Variation of the instability index on stationary curves of elliptic equations depending on a parameter

The set of solutions of the equation A(u, )=0 in the case of general position consists of smooth curves. Solutions of a quasilinear elliptic equation with large instability index are constructed and an estimate obtained for the number of turning points of the solution curves.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 12, pp. 47–58, 1987.     

Synthèse du mèthyl-1 diformyl-2,3 pyrrole,de la méthyl-1 pyrrolo[2,3-d] pyridazine et de la méthyl-1 oxo-6 (6h)cycloheptatrieno[b] pyrrole

This communication describes the synthesis of l-methyl-2,3-diformylpyrrole. This new compound is used to prepare a new heterocycle, l-methylcyclohepta[b]pyrrol-6-one and thus allows a new synthesis of l-methylpyrrolo[2,3-d]pyridazine.     

Instability criteria for steady flows of a perfect fluid

An instability criterion based on the positivity of a Lyapunov-type exponent is used to study the stability of the Euler equations governing the motion of an inviscid incompressible fluid. It is proved that any flow with exponential stretching of the fluid particles is unstable. In the case of an arbitrary axisymmetric steady integrable flow, a sufficient condition for instability is exhibited in terms of the curvature and the geodesic torsion of a stream line and the helicity of the flow.     

Stabilized Asymptotics of solutions to reaction-diffusion systems with a small parameter

For solutions of reaction-diffusion systems under Dirichlet or Neumann boundary conditions, having a small parameter ε as a coefficient to the time derivative of the first component, the principal term of the asymptotics with respect to ε is found for all t>0. This principal term is a solution of the system, obtained as a limit for ε=0, and has a finite number of discontinuities; the continuous parts, beginning from the second, are situated on finite-dimensional unstable manifolds passing through stationary points of the limit system. Bibliography: 4 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 17, pp. 128–152, 1994.     

Attractors of periodic processes and estimates of their dimension

This work was partially supported by grants from the Russian Fund for Basic Research and grant No. MR5000 from the International Scientific Fund.