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Часть 2. Дополнительные тексты с заданиями



 

Text 1. The Poincaré-Melnikov Method

The Forced Pendulum.To begin with a simple example, consider the equation of a forced pendulum:

Here is a constant angular forcing frequency and is a small parameter. Systems of this or a similar nature arise in many interesting situations. For example, a double planar pendulum and other "executive toys" exhibit chaotic motion that is analogous to the behavior of this equation; see Burov [1986] and Shinbrot, Grebogi, Wisdom, and Yorke [1992].

For = 0 (1.9.1) has the phase portrait of a simple pendulum (the same as shown later in Figure 2.8.2a). For small but nonzero, (1.9.1) possesses no analytic integrals of the motion. In fact, it possesses transversal inter­secting stable and unstable manifolds (separatrices); that is, the Poincaré map Pto : R2 → R2 defined as the map that advance solutions by one period T = starting at time to possess transversal homoclinic points. This type of dynamic behavior has several consequences, besides precluding the existence of analytic integrals, that lead one to use the term "chaotic." For example, (1.9.1) has infinitely many periodic solutions of arbitrarily high period. Also, using the shadowing lemma, one sees that given any bi-infinite sequence of zeros and ones, there exists a corresponding solu­tion of (1.9.1) that successively crosses the plane = 0 (the pendulum's vertically downward configuration) with > 0 corresponding to a zero and < 0 corresponding to a one. The origin of this chaos on an intuitive level lies in the motion of the pendulum near its unperturbed homoclinic orbit, the orbit that does one revolution in infinite time. Near the top of its motion (where ) small nudges from the forcing term can cause the pendulum to fall to the left or right in a temporally complex way.

The dynamical systems theory needed to justify the preceding statements is available in Smale [1967], Moser [1973], Guckenheimer and Holmes [1983], and Wiggins [1988, 1990]. Some key people responsible for the development of the basic theory are Poincaré, Birkhoff, Kolmogorov, Melnikov, Arnold, Smale, and Moser.

 

A. Are these statements true or false according to the text?

  1. Here is constant angular forcing frequency and is a big parameter.
  2. A double planar pendulum exhibits chaotic motion.
  3. small but nonzero possesses some analytic integrals of the motion.
  4. The origin of the chaos on an intuitive level lies in the destruction of the pendulum near its unperturbed homoclinic orbit.
  5. Some key people responsible for the development of the basic theory are Newton and Mozart.

B. What are the key words in the text?







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