School of Mathematical and Computational Sciences, Massey University（梅西大学数学与计算科学学院）；School of Mathematics and Statistics, University College Dublin（都柏林大学学院数学与统计学院）
In vibro-impact mechanics, the division between an impact and a near miss is a zero-velocity grazing event. Grazing bifurcations of stable periodic motions often produce complicated attractors when grazing generates a square-root term in the Poincaré map. This paper concerns codimension-two scenarios for which the square-root term vanishes in some iterate of the Poincaré map. For forced one-degree-of-freedom oscillators, this occurs when the forcing frequency is a certain rational multiple of the damped natural frequency, i.e., the system is in resonance. In two-parameter bifurcation diagrams, curves of saddle-node and period-doubling bifurcations of single-impact periodic motions emanate from the codimension-two points. In this paper we prove these curves are quadratically tangent to the curve of grazing bifurcations, and derive explicit formulas for their quadratic coefficients. This is achieved by modifying the Poincaré map in a way that circumvents the square-root singularity, enabling us to use the implicit function theorem to demonstrate smoothness and perform asymptotic calculations of the saddle-node and period-doubling bifurcation curves. In doing so we resolve a long-standing conjecture on the admissibility of single-impact periodic motions by supplementing raw asymptotic computations with geometric and topological arguments. We illustrate the results with a linear impact oscillator model, matching the theoretical unfolding to numerically computed bifurcation curves. The results explain why previously reported physical experiments reveal an absence of chaos shortly past the grazing bifurcation.
Bifurcations of grazing loops of arbitrary tangent multiplicity in piecewise-smooth systems
School of Mathematics, Sichuan University（四川大学数学学院）；School of Mathematics, China University of Mining and Technology（中国矿业大学数学学院）；School of Mathematics, Southwestern University of Finance and Economics（西南财经大学数学学院）
