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How Poincaré, Hadamard, and Perron contributed to the theory of invariant manifolds. (Als Poincaré, Hadamard and Perron die invarianten Mannigfaltigkeiten entdeckten.) (German) Zbl 0912.34038
The author explains the geometric and analytic approaches of Poincaré, 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.
MSC:
34C45 Invariant manifolds for ordinary differential equations
01A72 Schools of mathematics
34-03 History of ordinary differential equations
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