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    Analysis of discontinuous bifurcations in nonsmooth dynamical systems

    2012, Vol. 8, No. 2, pp.  231-247

    Author(s): Ivanov A. P.

    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
    Citation: Ivanov A. P., Analysis of discontinuous bifurcations in nonsmooth dynamical systems, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 2, pp.  231-247

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