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].
