The User's Approach to Topological Methods in 3-D Dynamical Systems

M. A NATIELLO, H. G. SOLARI

World Publisher

Lugar: Singapore; Año: 2007 p. 141

978 981 270 380 4

During most of the Twentieth Century, physicists have been mainly concernedwith linear dynamics. Despite the works of Poincar´e, Birkhoff and vonNeumann, the paradigm in physics was linear dynamics. Courses in ClassicalMechanics systematically ignored intrinsically non-linear phenomena and chaos,restricting Mechanics to Integrable Systems, i.e., dynamical systems with anunderlying Lie group structure, having dynamics that are exponentials of linearalgebras.During the second half of the 70´s the interest in nonlinear dynamics graduallyemerged in physics fueled by the possibility of enriching our intuition usingincreasingly powerful (as well as popular and affordable) computers. The {emchaos} paradigm took form, with new problems and new ways to analyze nature.An intense development followed the introduction of graphic workstations inthe 1980s. Questions such as: How to characterize systems presentingchaotic dynamics?  How to compare models with experiments? were then includedwithin the valid questions of the chaos paradigm.y that time it became clear that although there exist only a few differentways of displaying linear behaviour (always present in widely differentclasses of problems),nonlinear problems presented a large variety ofdifferent patterns, as well as other specific features such as sensitivity toinitial conditions. The urge to generate some comprehensive understanding ofchaos (are there different {em classes} of chaotic behaviour?) becameevident. During the ´80s, there were several attempts to solve theclassification problem. Earlier attempts focused in the{em routes to chaos}, the sequence of bifurcations as a function ofa single control parameter, that lead to chaos in a particular system.By the middle ´80s this attempt had proven to be of limited use: there wereinfinitely many routes to chaos in simple two-parameter systems. The chaoscommunity then turned its hopes towards {em fractal dimensions},i.e., a measure of the geometrical imprint (in phase-space) of achaotic attractor. By the end of the ´80s this path had also proven to bealmost useless for the characterization/classification problem (although someinteresting features such as Barnsley´s fractal pictures spun off thiseffort).The two main directions taken by the chaos community that we just describedwere not the only explored directions. Around 1987, a third programme aiming toclassify low dimensional ($3$-D) systems using topological orbit organizationbegan. This project in Physics was preceded by at least two importantdevelopments in Mathematics: (a) results from Birman-Williams-Holmes (1983--)developed to extract the knot content of hyperbolic attractors, introducing ageometrical construction that they named template or knot-holder, and (b)results due to Thurston (1979--) on the classification of $2$-Ddiffeomorphisms in terms of two main classes: rotation compatiblediffeomorphisms and pseudo-Anosov diffeomorphisms (the latter class admits afine structure) and the braid content of the diffeomorphism. Thurston´sresults appeared earlier than the template development, but they wereincorporated to the Physics project at a later stage.While the relation among the mathematical developments and the programme inphysics is direct and immediate, there also exist important differences amongthem. We have given the name {f The User´s Approach to Topological Methods in$3$-D Dynamical Systems} to the classification and recognition programme inPhysics, emphasizing that its aim is the use of the mathematical methods(emerging from Topology) in experimental situations. Unlike other programmes inchaos, the topological classification programme is still alive. In this book weintend to re-evaluate this programme.While writing this book we have come in contact with some difficult aspectsconcerning how to assess, prove or disprove a certain property in a system,that require a clear conception about how the programme relates to theory,experiment and numerical modeling. The readers will therefore finddiscussions on, and references to, epistemological matters. We have adoptedas much as possible a Popperian {em demarcationist} and{em fallibilistic} attitude, since we are dealing with experimentalscience, i.e., our interest is to {em induce} from experiments theoriginating properties of the underlying system in a scientifically validway. On the contrary, we have left outside this presentation topics thatare encompassed by the concept of {em normal science} in the senseof Khun (repetition of a paradigm with little variation) as well as someattempts which are still under development, but have not yet reachedthe level of an organized theory, at least in our understanding.This needs not be a serious loss, there are other sources where the materialcan be found. It is our hope that all existing proto-attempts will soonreach the mature level so that they can be thoroughly assessed.