Attractor and Vector Structure Analyses of Bursting Oscillation with Sliding Bifurcation in Filippov Systems

Abstract

The main purpose of this paper is to investigate the mechanism of sliding phenomenon in Filippov (nonsmooth) dynamical systems by attractor analysis and vector analysis. A corresponding simple model based on Chua’s circuit with periodic excitation was introduced as an example. The attractor analysis proposed in our previous work is used to discuss the complicated oscillations of the Filippov system. However, it failed to perfectly explain the sliding phenomena and establish an analytical method of constant voltage control. Therefore, the geometric structure and analytic conditions of sliding bifurcations in the general n-dimensional piecewise smooth system are discussed in detail by vector structure analysis. The prospects of practical application of this method are also discussed in the end.

Non-Smooth Dynamic Behaviors as well as the Generation Mechanisms in a Modified Filippov-Type Chua’s Circuit with a Low-Frequency External Excitation

“…The circuit as well as the various modifications are often taken as examples in the research on slow-fast dynamics. For instance, canards and chaotic bursting can be revealed in memristor-based Chua's circuits [26], or more to the point, those modified Chua's circuits with low-frequency excitations are employed for the investigation of various bursting oscillations as well as the generation mechanisms via slow-fast decomposition [27][28][29].…”

Bursting Oscillations and Sliding Motion in Permanent Magnet Synchronous Motor Systems with Friction Factor

“…It can be concluded that the vector field direction on each side of nonsmooth boundary is opposite in the sliding region, while the direction remains the same in the traversing region. e analytic expressions of the sliding region and the traversing region which were reported in our previous work [13] are omitted in this paper. 12(b)).…”

“…To be specific, the studies involving different types of oscillations resulting from external factors in PMSM system are still relatively lacking. Besides, the great majority of current researches on oscillations are still limited in smooth system [10,11], which indicates that the possible effects resulting from the introduction of nonsmooth factor [12,13] are still not clear. In engineering practice, the vibration with friction corresponding to the oscillation with nonsmooth bifurcation may aggravate the equipment damage.…”

Slow–Fast Dynamics Behaviors under the Comprehensive Effect of Rest Spike Bistability and Timescale Difference in a Filippov Slow–Fast Modified Chua’s Circuit Model

“…Riding on the nonconventional dynamics theories, scholars have proved that many nonconventional bifurcations may also affect the transition mechanisms of slow-fast dynamic behaviors, for instance, nonsmooth fold bifurcation and boundary homoclinic bifurcation can also cause the alternations between QSs and SPs [30][31][32]. Sliding bifurcations lead to unique sliding structures in SPs, although they do not result in transitions between QSs and SPs [33]. In [34], it was proved that boundary homoclinic bifurcation is a high codimension nonconventional bifurcation, which may result in different nonsmooth bursting patterns under different local structures of pseudoequilibrium.…”