To simulate the self excited torsional vibrations of rotating drill strings (DSs) in vertical bore-holes, the nonlinear wave models of homogeneous and sectional torsional pendulums are formulated. The stated problem is shown to be of singularly perturbed type because the coefficient appearing before the second derivative of the constitutive nonlinear differential equation is small. The diapasons ${omega}_bleq{omega}leq{omega}_l$ of angular velocity ${omega}$ of the DS rotation are found, where the torsional auto-oscillations (of limit cycles) of the DS bit are generated. The variation of the limit cycle states, i.e. birth (${omega}={omega}_b$), evolution (${omega}_b<{omega}<{omega}_l$) and loss (${omega}={omega}_l$), with the increase in angular velocity ${omega}$ is analyzed. It is observed that firstly, at birth state of bifurcation of the limit cycle, the auto-oscillation generated proceeds in the regime of fast and slow motions (multiscale motion) with very small amplitude and it has a relaxation mode with nearly discontinuous angular velocities of elastic twisting. The vibration amplitude increases as ${omega}$ increases, and then it decreases as ${omega}$ approaches ${omega}_l$. Sectional drill strings are also considered, and the conditions of the solution at the point of the upper and lower section joints are deduced. Besides, the peculiarities of the auto-oscillations of the sectional DSs are discussed.