Rotating speed of the blade is [mathematical expression not reproducible], where [[OMEGA].sub.0] is the steady-state rotating speed and [[OMEGA].sub.1] cos([??]t) is the

periodic perturbation. The shape of the blade is a rectangular plate, which is characterized by the span length L, the chord length C, and the thickness h.

It goes without saying that one of the main features of Theorem 3.7 regards the

periodic perturbation p(t) which is no longer required to be small, as shown in Figure 3.

However, it is not necessary that the pump generating the

periodic perturbation be absorbed in a process involving that particular transition to be probed in the experiment.

Various applications are described in the literature (10-12) where

periodic perturbation or symmetry breaking is introduced to cause chaotic particle trajectories in the deterministic flow patterns.

The interface of the guiding and metal layers has given a

periodic perturbation similar to grating structure.

The classical approach to the problem uses convex analysis (Rockafellar, 1970) to derive first-order conditions for improvement around an a priori known steady-state solution to the optimization problem using

periodic perturbation. Linearizing around a static optimal point, Gilbert (1977), and Bernstein and Gilbert (1980) developed linear second-order conditions for optimality of periodic operation, the [pi]-test.

It rotates at a varying rotating speed [OMEGA](t) around its polar axis where [OMEGA](t) = [[OMEGA].sub.0] + f cos [[OMEGA].sub.1] t, where [[OMEGA].sub.0] is the rotating speed at the steady-state and f cos [[OMEGA].sub.1] t is a

periodic perturbation. It is also allowed to vibrate flexurallyin the plane making an angle y, as shown in Figure 1(a).

By applying Melnikov method [15, 20, 21], we prove the criterion of existence of chaos under

periodic perturbation. Our interests have the following two points.

Laplace's

periodic perturbation and the secular tidal interaction both dwarf the orbital energy loss we now attribute to general relativity.

The periodically inserted B zone (no-barrier zone) is regarded as the

periodic perturbation in our modeling.

Melnikov, "On the stability of the center for time

periodic perturbations," Transactions of the Moscow Mathematical Society, vol.

But because the comet--and the meteoroid stream all along its orbit--have made hundreds of trips around the Sun over tens of thousands of years, slight

periodic perturbations can add up.