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Zbl 1164.34017
Hatvani, László
Stability problems for the mathematical pendulum.
(English)
[J] Period. Math. Hung. 56, No. 1, 71-82 (2008). ISSN 0031-5303; ISSN 1588-2829/e

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 $$\ddot x +a^2(t)x=0,\ a(t):=a_k\quad \text{if}\quad t_{k-1}\le t<t_k,\quad k=1,2,\dots\, .$$ Using a probability approach, we assume that $(a_k )^\infty_{k=1}$ is given, and $\{t_k\}_{k=1}^\infty$ 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 $a_k\nearrow \infty \text{ as } k \to \infty$. 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 $$\ddot x+10^{-1}\dot x +\sin x=\cos t.$$ {\it 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 2000:
*34C15 Nonlinear oscillations of solutions of ODE
34C28 Other types of "recurrent" solutions of ODE
65G30 Interval and finite arithmetic
34F05 ODE with randomness
34A36 Discontinuous equations

Keywords: random coefficients; parametric resonance; forced damped pendulum; computer-aided proof; interval arithmetic

Citations: Zbl 0989.70014

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