Modelling of physical systems with a Hopf bifurcation using mechanistic models and machine learning

Limit-cycle oscillations (LCOs) are periodic responses that can be observed in many systems such as aircraft wings [1], wheels [2], machine tools [3], and living cells [4, 5]. Self-excited systems are a common source of LCOs and are typically modelled using ordinary differential equations (ODEs) where the variation of a parameter beyond a critical value (a bifurcation point) triggers oscillations. Deriving a low-dimensional mathematical model that quantitatively captures the onset and amplitude of LCOs is usually a very challenging problem as self-excited systems are typically characterised by the interplay of several physical phenomena. Take, for example, the aforementioned fluid-structure and tyreground interactions in wings and wheels, respectively. For self-excited systems with a Hopf bifurcation, the change in parameter leads to a loss of stability of the equilibrium and the birth of a family of LCOs near the bifurcation point.