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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

$\stackrel{¨}{x}+{a}^{2}\left(t\right)x=0,\phantom{\rule{4pt}{0ex}}a\left(t\right):={a}_{k}\phantom{\rule{1.em}{0ex}}\text{if}\phantom{\rule{1.em}{0ex}}{t}_{k-1}\le t<{t}_{k},\phantom{\rule{1.em}{0ex}}k=1,2,\cdots \phantom{\rule{0.166667em}{0ex}}·$

Using a probability approach, we assume that ${\left({a}_{k}\right)}_{k=1}^{\infty }$ is given, and ${\left\{{t}_{k}\right\}}_{k=1}^{\infty }$ 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 $t\to \infty$, provided that ${a}_{k}↗\infty \phantom{\rule{4.pt}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}k\to \infty$. 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 $\infty$ in some (probabilistic) sense. In the second part we deal with the damped forced pendulum equation

$\stackrel{¨}{x}+{10}^{-1}\stackrel{˙}{x}+sinx=cost·$

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:
 34C15 Nonlinear oscillations, coupled oscillators (ODE) 34C28 Complex behavior, chaotic systems (ODE) 65G30 Interval and finite arithmetic 34F05 ODE with randomness 34A36 Discontinuous equations
##### References:
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