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We prove that all Galerkin truncations of the 2d stochastic Navier-Stokesequations in vorticity form on any rectangular torus subjected to hypoelliptic,additive stochastic forcing are chaotic at sufficiently small viscosity,provided the frequency truncation satisfies $N\geq 392$. By "chaotic" we meanhaving a strictly positive Lyapunov exponent, i.e. almost-sure asymptoticexponential growth of the derivative with respect to generic initialconditions. A sufficient condition for such results was derived in previousjoint work with Alex Blumenthal which reduces the question to thenon-degeneracy of a matrix Lie algebra implying H\"ormander's condition for theMarkov process lifted to the sphere bundle (projective hypoellipticity). Thepurpose of this work is to reformulate this condition to be more amenable forGalerkin truncations of PDEs and then to verify this condition using a) areduction to genericity properties of a diagonal sub-algebra inspired by theroot space decomposition of semi-simple Lie algebras and b) computationalalgebraic geometry executed by Maple in exact rational arithmetic. Note thateven though we use a computer assisted proof, the result is valid for allaspect ratios and all sufficiently high dimensional truncations; in fact,certain steps simplify in the formal infinite dimensional limit.

Chaos in stochastic 2d Galerkin-Navier-Stokes