By Francis C. Moon

A revision of a pro textual content at the phenomena of chaotic vibrations in fluids and solids. significant adjustments replicate the most recent advancements during this fast-moving subject, the creation of difficulties to each bankruptcy, extra arithmetic and functions, extra insurance of fractals, a variety of machine and actual experiments. includes 8 pages of 4-color images.

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Is a geometric factor. Note that the only nonlinear terms are xz and xy in the second and third equations. For u = 10 and /? = 8/3 (a favorite set of parameters for experts in the field), there are three equilibria for p > 1 for which the origin is an unstable saddle (Figure 1-26). When p > 25, the other two equilibria become unstable spirals and a complex chaotic trajectory moves between all three equilibria as shown in Figure 1-27. , see Sparrow, 1982). Since 1963, hundreds of papers have been written about these equations, and this example has become a classic model for chaotic dynamics.

The theory we have just described is called a local analysis because it only tells what happens dynamically in the vicinity of each equilibrium point. The piece de resistance in classical dynamical analysis is to piece together all the local pictures and describe a global picture of how trajectories move between and among equilibrium points. Such analysis is tractable when bundles of different trajectories corresponding to different initial conditions move more or less together as a laminar fluid flow.

More will be said about quasiperiodic vibrations later, but because they are not periodic, they may be mistaken for chaotic solutions, which they are not. [For one, the Fourier spectrum of Eq. 10) is not closed when o1 and o2are incommensurate, so another method is used to portray the quasiperiodic function graphically. 11) 01 and denote x(r,) = x,, k(t,) = u,. Then Eq. 12) "1 As n increases, the points x,, u, move around an ellipse in the stroboscopic phase plane (called a PoincarP map), as shown in Figure 1-12.