10, Ulyanova Str. 603005 Nizhny Novgorod, Russia
Research Institute of Applied Mathematics and Cybernetics Nizhny Novgorod State University
Lerman L. M., Turaev D. V.
On symmetry breaking bifurcations in reversible systems
2012, Vol. 8, No. 2, pp. 323-343
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).
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
Lerman L. M., Slinyakova N. A.
On the dynamics of the piecewise-linear model of the Swift–Hohenberg equation
2009, Vol. 5, No. 4, pp. 569-583
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.