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The frozen-in condition for a direction field, small denominators and chaotization of steady flows of a viscous liquid. (English. Russian original) Zbl 0937.37030
J. Appl. Math. Mech. 63, No. 2, 229-235 (1999); translation from Prikl. Mat. Mekh. 63, No. 2, 237-244 (1999).
The family of the integral curves is called frozen into the flow of system \(\dot x=V(x)\), \(x\in M\), if it passes into itself for all transformations \(g^t\). Let \(a\neq 0\) be a smooth vector field on \(M\). The author discusses the freezing-in criterion for the integral curves of the field \(a\) in the form \([a,v]=\lambda a\), where \([\;,\;]\) is the commutator of vector fields, \(\lambda\) is some smooth function on \(M\), as well as its relationship with the small denominators problem.

MSC:
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37N05 Dynamical systems in classical and celestial mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
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