The divergence of the vector field of a

dynamical system is identical to the sum of all its Lyapunov exponents (13).

Let (X, f) be a

dynamical system and be defined by the map.

If there exists a unique solution e [member of] [R.sup.J.sub.+]+ to the system y = Ax with the matrix A satisfying the condition of (5), the equilibrium e of the

dynamical system in (4) is asymptotically stable.

The key idea to prove this result consists in using the Stone Representation Theorem for Boolean Algebras in a suitable way to decompose this general

dynamical system into p PDS where the entities take values in {0,1} (see [1, Section 3] for the details).

To our knowledge, we summarize four criteria for the existence of chaos in the investigation of

dynamical systems. The first one is the well-known Lyapunov exponents [15].

Dynamical systems. Let [sigma] [member of] [S.sub.n], and X = [n].

Let X := W x [OMEGA] and (X, [Z.sub.+], [pi]) be a semigroup

dynamical system on X, where [pi](n, (u, [omega])) := ([phi](n, u, [omega]), [sigma](n, [omega])) for all u [member of] W and u [member of] [OMEGA], then (X, [Z.sub.+], [pi]) is called [19] a skew-product

dynamical system, generated by the cocycle <W, [phi], ([OMEGA], [phi], [Z.sub.+], a)>.

Random Number Generators as Chaotic

Dynamical SystemsThey are both a consequence of what we might call the complex underlying geometrical structure of the phase space of a nonlinear

dynamical system. This structure can be analytically quantified (see, for example, Smale [1963] and Schuster [1989]), and many physically important classes of systems can be shown to possess the same dynamics.

is a generalized nonautonomous

dynamical system on X.

(3) Trapping time (TT) is related with the laminarity time of the

dynamical system, that is, how long the system remains in a specific state.

Because of the existence of some nonlinear factors, such as multiple gear backlashes and time-varying mesh stiffness, planetary gear train is an inherent strongly nonlinear

dynamical system. Generally speaking, a nonlinear

dynamical system with a certain set of parameters may have several periodic orbits [1], and different orbits may have different domains of attraction [2].