Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]