Neimark-Sacker Bifurcations Near Degenerate Grazing Point in a Two Degree-of-Freedom Impact Oscillator

Abstract

Saddle-node or period-doubling bifurcations of the near-grazing impact periodic motions have been extensively studied in the impact oscillators, but the near-grazing Neimark-Sacker bifurcations have not been discussed yet. For the first time, this paper uncovers the novel dynamic behavior of Neimark-Sacker bifurcations, which can appear in a small neighborhood of the degenerate grazing point in a two degree-of-freedom impact oscillator. The higher order discontinuity mapping technique is used to determine the degenerate grazing point. Then, shooting method is applied to obtain the one-parameter continuation of the elementary impact periodic motion near degenerate grazing point and the peculiar phenomena of Neimark-Sacker bifurcations are revealed consequently. A two-parameter continuation is presented to illustrate the relationship between the observed Neimark-Sacker bifurcations and degenerate grazing point. New features that differ from the reported situations in literature can be found. Finally, the observed Neimark-Sacker bifurcation is verified by checking the existence and stability conditions in line with the generic theory of Neimark-Sacker bifurcation. The unstable bifurcating quasi-periodic motion is numerically demonstrated on the Poincaré section.

Multistability in a quasiperiodically forced piecewise smooth dynamical system

“…Up to now, most of the studies on SNAs has been confined to smooth dynamic systems. The nonsmooth systems have abundant nontypical dynamics [28,29]. It has been found that SNAs can also be generated in the nonsmooth system in several past works [30,31,32,33].…”

Dynamic Behaviors of a Two‐Degree‐of‐Freedom Impact Oscillator with Two‐Sided Constraints

“…As a special bifurcation of nonsmooth systems, the grazing bifurcation is a qualitative transition between impact and nonimpact motion [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]. Nordmark [31,32] studied the dynamic behavior near the grazing trajectory and a nonlinear mapping is obtained.…”

“…e existence condition and stability of the grazing periodic orbit are derived in a two degree-of-freedom vibroimpact system with one-sided constraint in [36]. Yin et al [37][38][39][40][41][42] analyzed and measured the degenerate grazing bifurcations of the impact systems.…”