コロキウム

We consider systems of differential equations with equilibrium points. These systems are formally transformed to Poincaré-Dulac normal forms, but the normalization may not be analytic. Such an analytic normalization is a classical problem that dates back to Poincaré. Zung [2] proved that analytically integrable systems of differential equations have analytic normalizations. Nonintegrability and nonexistence of analytic normalizations are equivalent when the resonance degree is less than two since their Poincaré-Dulac normal forms are integrable [1]. In this talk, we prove that a special class of two-dimensional systems whose resonance degree is one have no analytic normalizations so that they are not integrable.

[1] S. Yamanaka, Local integrability of Poincaré-Dulac normal forms, Regul. Chaotic. Dyn. 23 (2018), 933–947.
[2] N. T. Zung, Convergence versus integrability in Poincaré-Dulac normal forms, Math. Res. Lett. 9 (2002), 217–228.