergodic theorem, with a view to operator-valued extensions. We will also address stability of the Lyapunov exponents along the way, and briefly describe applications to environmental dynamical systems.