Qualitative bifurcation analysis of a non-smooth friction-oscillator model
Originally, the equation of motion of the investigated system is \(\ddot x(t)+ x(t)+ a[\mu(1)+ \mu(\dot x(t)- 1)\cdot\text{sgn}(\dot x(t)- 1)]= \gamma\cos(\eta t)\), \(t\in\mathbb{R}\), \(a> 0\), \(\gamma> 0\), \(\eta> 0\). Here \(\mu\) is the friction coefficient taken as \(\mu(v)= {\mu_0- \mu_1\over 1+\lambda_0| v|}+ \mu_1+ \lambda_1| v|^2\), where \(\mu_0> \mu_1\). To avoid contradictions, the sign-function is replaced by the function \(\text{Sgn}(v)= \begin{cases} \{v/| v|\}, & v\neq 0\\ [-1,1], & v=0\end{cases}\), and the above differential equation can be reduced to the differential inclusion \(\ddot x(t)+ x(t)+ a\mu(1)\in \gamma\cos(\eta t)- a\mu(\dot x(t)- 1)\cdot\text{Sgn}(\dot x(t)- 1)\). For \(\dot x(t)\neq 1\), this inclusion coincides with the above differential equation, but for \(\dot x(t)= 1\) it only requires that \(\ddot x(t)+ x(t)+ a\mu(1)\in [-a\mu_0+ \mu\cos(\eta t), a\mu_0+ \gamma\cos(\eta t)]\). Solution of this inclusion is shown to be of class \(C^1\) with absolutely continuous \(\dot x\) and with locally integrable \(\ddot x\). The main reason of using the differential inclusion is the consideration of the stick-slip phenomenon. This paper gives a thorough numerical study of non-smooth bifurcations of this system, identifies a lot of new bifurcation structures, and describes some new phenomena. A large number of figures show the most important and interesting features.
scientific article; zbMATH DE number 1059767