By Francis C. Moon
The booklet that brings order to chaos
Fifteen years in the past, Francis Moon’s Chaotic Vibrations brought, in functional language, the recent principles of nonlinear dynamics and chaos. due to the fact that then, the sector has grown enormously, and "chaos" has entered the vocabulary of not just physicists and mathematicians but in addition most of the people. Researchers in nonlinear dynamics have broadened their scope to enquire principles of complexity in traditional and human-made platforms, together with networks within the mind, electrical power grids, and the web.
With this paperback reissue of Professor Moon’s vintage, we are hoping to curiosity a brand new iteration of readers intrigued by means of unpredictability within the legislation of physics and its manifestation within the actual international within the kind of chaotic dynamics.
Written for engineers and utilized scientists, Chaotic Vibrations offers particular examples and purposes of chaotic dynamics within the actual global. It additionally describes easy methods to practice either computing device and actual experiments in chaotic dynamics. themes coated comprise:
- Poincaré maps
- Fractal dimensions
- Lyapunov exponents
- Experiments in chaos
- Chaos in engineering
Similar in its wide scope to James Gleick’s Chaos, merely geared to a extra technically curious reader, Chaotic Vibrations positive factors an intensive consultant to the literature, specifically because it pertains to extra mathematically-oriented works; a word list of nonlinear dynamics phrases; a listing of laptop experiments; and information for an indication test on chaotic vibrations. this useful paperback model of Chaotic Vibrations palms engineers and researchers with the hot instruments of dynamical structures and prepares them to make their very own contributions to this interesting and speedily constructing box.
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Extra resources for Chaotic vibrations: an introduction for applied scientists and engineers
This fractal property of a strange attractor is illustrated in the Henon map, Figure 1-22. Blowups of small regions of the Henon attractor reveal finer and finer structure. This self-similar structure of chaotic attractors can often be revealed by taking Poincare maps of experimental chaotic oscillators (see Chapter 4). The fractal property of self-similarity can be measured using a concept of fractal dimension, which is discussed in Chapter 6. , see Guckenheimer and Holmes, 1983). This idea is the centerpiece of a method developed to find a criterion for when chaotic vibrations are possible in a dynamical system and when predictability of future time history becomes sensitive to initial conditions.
There are also certain narrow windows AX for X^ < X < 4 for which periodic orbits exist. Periodic and chaotic orbits of the logistic map are shown in Figure 1-19 by plotting xn+l versus xn. 7). Thus, the phenomenon of period doubling or bifurcation parameter scaling has been called a universal property for certain classes of one-dimensional difference equation models of dynamical processes. 7) have been observed in many physical experiments (see Chapter 3). 4): hence the importance of maps to the study of differential equations.
18 Introduction: A New Age of Dynamics Au u=o. 9) in the phase plane. The phase plane portrait of Eq. 10) is not closed when co1 and co2 are incommensurate, so another method is used to portray the quasiperiodic function graphically. 11) 1 and denote x(tn) = xn, x(tn) = vn. 12) v / As n increases, the points (xn, vn) move around an ellipse in the Stroboscopic phase plane (called a Poincare map), as shown in Figure 1-13. When coo/co! 13) Quasiperiodic oscillations also occur in systems with more than one degree of freedom.
Chaotic vibrations: an introduction for applied scientists and engineers by Francis C. Moon