Stability problems for the mathematical pendulum

Abstract

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

$$ \ddot x + a^2 (t)x = 0,a(t): = a_k ift_{k - 1} \leqslant t < t_k ,k = 1,2,.... $$

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 ↗ ∞ (k → ∞). In the problem of swinging the coefficient a ² takes only two different values alternating each others, 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

$$ \ddot x + 10^{ - 1} \dot x + sinx = cost. $$

In 1999 J. Hubbard discovered that some motions of this simple physical model are chaotic. Recently, using also the computer (the method of interval arithmetic), we gave a proof for Hubbard’s assertion. Here we show some tools of the proof.