Bifurcations and chaos of a parametrically excited pendulum. (English) Zbl 0816.34031
Ladde, G. S. (ed.) et al., Dynamic systems and applications. Vol. 1. Proceedings of the 1st international conference, held at Morehouse College, Atlanta, GA, USA, May 26-29, 1993. Atlanta, GA: Dynamic Publishers, Inc. 307-314 (1994).
Some results on the numerical integration of the equation \(\ddot \theta + k \dot \theta + (1 + 2q \cos \Omega t) \sin \theta = 0\) are presented. The damping \(k\) has been chosen as \(k = 0.2\) for a reasonable approximation to the effect of friction. The parameters \(q\) and \(\Omega\) belong to the rectangle \(0.8 \leq \Omega \leq 2.4\); \(0 \leq q \leq 1.6\); containing the principal resonant frequencies \(\Omega = 1,2\). It has been found out that there are three general categories of motions: a) no rotation, b) rotation changing its direction with time, c) rotation in a single direction. Special attention is paid to oscillatory responses. Results are exposed via numerous diagrams.
For the entire collection see [Zbl 0802.00023].
For the entire collection see [Zbl 0802.00023].
Reviewer: Yu.N.Bibikov (St.Peterburg)
MSC:
34C23 | Bifurcation theory for ordinary differential equations |
34C25 | Periodic solutions to ordinary differential equations |
34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
65J99 | Numerical analysis in abstract spaces |