The inherent complex dynamics of a parametrically excited pendulum
is of great interest in nonlinear dynamics, which can help one
better understand the complex world. Even though the parametrically
excited pendulum is one of the simplest nonlinear systems, until
now, complex motions in such a parametric pendulum cannot be
achieved. In this book, the bifurcation dynamics of periodic
motions to chaos in a damped, parametrically excited pendulum is
discussed. Complete bifurcation trees of periodic motions to chaos
in the parametrically excited pendulum include: period-1 motion
(static equilibriums) to chaos, and period- motions to chaos ( = 1,
2, ***, 6, 8, ***, 12). The aforesaid bifurcation trees of periodic
motions to chaos coexist in the same parameter ranges, which are
very difficult to determine through traditional analysis. Harmonic
frequency-amplitude characteristics of such bifurcation trees are
also presented to show motion complexity and nonlinearity in such a
parametrically excited pendulum system. The non-travelable and
travelable periodic motions on the bifurcation trees are
discovered. Through the bifurcation trees of travelable and
non-travelable periodic motions, the travelable and non-travelable
chaos in the parametrically excited pendulum can be achieved. Based
on the traditional analysis, one cannot achieve the adequate
solutions presented herein for periodic motions to chaos in the
parametrically excited pendulum. The results in this book may cause
one rethinking how to determine motion complexity in nonlinear
dynamical systems.
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