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Abstract: We study the normal forms for incompressible flows and maps in theneighborhood of an equilibrium or fixed point with a triple eigenvalue. Weprove that when a divergence free vector field in $\mathbb{R}^3$ has nilpotentlinearization with maximal Jordan block then, to arbitrary degree, coordinatescan be chosen so that the nonlinear terms occur as a single function of twovariables in the third component. The analogue for volume-preservingdiffeomorphisms gives an optimal normal form in which the truncation of thenormal form at any degree gives an exactly volume-preserving map whose inverseis also polynomial inverse with the same degree.



Author: H.R. Dullin, J.D. Meiss

Source: https://arxiv.org/







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