×

zbMATH — the first resource for mathematics

Chaotic orbits of a pendulum with variable length. (English) Zbl 1060.34023
The authors show that a pendulum, whose pivot oscillates vertically in a periodic fashion, has uncountable many chaotic orbits that start, with zero velocity, from positions sufficiently close to the unstable equilibrium.
The main result is given as the theorem: Let \(r\in(0,1)\) be given. Select any infinite sequence of entries from the symbols 1, \(-1\), 0 or any finite sequence of entries from the same symbols and ending with \(\omega\). Then there are infinitely many initial conditions \((\upsilon_0,0)\) such that the given sequence of symbols corresponds to the solution of the initial value problem \[ \ddot x(t)+(1+r\sin t)\sin x(t)=0,\quad x(0)=\upsilon_0,\quad \dot x(0)=0. \]

MSC:
34C28 Complex behavior and chaotic systems of ordinary differential equations
PDF BibTeX XML Cite
Full Text: EMIS EuDML