Nonlinear dynamics

HERNAN GUSTAVO SOLARI; NATIELLO, MA; MINDLIN, GABRIEL BERNARDO

Institute of Physics

Lugar: London; Año: 1996 p. 1

0 7503 0379 4

The aim of this book is to render available to the readers the tools thatnonlinear dynamics provides for the exploration of new problems in all fieldsof physics. In research we deal with open problems: problems for which, atthe beginning, we have no solutions but at most a set of hunches, feelings andguesses based on our previous experience with other problems. From there, wework our path to the solutions (though we do not always succeed). In findingour way we take what seems to us the {\it most natural} approach just as wewould bush-walk in the forest avoiding as much as possible the difficult pathsin our hike for an interesting place.We will follow in the presentation the same procedure that we follow whendoing research, i.e., we begin with the problem and find one path to thesolution, we work inductively proposing new paths, checking them and redrawingour route according to the experience we gain in the successive efforts.We will avoid the temptation of selecting the problems according to the toolswe possess. Rather, we prefer to construct the tools along with the problems.We sustain the idea that our tools (theories) and our problems evolve hand inhand.The best pages of physics have been written in this way. Consider for examplethe pairs calculus--mechanics (Newton) or Hilbert spaces--quantum theory(Von Neumann) and as we will see, nonlinear dynamics--topology (Poincar{\'e}).This is a book to be read with paper and pencil at hand. Our intention is tofurnish the readers with enough knowledge to be able to do research innonlinear dynamics after having read the book (or better, {\em while} they arereading the book). We prefer to convey the key ideas within their mathematicalframework rather than doing lengthy demonstrations. Therefore, thecalculations around the results presented in the book are usually onlysketched or left as more or less guided exercises.At the end, we would like the reader to finish this book with the feeling (orthe certainty) that, given enough time, she/he would have come up with thesame answers to the problems than those that we have shown (well ... perhapsjust better answers). After all {\em the answers are dictated by theproblems}, the two of them evolve in interaction, and our task is to readthem from nature.