Occurrence of non-smooth bursting oscillations in a Filippov system with slow-varying periodic excitation

The main purpose of this paper is to explore the mechanism of the non-smooth bursting oscillations in a Filippov neuronal model with the coupling of two scales and to try to explain some special phenomena appearing on the attractors. Based on a typical Hindmarsh–Rose neuronal model, when the recovery variable of the slow current is replaced by a slow-varying periodic excitation, which means the exciting frequency is far less than the natural frequency, the coupling of two scales in frequency exists, leading to the non-smooth bursting oscillations. By regarding the whole exciting term as a slow-varying parameter, we can define the full subsystem as Filippov type, which appears in generalised autonomous form. Equilibrium branches and their bifurcations of the fast subsystem can be derived by varying the slow-varying parameter. With the increase of the exciting amplitude, different types of equilibrium branches and the bifurcations may involve the slow–fast vector field, which may cause qualitative change of the bursting attractors, resulting in several types of periodic non-smooth bursting oscillations. By employing the modified slow–fast analysis method, the mechanism of the bursting oscillations is presented upon overlapping the transformed phase and the equilibrium branches as well as their bifurcations of the generalised autonomous system. The sliding phenomenon in the bursting oscillations may occur since the governing system with different stable attractors may alternate between two subsystems located in two neighbouring regions divided by the boundary. Furthermore, the inertia of the movement along an equilibrium branch increases with the increase of the exciting amplitude, leading to the disappearance of the influence of the associated bifurcations on the attractors.