By Robert Gilmore
A brand new method of figuring out nonlinear dynamics and unusual attractors The habit of a actual procedure might seem abnormal or chaotic even if it really is thoroughly deterministic and predictable for brief sessions of time into the longer term. How does one version the dynamics of a approach working in a chaotic regime? Older instruments similar to estimates of the spectrum of Lyapunov exponents and estimates of the spectrum of fractal dimensions don't sufficiently resolution this question. In an important evolution of the sector of Nonlinear Dynamics, The Topology of Chaos responds to the elemental problem of chaotic structures by way of introducing a brand new research method-Topological Analysis-which can be utilized to extract, from chaotic info, the topological signatures that be sure the stretching and squeezing mechanisms which act on flows in part area and are answerable for producing chaotic info. starting with an instance of a laser that has been operated lower than stipulations during which it behaved chaotically, the authors exhibit the technique of Topological research via distinct chapters on: * Discrete Dynamical platforms: Maps * non-stop Dynamical platforms: Flows * Topological Invariants * Branched Manifolds * The Topological research application * Fold Mechanisms * Tearing Mechanisms * Unfoldings * Symmetry * Flows in larger Dimensions * A application for Dynamical platforms conception compatible this day for reading ''strange attractors'' that may be embedded in 3-dimensional areas, this groundbreaking process bargains researchers and practitioners within the self-discipline a whole and pleasurable answer to the basic questions of chaotic structures.
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Additional resources for The Topology of Chaos: Alice in Stretch and Squeezeland
Sample text
25) a piecewise linear map known as the tent map. 5 1 -. 5 2 xn Fig. 25). 9 shows that the graph of the tent map is extremely similar to that of the logistic map (Fig. 1). In both cases, the interval I is decomposed into two subintervals: I = I0 U 11,such that each restriction fk : I k + f ( I k ) o f f is a homeomorphism,with f ( I 0 ) = f(I1) = I . , orientation-reversing). In fact, these topological properties suffice to determine the dynamics completely and are characteristic features of what is often called a topological horseshoe.
At the accumulation point am, the period of the solution has become infinite. Right of this point, the system can be found in chaotic regimes, as can be guessed from the abundance of dark regions in this part of the bifurcation diagram, which indicate that the system visits many different states. The period-doubling cascade is one of the best-known routes to chaos and can be observed in many low-dimensional systems [34]. It has many universal properties that are in no way restricted to the case of the logistic map.
On the one hand, it can be shown that maps based on a one-dimensional homeomorphismcan only display stationaryor periodic regimes, and hence cannot be chaotic. On the other hand, if we sacrifice invertibilitytemporarily,thereby introduc- + LOGISTIC MAP 79 ing singularities, one-dimensionalchaotic systems can easily be found, as illustrated by the celebrated logistic map. Indeed, this simple system will be seen to display many of the essential features of deterministic chaos. It is, in fact, no coincidence that chaotic behavior appears in its simplest form in a noninvertible system.