## The Topology of Chaos: Alice in Stretch and SqueezelandA new approach to understanding nonlinear dynamics and strange attractors The behavior of a physical system may appear irregular or chaotic even when it is completely deterministic and predictable for short periods of time into the future. How does one model the dynamics of a system operating in a chaotic regime? Older tools such as estimates of the spectrum of Lyapunov exponents and estimates of the spectrum of fractal dimensions do not sufficiently answer this question. In a significant evolution of the field of Nonlinear Dynamics, The Topology of Chaos responds to the fundamental challenge of chaotic systems by introducing a new analysis method-Topological Analysis-which can be used to extract, from chaotic data, the topological signatures that determine the stretching and squeezing mechanisms which act on flows in phase space and are responsible for generating chaotic data. Beginning with an example of a laser that has been operated under conditions in which it behaved chaotically, the authors convey the methodology of Topological Analysis through detailed chapters on: * Discrete Dynamical Systems: Maps * Continuous Dynamical Systems: Flows * Topological Invariants * Branched Manifolds * The Topological Analysis Program * Fold Mechanisms * Tearing Mechanisms * Unfoldings * Symmetry * Flows in Higher Dimensions * A Program for Dynamical Systems Theory Suitable at the present time for analyzing "strange attractors" that can be embedded in three-dimensional spaces, this groundbreaking approach offers researchers and practitioners in the discipline a complete and satisfying resolution to the fundamental questions of chaotic systems. |

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The Topology of Chaos: Alice in Stretch and Squeezeland Robert Gilmore,Marc Lefranc Limited preview - 2012 |

The Topology of Chaos: Alice in Stretch and Squeezeland Robert Gilmore,Marc Lefranc Limited preview - 2012 |

### Common terms and phrases

algebraic algorithm amplitude axis bifurcation diagram braid branch line branched manifold chaos chaotic behavior classiﬁcation close returns computed constructed coordinates corresponding critical point crossings cusp data set deﬁned deformation described determined diffeomorphism dimension Dufﬁng dynamical system eigenvalues embedding example experimental ﬁeld ﬁgure ﬁnd ﬁnite ﬁxed points ﬂow function global torsion Hénon map horseshoe map identiﬁed inﬁnite initial conditions integer intersections interval inverse iterate knots laser linking numbers logistic map Lorenz equations Lyapunov exponents matrix nonlinear occur one-dimensional orbits of period oscillator partition period-doubling bifurcation period-doubling cascade period-l periodic points phase space Phys plane Poincaré section projection relative rotation rates return map Rossler saddle-node bifurcation segments shown in Fig singularities Smale horseshoe template Speciﬁcally spectrum squeezing mechanisms strange attractor stretching and squeezing surrogate symbol sequence symbolic dynamics symbolic names symmetry theorem theory topological analysis topological entropy topological invariants topological organization torus two-dimensional unstable periodic orbits variables