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Tomlinson’s model is often used to describe the friction of a single asperity or of a scanning force probe sliding over an atomic lattice. We present results on the complex dynamic behavior found in this model using a combination of continuation methods, perturbation techniques, and numerical simulations. Specifically, periodic stick-slip motions and their bifurcations and stability are investigated in the slow-sliding speed range and in higher speed ranges at which fundamental and parametric resonances set in. The results predict a complex range of bifurcations, superharmonic and subharmonic motions, and possibly chaotic dynamics which bear significant implications for understanding single-asperity friction or the dynamic response in friction force microscopy.