Abstract

This paper presents results of the investigation of bifurcations of stationary solutions of the Swift--Hohenberg equation and dynamic descent to the points of minimal values of the functional of energy for this equation, obtained with the use of the modification of the Lyapunov--Schmidt variation method and some methods from the theory of singularities of smooth functions. Nonstationary case is investigated by the construction of paths of descent along the trajectories of the infinite-dimensional SH dynamical system from arbitrary initial states to points of the minimum energy.

Keywords: Lyapunov--Schmidt variation method, Swift--Hohen\-berg equation, bifurcation analysis.

MSC: 37M20, 35Q99, 34K18, 34C25

Pages:  26$-$39     

Volume  70 ,  Issue  1 ,  2018