Resonant normal forms as constrained linear systems

Details Resonant normal forms as constrained linear systems Resonant normal forms as constrained linear systems

2001-06-19

arxiv.org/abs/math-ph/0106017v2

Mod. Phys. Lett. A 17 (2002), 583-597

Physics

Mathematical Physics

15 pages; revised version (with revised title)

Scientific paper

10.1142/S0217732302006825

We show that a nonlinear dynamical system in Poincare'-Dulac normal form (in $\R^n$) can be seen as a constrained linear system; the constraints are given by the resonance conditions satisfied by the spectrum of (the linear part of) the system and identify a naturally invariant manifold for the flow of the ``parent'' linear system. The parent system is finite dimensional if the spectrum satisfies only a finite number of resonance conditions, as implied e.g. by the Poincare' condition. In this case our result can be used to integrate resonant normal forms, and sheds light on the geometry behind the classical integration method of Horn, Lyapounov and Dulac.

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