コロキウム

We consider systems of differential equations with equilibrium points. Such systems are formally transformed to Birkhoff or Poincar\'e-Dulac normal forms, but the normalization may not be analytic. Zung [3, 4] proved that the analytically integrable systems have analytic normalizations. On the other hand, Birkhoff normal forms are not necessarily integrable, but they are integrable if their resonance degrees are not greater than one [1]. This means that 4-dimensional Hamiltonian systems in Birkhoff normal form are integrable. After reviewing the theory of Birkhoff normal forms and Poincar\'e-Dulac normal forms, we discuss relation between Poincar\'e-Dulac normal forms and resonance degrees [2]. Moreover, we give an example of rationally nonintegrable, 4-dimensional normal forms.

[1] O. Christov, Non-integrability of first order resonances in Hamiltonian systems in three degrees of freedom, Celestical Mech. Dynam. Astronom. 112 (2012), no.2, 149–167.
[2] S. Yamanaka, Local analytic integrability of Poincar\'e-Dulac normal forms, in preparation.
[3] N. T. Zung, Convergence versus integrability in Poincar\'́e-Dulac normal forms, Math. Res. Lett. 9 (2002), no. 2-3, 217–228.
[4] N. T. Zung, Convergence versus integrability in Birkhoff normal forms, Ann. Math. 161 (2005), no. 1, 141–156.