By contrast, for [lambda] = 1000 shown in Figure 3(b), the nonlinear aerodynamic theory obtains a quasi-periodic motion in contrast to the LCO periodic motion from the linear aerodynamics.

For 435 < [lambda] < 455 a quasi-periodic motion is observed with the linear aerodynamics.

Although only four scattered and isolated points are observed in the Poincare maps shown in Figures 10(d), 11(d), 16(d), and 17(d), the enlarged graphic circulated by the red circle shows that the isolated "points" are actually closed loops which indicate quasi-periodic motion. Besides, by comparison between Figures 10 and 11 it can be seen that both the counter- and corotating systems execute four-period orbital motion although their orbits show a little difference--the amplitude for the orbit of disk 2 of the counterrotating system is slightly larger than that of the corotating system.

The Poincare map shown in Figure 12(d) suggests that disk 2 executes four-period quasi-periodic motion under counterrotating condition while no obvious periodicity can be observed from the random-like and irregularly distributed points shown in Figure 13(d), the corotating condition.

The above bifurcation diagrams show that breathing crack plate has complex nonlinear phenomenon:

quasi-periodic motion, bifurcation motion, and chaotic motion.

In our recent study (Vondrak and Ron, 2010) we demonstrated that the atmospheric and oceanic excitations, playing a dominant role in polar motion and rotational velocity of the Earth, have also a non-negligible and observable effect in nutation, i.e., the

quasi-periodic motion of Earth's axis of rotation in space, especially at annual and semi-annual frequencies.

According to these figures, a transition from

quasi-periodic motion to the chaotic dynamics is seen with the increase of b.

It can be seen that the motions of the one-way clutch, two-shaft assemblies system undergo period-1 motion, chaotic motion, period-2 motion,

quasi-periodic motion, and period-7 motion.

Conventionally, the steady-state condition of a deterministic nonlinear dynamic system can be regarded as being in one of three main states: equilibrium, periodic motion, and

quasi-periodic motion. These may collectively be described as the "attractor" because, after a transient-state response, a system in a steady state can be attracted towards another state.

Pinzari proves the existence of an almost full measure set of (3n-2)-dimensional

quasi-periodic motions in the planetary problem with (1+ n) masses, with eccentricities arbitrarily close to the Levi-Civita limiting value and relatively high inclinations.

According to Poincare the main problem of mechanics is to study the perturbation of

quasi-periodic motions in the system given by the Hamiltonian

In particular, the stability of periodic and

quasi-periodic motions of a solid body [1] will be analyzed in the sequel by using these results.