Bifurcation, Stability and Applications - Part I of II
For Part II, see MS-Th-D-12

Date: August 13
Time: 10:00--12:00
Room: 208B

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Loginov, Boris (Ulyanovsk State Technical Univ.)

Abstract: In applications of bifurcation theory the situation arises when the finite-dimensional branching equation (BEqs) is potential, while the original nonlinear equation havení»t this property. Three articles are devoted to this phenomenon. Here sufficient conditions for BEq potentiality and pseudopotentiality are obtained, particularly under group symmetry conditions, when the bifurcation point has nontrivial stationary subgroup. For stationary and dynamic bifurcation problems general theorems are proved about the inheritance of the group symmetry of original nonlinear equation by the relevant Lyapounov and Schmidt BEqs moving along the trajectory of the branching point, taking into account the presence of stationary subgroup of the branching point. Theorems on the BEqs reduction (its order lowering) are proved at the action of continuous group symmetry, G-invariant implicit operators theorems are proved for stationary and dynamic bifurcation. Simple, but very technical examples of SO(2) and SH(2) symmetries are considered with the general form of C1-smooth BEq construction on allowed group symmetry. With the aid of Morse-Conley topological index theory it is proved the bifurcation existence theorem for Andronov-Hopf bifurcation. Sufficient conditions for the linearized stability of bifurcating solutions are obtained.
The obtained results are applied to bifurcation problems with E. Schmidt spectrum in the linearization.
Three communications are devoted to nonlinear equations, their solutions stability and bifurcation theory to problems of hydroaeroelasticity . One of them considers the multiparameter bifurcation problems on the divergence of the elongated plate in supersonic gas flow compressed or extended by external boundary conditions in the exact statement, that is achieved by the representation of the bifurcation manifold through the roots of the characteristic equation of the linearized ODE. Here the most difficulties arise at the analytical proof of the divergence absence. The Fredholm property of these problems is proved also on the base of the usage of the roots of characteristic equations of the linearization.
Lyapounov functions and functionals, Lyapounov vector- functions techniques is applied to the investigation of solutions stability in two reports to hydroaeroelasticity problems and two articles on the stabilization of nonlinear systems motions (with digital control and with aftereffect.)

Symmetry and potentiality in branching theory problems with stationary subgroup of branching point
Loginov, Boris (Ulyanovsk State Technical Univ.)
Konopleva, Irina (Ulyanovsk State Technical Univ.)

Bifurcation problem on Schmidt spectrum under group symmetry conditions
Loginov, Boris (Ulyanovsk State Technical Univ.)
Konopleva, Irina (Ulyanovsk State Technical Univ.)

Andronov-Hoph bifurcation in equations with symmetrizable operators
Kim-Tyan, Luiza (National Research Technological Univ. (Moscow Inst. of Steel & Alloys))

Multiparameter bifurcation problems on divergence of elongated plate in supersonic gas flow
Badokina, Tatyana (Mordovian State Univ. of N. P. Ogarev)
Loginov, Boris (Ulyanovsk State Technical Univ.)


Code: Type-Date-Time-Room No.
Type : IL=Invited Lecture, SL=Special Lectures, MS=Minisymposia, IM=Industrial Minisymposia, CP=Contributed Papers, PP=Posters
Date: Mo=Monday, Tu=Tuesday, We=Wednesday, Th=Thursday, Fr=Friday
Time : A=8:30-9:30, B=10:00-11:00, C=11:10-12:10, BC=10:00-12:10, D=13:30-15:30, E=16:00-18:00, F=19:00-20:00, G=12:10-13:30, H=15:30-16:00
Room No.: TBA