Geodesic
Analysis of discontinuous bifurcations in nonsmooth dynamical systems
Russian journal of nonlinear dynamics, Tome 8 (2012) no. 2, pp. 231-247

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Dynamical systems with discontinuous right-hand sides are considered. It is well known that the trajectories of such systems are nonsmooth and the fundamental solution matrix is discontinuous. This implies the presence of the so-called discontinuous bifurcations, resulting in a discontinuous change in the multipliers. A method of stepwise smoothing is proposed allowing the reduction of discontinuous bifurcations to a sequence of typical bifurcations: saddle-node, period doubling and Hopf bifurcations. The results obtained are applied to the analysis of the well-known system with friction a block on the moving belt, which serves as a popular model for the description of selfexcited frictional oscillations of a brake shoe. Numerical techniques used in previous investigations of this model did not allow general conclusions to be drawn as to the presence of self-excited oscillations. The new method makes it possible to carry out a complete qualitative investigation of possible types of discontinuous bifurcations in this system and to point out the regions of parameters which correspond to stable periodic regimes.
Keywords: non-smooth dynamical systems, discontinuous bifurcations, oscillator with dry friction. 
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     author = {A. P. Ivanov},
     title = {Analysis of discontinuous bifurcations in nonsmooth dynamical systems},
     journal = {Russian journal of nonlinear dynamics},
     pages = {231--247},
     publisher = {mathdoc},
     volume = {8},
     number = {2},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ND_2012_8_2_a2/}
}
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A. P. Ivanov. Analysis of discontinuous bifurcations in nonsmooth dynamical systems. Russian journal of nonlinear dynamics, Tome 8 (2012) no. 2, pp. 231-247. http://geodesic.mathdoc.fr/item/ND_2012_8_2_a2/