an:01211605
Zbl 0912.34038
Kirchgraber, Urs
How Poincar??, Hadamard, and Perron contributed to the theory of invariant manifolds
DE
Math. Semesterber. 44, No. 2, 153-171 (1997).
00039880
1997
j
34C45 01A72 34-03
invariant manifolds; autonomous differential systems; dynamical systems
The author explains the geometric and analytic approaches of Poincar??, Hadamard and Perron in studying autonomous differential systems by means of the discrete dynamical system \((*)\) \(\bar{u} =F(u,v)\), \(\bar{v} = G(u,v)\) having at \(u=v=0 \in \mathbb{R}\) a hyperbolic fixed point. He introduces the concept of stable and unstable invariant manifolds of \((*)\) through the origin, and considers the (different) method of Hadamard and Perron to construct these manifolds. This is a well-written introduction into the geometric ideas of the qualitative theory of dynamical systems. Unfortunately, the author did not mention the contribution of A. M. Lyapunov to invariant manifolds.
K.R.Schneider (Berlin)