Isolating Blocks For Periodic Orbits

Isolating Blocks for Periodic Orbits

Periodic Orbits, Stability and Resonances

The subjects of resonance and stability are closely related to the problem of evolution of the solar system. It is a physically involving problem and the methods available to mathematics today seem unsatisfactory to produce pure non linear ways of attack. The linearization process in both subjects is clearly of doubtful significance, so that, even if very restrictive, numerical solutions are still the best and more valuable sources of informations. It is quite possible that we know now very little more of the entire problem that was known to Poincare, with the advantage that we can now compute much faster and with much more precision. We feel that the papers collected in this Symposium have contributed a step forward to the comprehension of Resonance, Periodic Orbits and Stability. In a field like this, it would be a surprise if one had gone a long way toward that comprehension, during the short time of two weeks. But we are sure that the joint efforts of all the scientists involved has produced and will produce a measurable acceleration in the process. If this is true it will be a great satisfaction to us that this has happened in Brasil. The Southern Hemisphere in America has now begun to participate actively in the Astro nomical Society and for this, we are grateful to everyone who has helped.

Isolated Invariant Sets and the Morse Index

This volume contains lectures from the Conference Board of Mathematical Sciences meeting held at the University of Colorado on May 31-June 4, 1976. The lectures consist of an expository discussion of basic results for topological flows and a somewhat more detailed discussion of isolated invariant sets and continuation. The construction of the index for isolated invariant sets is new and allows more general application than previous ones. Also, the index itself is endowed with more structure and the continuation theorem is modified to take this new structure into account. Some elementary applications are given, but the main emphasis is on the abstract theory.