▁mechanism ▁provided ▁by ▁, ▁the ▁oscill ation ▁amplitude ▁grows ▁expon entially . ▁( This ▁phenomen on ▁is ▁called ▁paramet ric ▁exc itation , ▁paramet ric ▁reson ance ▁or ▁paramet ric ▁p ump ing .) ▁However , ▁if ▁the ▁initial ▁amplitude ▁is ▁zero , ▁it ▁will ▁remain ▁so ; ▁this ▁distingu ishes ▁it ▁from ▁the ▁non - param et ric ▁reson ance ▁of ▁driven ▁simple ▁harm onic ▁oscill ators , ▁in ▁which ▁the ▁amplitude ▁grows ▁linear ly ▁in ▁time ▁regardless ▁of ▁the ▁initial ▁state . ▁ ▁A ▁familiar ▁experience ▁of ▁both ▁paramet ric ▁and ▁driven ▁oscill ation ▁is ▁playing ▁on ▁a ▁swing . ▁Rock ing ▁back ▁and ▁forth ▁p umps ▁the ▁swing ▁as ▁a ▁driven ▁harm onic ▁oscill ator , ▁but ▁once ▁moving , ▁the ▁swing ▁can ▁also ▁be ▁param etr ically ▁driven ▁by ▁altern ately ▁standing ▁and ▁squ atting ▁at ▁key ▁points ▁in ▁the ▁swing ▁arc . ▁This ▁changes ▁moment ▁of ▁in ert ia ▁of ▁the ▁swing ▁and ▁hence ▁the ▁reson ance ▁frequency , ▁and ▁children ▁can ▁quickly ▁reach ▁large ▁amplit udes ▁provided ▁that ▁they ▁have ▁some ▁amplitude ▁to ▁start ▁with ▁( e . g ., ▁get ▁a ▁push ). ▁Stand ing ▁and ▁squ atting ▁at ▁rest , ▁however , ▁leads ▁now here . ▁ ▁Trans formation ▁of ▁the ▁equation ▁ ▁We ▁begin ▁by ▁making ▁a ▁change ▁of ▁variables ▁ ▁where ▁ ▁is ▁a ▁time ▁integral ▁of ▁the ▁d amp ing ▁. ▁ ▁This ▁change ▁of ▁variables ▁elimin ates ▁the ▁d amp ing ▁term ▁ ▁where ▁the ▁transformed ▁frequency ▁is ▁defined ▁. ▁