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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
##### Keywords:
pendulum; chaotic orbits
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