PhD: Characterisation of out-of-equilibrium phase transitions with the emergence of limit cycles

Systems held out of equilibrium by forcing often exhibit a rich phenomenology. Examples include the synchronisation transition, in which oscillators of different frequencies begin to oscillate in phase if their interactions are sufficiently strong, or the appearance of oscillations in highly dissipative systems, such as chemical clocks (Belousov-Zhabotinsky reaction). Such systems are archetypes of non-equilibrium statistical physics, which has developed methodologies adapted to their study. In this thesis, we develop theoretical tools to study out-of-equilibrium phase transitions with the emergence of a limit cycle, in the case where the microscopic degrees of freedom are not constrained to oscillate as in standard studies of the synchronisation transition but can lead to a form of oscillation in certain regimes.