On Symmetric Periodic Motions With Different Excitation Periods in a Discontinuous System With a Hyperbolic Boundary

This paper studies periodic motions in a discontinuous dynamic system with three vector fields switching at two hyperbolic curve boundaries. The switchability conditions of flows at the two hyperbolic curve boundaries are presented through G-functions. The passible motions and grazing motions at the boundary plus sliding motions on the boundary are presented. Periodic motions with specific mapping structures are analytically predicted, and the corresponding stability and bifurcation are presented through eigenvalue analysis. Numerical simulations of periodic motions are carried out.  The corresponding G-functions are presented for illustration of motion switchability at the hyperbolic boundaries.

The discontinuous systems exist extensively in mechanical and control systems. The early discussion on discontinuous dynamical systems was from den Hartog [1] in 1931. He discussed a periodically driven oscillator with coulomb friction and viscous damping. Such an oscillator is a discontinuous dynamical system. One tried to use the methods for continuous dynamical systems to get approximate solutions for such discontinuous dynamical systems. In 1960, Levitan [2] further studied the frictional, mass-spring-damper oscillator, and the frictional force was approximated by a Fourier series. Once the frictional force is expressed by the Fourier series, the traditional analysis can be applied to such a discontinuous dynamical system. However, such approximate solutions cannot find the generic characteristics caused by the discontinuity in such discontinuous dynamical systems. Thus, in 1964, Filippov [3] considered mathematically the dynamics of the aforementioned frictional oscillator, and a theory for differential equations with the discontinuous right-hand side was presented. The comprehensive discussion on the theory of differential equations with discontinuous vector fields was presented in Filippov [4]. In 1974, Aizerman and Pyatnisky [5,6] presented a review on the Filippov theory of discontinuous dynamical systems and the initial sliding mode controls. From the Filippov theory, in 1977, Utkin [7] gave a survey on sliding mode controls along the boundary in discontinuous systems. The Filippov theory based on the convex function cannot be used for sliding motions in discontinuous dynamical systems.

In 2005, Luo [8] developed a singularity theory for discontinuous dynamical systems at the discontinuous boundary. In 2006, Luo and Gegg [9] applied the local singularity theory to the frictional oscillator. In 2006, Luo and Gegg [10] discussed a periodically forced frictional oscillator on an oscillating belt. The discontinuous boundary for such a frictional oscillator is time-varying. The Filippov theory cannot be applied to such a frictional oscillator. The local singularity theory presented in Luo [8] cannot be applied curved surfaces. In 2008, Luo [11] developed a generalized theory for flow switchability in discontinuous dynamical systems. The G-functions was introduced for determining the switchability of flows at the boundary. The corresponding switching bifurcations including sliding and grazing bifurcations were developed. In 2009, Luo and Rapp [12] applied the generalized theory of flow switchability to discuss the sliding motions along the discontinuous boundary in discontinuous dynamic system. All achieved results were summarized in Luo [13] on discontinuous dynamical systems on time-varying domains. In 2010, Luo and Rapp [14] discussed a discontinuous dynamic system with a parabolic boundary, and the analytical conditions of the flow on the parabolic boundary were developed. The stable and unstable periodic motions in such a discontinuous dynamical system were predicted analytically. In 2012, the discontinuous dynamical systems with multivalued vector fields and flow barrier vector fields were discussed in Luo [15]. Such a theory for discontinuous dynamical systems was applied to the dynamical system synchronization. In 2017, Luo and Huang [16] used the multivalued vector fields in a domain to discuss periodic motions in a generalized billiard system. In 2016, Li and Luo [17] initially discussed the analytical conditions of flow switchability at the hyperbolic boundary in a discontinuous dynamical system, and the corresponding periodic motions for such a discontinuous system was numerically simulated. Herein, the motion complexity in such a discontinuous dynamical system with the hyperbolic boundary will be discussed.