# Dynamical Systems

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## Dynamical Systems

A system of equations where the output of one equation is part of the input for another. A simple version of a dynamical system is linear simultaneous equations. Non-linear simultaneous equations are nonlinear dynamical systems.

## Dynamical Systems

A series of equations in which the output of one becomes the input of another. One equation may determine a company's earnings for a particular period. The earnings, then, may be put into another equation to determine the earnings per share. This is a simple example of dynamical systems. It may (and often does) include a long string of equations.
References in periodicals archive ?
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 .
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  a skew-product dynamical system, generated by the cocycle <W, [phi], ([OMEGA], [phi], [Z.sub.+], a)>.
Random Number Generators as Chaotic Dynamical Systems
They 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  and Schuster ), and many physically important classes of systems can be shown to possess the same dynamics.
(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 , and different orbits may have different domains of attraction .

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