Bursting oscillations as well as the mechanism in a Filippov system with parametric and external excitations. (English) Zbl 1511.34013
Summary: The main purpose of this paper is to explore the bursting oscillations as well as the mechanism of a parametric and external excitation Filippov type system (PEEFS), in which different types of bursting oscillations such as fold/nonsmooth fold (NSF)/fold/NSF, fold/NSF/fold and fold/fold bursting oscillations can be observed. By employing the overlap of the transformed phase portrait and the equilibrium branches of the generalized autonomous system, the mechanisms of the bursting oscillations are investigated. Our results show that the fold bifurcation and the boundary equilibrium bifurcation (BEB) can cause the transitions between the quiescent states and repetitive spiking states. The oscillating frequencies of the spiking states can be approximated theoretically by their occurring mechanisms, which agree well with the numerical simulations. Furthermore, some nonsmooth evolutions are investigated by employing differential inclusions theory, which reveals that the positional relationship between the points of the trajectory interacting with the nonsmooth boundary and the related sliding boundary of the nonsmooth system may affect the nonsmooth evolutions.
MSC:
| 34A36 | Discontinuous ordinary differential equations |
| 34A60 | Ordinary differential inclusions |
| 34C25 | Periodic solutions to ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
Keywords:
Filippov type system; boundary equilibrium bifurcation; transformed phase portrait; nonsmooth evolution; differential inclusions theory; slidingReferences:
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