A Stiction Oscillator under Slowly Varying Forcing: Uncovering Small Scale Phenomena using Blowup\ast

In this paper, we analyze a mass-spring-friction oscillator with the friction modeled by a regularized stiction model. We do so in the limit where the ratio of the natural spring frequency and the forcing frequency is on the same order of magnitude as the scale associated with the regularized stiction model. The motivation for studying this special parameter regime (which can be interpreted as a rigid body limit) comes from [E. Bossolini, M. Brons, and K. U. Kristiansen, SIAM J. Appl. Dyn. Syst., 16, (2017), pp. 2233--2258] which demonstrated new friction phenomena in this regime. The results of this paper led to some open problems; see also [E. Bossolini, M. Brons, and K. U. Kristiansen, SIAM Rev., 62 (2020), pp. 869--897], that we resolve in this paper. In particular, using GSPT and blowup [C. K. R. T. Jones, in Dynamical Systems, Lecture Notes in Math. 1609, Springer, Berlin, 1995, pp. 44--118; M. Krupa and P. Szmolyan, SIAM J. Math. Anal., 33, (2001), pp. 286--314], we provide a simple geometric description of the bifurcation of stick-slip limit cycles through a combination of a canard and a global return mechanism. We also show that this combination leads to a canard-based horseshoe and are therefore able to prove existence of chaos in this fundamental oscillator system.

Automated summary

This paper analyzes a mass-spring-friction oscillator in a special parameter regime, showcasing new friction phenomena, resolving some open problems, and proving the existence of chaos in the fundamental oscillator system.