Bursting oscillations with a codimension-2 non-smooth bifurcation in a piecewise-smooth system of Filippov type

Zhixiang Wang, Chun Zhang and Qinsheng Bi

Chaos, Solitons & Fractals, 2024, vol. 184, issue C

Abstract: This paper focus on the effects of a codimension-2 non-smooth bifurcation on bursting behaviors in piecewise-smooth systems. A passive circuit with a switched power source is slightly modified by introducing a periodic exciting voltage to establish an example system of Filippov-type. By using Filippov’s convex method, the sliding vector field is obtained, and the analytical solution of the sliding motion equation is derived. A codimension-2 non-smooth bifurcation, called “catastrophic boundary focus and catastrophic crossing-sliding bifurcation”, is observed, and the unfolding of the bifurcation is discussed. Based on the bifurcation analysis, five bursting oscillations associated with the codimension-2 bifurcation are observed, and the dynamical mechanism is revealed. The study suggests that the bifurcation of boundary equilibrium can be neither a non-smooth fold one nor a persistence one if the sliding vector field is degenerate, and this bifurcation may also lead to jumping behaviors in a bursting. A non-smooth limit cycle may cross the switching manifold transversely, precisely at the boundary of the escaping subregion, causing the limit cycle to disappear catastrophically. This bifurcation of non-smooth limit cycle controls the transition between a quiescent state and a spiking state in a bursting. A grazing-sliding bifurcation in a slow–fast system can form “reentry sliding structures” in a bursting.

Keywords: Piecewise-smooth system; Co-dimension-2 non-smooth bifurcation; Slow–fast analysis method; Bursting oscillations (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:184:y:2024:i:c:s0960077924005927

DOI: 10.1016/j.chaos.2024.115040

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