▁( r - 1 )/ r ▁is ▁an ▁attract ing ▁periodic ▁point ▁of ▁period ▁ 1 . ▁With ▁r ▁greater ▁than ▁ 3 ▁but ▁less ▁than ▁ 1 ▁+ ▁, ▁there ▁are ▁a ▁pair ▁of ▁period - 2 ▁points ▁which ▁together ▁form ▁an ▁attract ing ▁sequence , ▁as ▁well ▁as ▁the ▁non - att ract ing ▁period - 1 ▁points ▁ 0 ▁and ▁( r - 1 )/ r . ▁As ▁the ▁value ▁of ▁parameter ▁r ▁r ises ▁toward ▁ 4 , ▁there ▁arise ▁groups ▁of ▁periodic ▁points ▁with ▁any ▁positive ▁integer ▁for ▁the ▁period ; ▁for ▁some ▁values ▁of ▁r ▁one ▁of ▁these ▁repeating ▁sequences ▁is ▁attract ing ▁while ▁for ▁others ▁none ▁of ▁them ▁are ▁( with ▁almost ▁all ▁or bits ▁being ▁cha otic ). ▁ ▁Dynam ical ▁system ▁ ▁Given ▁a ▁real ▁global ▁dynam ical ▁system ▁( R , ▁X , ▁ Φ ) ▁with ▁X ▁the ▁phase ▁space ▁and ▁ Φ ▁the ▁evolution ▁function , ▁ ▁a ▁point ▁x ▁in ▁X ▁is ▁called ▁periodic ▁with ▁period ▁t ▁if ▁there ▁exists ▁a ▁t ▁> ▁ 0 ▁so ▁that ▁ ▁The ▁smallest ▁positive ▁t ▁with ▁this ▁property ▁is ▁called ▁prime ▁period ▁of ▁the ▁point ▁x . ▁ ▁Properties ▁▁ ▁Given ▁a ▁periodic ▁point ▁x ▁with ▁period ▁p , ▁then ▁ ▁for ▁all ▁t ▁in ▁R ▁ ▁Given ▁a ▁periodic ▁point ▁x ▁then ▁all ▁points ▁on ▁the ▁orbit ▁ ▁through ▁x ▁are ▁periodic ▁with ▁the ▁same ▁prime ▁period . ▁ ▁See ▁also ▁ ▁Lim it ▁cycle ▁ ▁Lim it ▁set ▁ ▁St able