An analytical approach to codimension-2 sliding bifurcations in the dry friction oscillator

In this paper, we analytically consider sliding bifurcations of periodic orbits in the dry-friction oscillator. The system depends on two parameters: $F$, which corresponds to the intensity of the friction, and $\omega$, the frequency of the forcing. We prove the existence of infinitely many codimension-2 bifurcation points and focus our attention on two of them: $A_1:=(\omega\ii, F)=(2, 1/3)$ and $B_1:=(\omega\ii, F)=(3,0)$. We derive analytic expressions in ($\omega\ii$, $F$) parameter space for the codimension-1 bifurcation curves that emanate from $A_1$ and $B_1$. Our results show excellent agreement with the numerical calculations of Kowalczyk and Piiroinen.