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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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