An analytic form of differential equations

Normal forms of vector fields on Poisson manifolds

“…Example 3.4. g = sl (2). In this case, a Cartan subalgebra h of g is of dimension 1 and there are only two roots {α, −α}.…”

“…Using the same analytical tools as in the proof of Bruno's theorems about linearization of analytic vector fields [2,3], we will show the following theorem: Theorem 2.7. Let (Π, X) be an analytic homogeneous Poisson structure on (K n , 0) (where K is C or R) such that the linear part Π 1 of Π corresponds to a semisimple Lie algebra.…”

“…the vector field X is supposed to be analytic on (K n , 0). In order to show that the algorithm given in the previous subsection leads to a local analytic linearization, in addition to the nonresonance condition we will need a Diophantine-type condition, similar to Bruno's ω-condition for the analytic linearization of vector fields [2,3]. Keeping the notations of the previous subsection, for each d ≥ 1, put…”

“…Here we choose to work with a modified Bruno's ω-condition [2,3] adapted to our case. See Definition 2.5 for the precise definition of our ω-condition.…”

“…Repeating the above process, with the help of the time-1 flow ϕ (2) = exp X {G 1,1 (2) ,F 1 } of the Hamiltonian vector field of {G 1,1 (2) , F 1 }, we can replace 15) and so on. Since F 1 is "nilpotent", by iterating the above process a finite number of times, we can replace H 1,1 by G 1,1 (1) , i.e.…”