Final Report Summary - BREUDS (Brazilian-European partnership in Dynamical Systems) BREUDS is a research partnership between leading European and Brazilian research groups in dynamical systems, a prominent subject in mathematics. An extensive consortium of European and Brazilian institutions collaborates to provide world leading critical mass and support for research on the very forefront of the field. Work Packages reflect parallel priorities in the research. Transfer of knowledge is facilitated by conferences and five workshops. The project has excellent strategic value in view of the development of closer ties in higher education and research between the European Research Area and Brazil. While the research is predominantly abstract and theoretical in nature, the understanding of dynamical systems mathematics is essential to improve the quality of models and our ability to extract useful information from models and data. In a data-driven society with increasing importance of quantitative decision making, a good understanding of mathematical foundations is becoming increasingly practically relevant in the longer term. In recent years and decades important progress has been made to understand the (generic) dynamics of low- dimensional (intrinsically autonomous, i.e. time-independent) dynamical systems, leading to the identification of new challenges, including high-dimensional dynamical systems (including PDEs and other types of functional differential equations), dynamical systems with particular structure (such as Hamiltonian systems, systems with symmetry, and systems on networks), and intrinsically nonautonomous dynamical systems (including dynamical systems with random elements). This joint research programme between Europe and Brazil relies on strong traditions in dynamical systems at both ends, and complementary expertise that will drive innovation and resolve open problems at the forefront of modern dynamical systems research. Typically for the multifaceted field of dynamical systems, the identified work packages have many places of contact among each other, and thus naturally form part of one scientific community, inspiring and challenging each other in many ways. During the existence of this IRSES project, excellent progress has been made in the research projected for each of the five main Work Packages, concerning non-autonomous dynamical systems, ergodic theory, low-dimensional dynamics, Hamiltonian dynamics and bifurcation theory, in terms of their objectives, tasks, deliverables and milestones. BREUDS has supported internationally leading research in all these directions, as well a number of workshops and conferences were dissemination, transfer of knowledge and training has taken place. Research collaborations have already led to many publications in international peer reviewed journals. It also supported the work of many ESRs doing their PhD research. BREUDS has celebrated the 2014 Fields medal laureats Prof Martin Hairer (Warwick) and Prof Artur Avila (IMPA, CNRS) who are directly involved with and beneficiaries of this IRSES. For more information on BREUDS, please see the project website at (http://www.ma.ic.ac.uk/DynamIC/BREUDS).
