periods specified sharkovskii 's theorem sharkovskii 's theorem basis li yorke proof continuous one dimensional system exhibits regular cycle period three also display regular cycles every length well completely chaotic orbits strange attractors lorenz attractor displays chaotic behavior two plots demonstrate sensitive dependence initial conditions within region phase space occupied attractor dynamical systems like one dimensional logistic map defined span style white space nowrap ''x'' ''x'' ''x'' span chaotic everywhere
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