By Colonel Wm. T. McLyman
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Additional resources for High-Dimensional Chaotic and Attractor System - A Comprehensive Introduction
He discussed convergence and uniform convergence of the series solutions discussed by earlier mathematicians and proved them not to be uniformly convergent. The stability proofs of Lagrange and Laplace became inconclusive after this result. Poincar´e introduced further topological methods in 1912 for the theory of stability of orbits in the 3–body problem. It fact Poincar´e essentially invented topology in his attempt to answer stability questions in the three body problem. He conjectured that there are inﬁnitely many periodic solutions of the restricted problem, the conjecture being later proved by George Birkhoﬀ .
For example, the map on the right needs to be extended to a diﬀeomorphism of the sphere by using a ‘cap’ that wraps around the equator. The horseshoe map is an Axiom A diﬀeomorphism that serves as a model for the general behavior at a transverse homoclinic point, where the stable and unstable manifold s of a periodic point intersect. The horseshoe map was designed by Smale to reproduce the chaotic dynamics of a ﬂow in the neighborhood of a given periodic orbit. The neighborhood is chosen to be a small disk perpendicular to the orbit.
According to the basic insights of Marston Morse, a diﬀerentiable function on a manifold will, in a typical case, reﬂect the topology quite directly. Morse theory allows one to ﬁnd the so–called CW–structures and handle decompositions on manifolds and to obtain substantial information about their homology. Before Morse, Arthur Cayley and James Clerk Maxwell developed some of the ideas of Morse theory in the context of topography. Morse originally applied his theory to geodesics (critical points of the energy functional on paths).