Travelable Periodic Motions

Abstract

In this chapter, travelable asymmetric periodic motions on the bifurcation trees will be presented for motion complexity. There exist no any centers for such periodic motions, and the symmetric periodic motions do not exist. The travelable period-m motion has \(\bmod \left( {{x_{0,}}2\pi } \right) = \bmod \left( {{x_{mT,}}2\pi } \right)\) but \({x_0} \ne {x_{mT}}\) with \({y_0} = {y_{mT}}\) Thus, the Fourier series of displacement cannot exist. Herein, the Fourier series of velocity will be presented to show harmonic effects on such travelable period-m motion. To demonstrate travelable periodic motions, the following formula for coordinates in the physical model are expressed by the displacement as $$\matrix{ {{X_k} = \left( {{l_0} + ka} \right)\cos xk,\quad {\kern 1pt} {Y_k} = \left( {{l_0} + ka} \right)\sin xk,} \cr {k = 1,2,3....\,;\,\quad a = {1 \over {1200}},} \cr } $$ ((7.1)) where l0 is the pendulum length, ka is the fictitious function with node number k for illustration of pendulum rotation motions in the physical model. For the real physical model, we have a = 0. However, we cannot see the rotation complexity of displacement x. The coordinates X, Y in physical model is locations of the pendulum. Using such fictitious functions, we can easily observe the motion complexity of angular displacement, and the coefficient a>0 in the fictitious function is arbitrarily chosen. Without loss of generality, for the Fourier series of velocity, the symbols for harmonic amplitudes and phases will use the same as for displacement for the non-travelable periodic motions. The periodic motions in pendulum can be characterized by the rotation and libration numbers as follows: $$\left( {{R_ + }\,:\,{R_ - }\,:\,L} \right),$$ ((7.2)) where R+ is the number of positive rotations, R- is the number of negative rotations, and L is the number of librations. The libration number consists of positive librations and negative librations. L = L++L- and L+ = L-for periodic motions. For non-travelable period-1 motions, we have R+ = R-.