The main objective of this project is to create fundamental understanding in dynamical systems theory and to apply this theory in formulating and analyzing real world models met especially in Neuroscience, Plasma Physics and Medicine. The specific objectives, tasks and methodology of this proposal are contained in the 5 WPs of the project. In WP1 we want to develop new methods for the center and isochronicity problems for analytic and non-analytic systems, study bifurcations of limit cycles and critical periods, including time-reversible systems with perturbations, and investigate reaction-diffusion and fractional differential equations. In WP2 we deal with the problem of integrability for some differential systems with invariant algebraic curves, classification of cubic systems with a given number of invariant lines, study global attractors of almost periodic dynamical systems and their topological structure, respectively, Levitan/Bohr almost periodic motions of differential/difference equations. The main objective of WP3 is to study dynamics of some classes of continuous and discontinuous vector fields, preserving, respectively, breaking some symmetries, study of their singularities and closed orbits for classes of piecewise linear vector fields. WP4 deals with Hamiltonian systems in Plasma Physics, twist and non-twist area preserving maps, further studies of a recent model proposed to study some phenomena occurring in the process of plasma’s fusion in Tokamaks, numerical methods, and the study of symmetries of certain kinds of k-cosymplectic Hamiltonians. The last WP tackles mathematical models in Neuroscience and Medicine. Firstly, we study several ODE-based and map-based neuronal models, survey in vivo results with respect to Autism Spectrum Disorder (ASD) and propose a model for ASD. Secondly, we study several approaches to mathematical models for diabetes. Finally, bone remodeling by means of convection-diffusion-reaction equations is our last task.