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    Lev Lerman

    10, Ulyanova Str. 603005 Nizhny Novgorod, Russia
    Research Institute of Applied Mathematics and Cybernetics Nizhny Novgorod State University


    Lerman L. M., Turaev D. V.
    We review results on local bifurcations in reversible systems (flows and diffeomorphisms) which lead to the creation of pairs attractor-repellor at bifurcations from symmetric equilibria (for flows) and fixed points (for diffeomorphisms). We consider bifurcations of co-dimension 1 in systems of small dimensions (2,3, and 4).
    Keywords: reversible system, reversible diffeomorphism, bifurcation, symmetric equilibrium, symmetric fixed point, loss of symmetry
    Citation: Lerman L. M., Turaev D. V.,  On symmetry breaking bifurcations in reversible systems, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 2, pp.  323-343
    Afraimovich V. S., Belyakov L. A., Bykov V. V., Gonchenko S. V., Lerman L. M., Lukyanov V. I., Malkin M. I., Morozov A. D., Turaev D. V.
    Leonid Pavlovich Shilnikov (17.12.1934–26.12.2011)
    2012, Vol. 8, No. 1, pp.  183-186
    Citation: Afraimovich V. S., Belyakov L. A., Bykov V. V., Gonchenko S. V., Lerman L. M., Lukyanov V. I., Malkin M. I., Morozov A. D., Turaev D. V.,  Leonid Pavlovich Shilnikov (17.12.1934–26.12.2011), Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp.  183-186
    Lerman L. M., Slinyakova N. A.
    In this paper we study the piecewise-linear model of the stationary Swift—Hohenberg equation well known in mathematical physics, which provides explicit front type solutions. Due to the reversibility relative to two involutions of the corresponding Hamiltonian system, this involves the existence of a heteroclinic contour connecting two saddle-foci. Using methods of symbolic dynamics, we give a description of all solutions lying in the neighborhood of the contour at the level of the Hamiltonian containing the contour.
    Keywords: Swift–Hohenberg equation, fronts, heteroclinic contour, Hamiltonian system, saddle focus, symbolic dynamics
    Citation: Lerman L. M., Slinyakova N. A.,  On the dynamics of the piecewise-linear model of the Swift–Hohenberg equation, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp.  569-583

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