Bifurcation and stability in some problems of continua mechanics

Construction of asymptotics of periodic solutions and investigation of their stability for the problem about capillary-gravity waves on the free boundary of potential flow of the fluid in spatial layer over the flat bottom with generalized Bernoulli's law on free surface and the problem about interface of ferrofluid in magnetic field is considered. Bifurcation parameters are, correspondingly,the Froud number and the strength of magnetic field. For the construction and investigation of bifurcation equations group analysis methods in branching theory are used. The group transformations method allowing to reduce two-point boundary value problems (BVP) for ordinary differential equations to Cauchy problem is suggested. On its base the definition of small solutions of two-point BVP for ordinary differential equations of the fourth order describing some problems of hydroaeroelasticity. Stability of solutions for some initial BVP to partial integrodifferential equations,describing oscilations of viscoelastic plates and shells,accounting interaction with transonic flow is investigated.