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1 Flows vs. Maps     continued...

Chaotic effects can arise from Eq. (1) if two conditions are satisfied. First, at least one of the functions must contain a nonlinear term (e.g., , , ). Second, there is a theorem [1] which states that chaos only occurs in Eq. (1) if . For example, the famous Lorenz equations are given by [1,2]

where , r, and b are all constants. Note the nonlinear terms and in the second and third equations, respectively, of Eq. (5). For , , and r=28, one finds that the system of equations in (5) gives rise to the so-called Lorenz attractor [1,2] (a ``strange attractor''), which in 1963 was historically the first evidence of chaos in a dissipative system. (Shortly before 1900, Henri Poincaré discovered chaos in a conservative system.)