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Poincar\'e gave a criterion which determines the shape of equilibrium for planar differential equations. In his statement, he excluded the case of repeated eigenvalues. In fact, in such a case, we can give a $C^1$ counter-example to his assertion. In this note, we show that if we strengthen the condition to $C^{1,\alpha}$ ($0<\alpha<1$), his assertion becomes true even in case of repeated eigenvalues.

Author

Odani, Kenzi

Published

2025

Language

EN