▁is ▁called ▁periodic ▁with ▁period ▁n . ▁ ▁If ▁there ▁exist ▁distinct ▁n ▁and ▁m ▁such ▁that ▁ ▁then ▁x ▁is ▁called ▁a ▁pre period ic ▁point . ▁All ▁periodic ▁points ▁are ▁pre period ic . ▁ ▁If ▁f ▁is ▁a ▁dif fe omorphism ▁of ▁a ▁differenti able ▁manifold , ▁so ▁that ▁the ▁derivative ▁ ▁is ▁defined , ▁then ▁one ▁says ▁that ▁a ▁periodic ▁point ▁is ▁hyper bol ic ▁if ▁ ▁that ▁it ▁is ▁attract ive ▁if ▁ ▁and ▁it ▁is ▁rep elling ▁if ▁ ▁If ▁the ▁dimension ▁of ▁the ▁stable ▁manifold ▁of ▁a ▁periodic ▁point ▁or ▁fixed ▁point ▁is ▁zero , ▁the ▁point ▁is ▁called ▁a ▁source ; ▁if ▁the ▁dimension ▁of ▁its ▁un stable ▁manifold ▁is ▁zero , ▁it ▁is ▁called ▁a ▁sink ; ▁and ▁if ▁both ▁the ▁stable ▁and ▁un stable ▁manifold ▁have ▁non zero ▁dimension , ▁it ▁is ▁called ▁a ▁s addle ▁or ▁s addle ▁point . ▁ ▁Ex amples ▁ ▁A ▁period - one ▁point ▁is ▁called ▁a ▁fixed ▁point . ▁ ▁The ▁log istic ▁map ▁▁ ▁exhib its ▁period icity ▁for ▁various ▁values ▁of ▁the ▁parameter ▁r . ▁For ▁r ▁between ▁ 0 ▁and ▁ 1 , ▁ 0 ▁is ▁the ▁sole ▁periodic ▁point , ▁with ▁period ▁ 1 ▁( g iving ▁the ▁sequence ▁ 0 , ▁ 0 , ▁ 0 , ▁... , ▁which ▁attract s ▁all ▁or bits ). ▁For ▁r ▁between ▁ 1 ▁and ▁ 3 , ▁the ▁value ▁ 0 ▁is ▁still ▁periodic ▁but ▁is ▁not ▁attract ing , ▁while ▁the ▁value