Sliding Bifurcation in Planar Piecewise Smooth Systems with Two Zones Separated by an Ellipse

The objective of this paper is to study the sliding bifurcation in a planar piecewise smooth system with an elliptic switching curve. Some new phenomena are observed, such as a crossing limit cycle containing four intersections with the switching curve, sliding cycles having four sliding segments, and sliding cycles consisting of the entire switching curve. Firstly, we investigate the bifurcation of sliding cycle from a sliding heteroclinic connection to two cusps and show the appearance of one sliding cycle with two folds. To plot the bifurcation diagram, a planar piecewise linear system with two zones separated by an ellipse are considered. Moreover, we study in more detail the unfolding of a sliding cycle connecting four cusps by exhibiting its complete bifurcation diagram. More precisely, we explore the necessary and sufficient conditions for the existence of limit cycles and derive the concrete bifurcation curves. Additionally, a simple piecewise smooth system with nonlinear subsystems is studied, which shows the possibility of the existence of two nested limit cycles. Finally, numerical simulations are given to confirm the theoretical analysis.

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

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.

Bifurcation Analysis of Stick-Slip Motion of the Vibration-Driven System with Dry Friction

This paper is concerned with the vibration-driven system which can move due to the periodic motion of the internal mass and the dry friction; the system can be modeled as Filippov system and has the property of stick-slip motion. Different periodic solutions of stick-slip motion can be analyzed through sliding bifurcation, two-parameter numerical continuation for sliding bifurcation is carried out to get the different bifurcation curves, and the bifurcation curves divide the parameters plane into different regions which stand for different stick-slip motion of the periodic solution. Furthermore, continuations with additional condition v=0 are carried out for the directional control of the vibration-driven system in one period; the curves divide the parameter plane into different progressions.

Grazing-Sliding Bifurcations Creating Infinitely Many Attractors

As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation. Near grazing-sliding bifurcations, structurally stable dynamics are captured by piecewise-linear continuous maps. Recently it was shown that maps of this class can have infinitely many asymptotically stable periodic solutions of a simple type. Here this result is used to show that at a grazing-sliding bifurcation an asymptotically stable periodic orbit can bifurcate into infinitely many asymptotically stable periodic orbits. For an abstract ODE system the periodic orbits are continued numerically revealing subsequent bifurcations at which they are destroyed.

Bifurcation Analysis of Planar Piecewise Linear System with Different Dynamics

In this paper, we investigate the bifurcation phenomena of a planar piecewise linear system. This piecewise linear system comprises two linear subsystems. The two linear subsystems have different types of dynamics. One subsystem has node or saddle dynamic and the other has focus dynamic. Some sufficient and necessary conditions for the existence of periodic orbit are given by studying the properties of Poincaré maps. Our results show that two crossing periodic orbits can bifurcate from this piecewise linear system. Moreover, we establish some sufficient and necessary conditions for the existence of sliding periodic orbit, crossing–sliding periodic orbit and sliding homoclinic orbit passing through a pseudo saddle and so on. We find that this piecewise system can appear multiply as two limit cycle bifurcation, buckling bifurcation, critical crossing cycle bifurcation, sliding homoclinic bifurcation, pseudo homoclinic bifurcation and so on. To our knowledge, sliding bifurcation phenomena are usually ignored when people study piecewise linear systems.