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Stability problems for the mathematical pendulum. (English) Zbl 1164.34017

Summary: The first part of this review paper is devoted to the simple (undamped, unforced) pendulum with a varying coefficient. If the coefficient is a step function, then small oscillations are described by the equation

x ¨+a 2 (t)x=0,a(t):=a k ift k-1 t<t k ,k=1,2,·

Using a probability approach, we assume that (a k ) k=1 is given, and {t k } k=1 is chosen at random so that t k -t k-1 are independent random variables. The first problem is to guarantee that all solutions tend to zero as t, provided that a k ask. In the problem that the coefficient a 2 takes only two different (alternating) values, and t k -t k-1 are identically distributed, one has to find the distributions and their critical expected values such that the amplitudes of the oscillations tend to in some (probabilistic) sense. In the second part we deal with the damped forced pendulum equation

x ¨+10 -1 x ˙+sinx=cost·

J. H. Hubbard [Am. Math. Mon. 106, No. 8, 741-758 (1999; Zbl 0989.70014)] discovered that some motions of this simple physical model are chaotic. Recently, also using the computer (the method of interval arithmetic), we gave a proof for Hubbard’s assertion. Here we show some tools of the proof.

MSC:
34C15Nonlinear oscillations, coupled oscillators (ODE)
34C28Complex behavior, chaotic systems (ODE)
65G30Interval and finite arithmetic
34F05ODE with randomness
34A36Discontinuous equations
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