ify ▁, ▁and ▁is ▁proportional ▁to ▁the ▁amplitude ▁. ▁Hence , ▁the ▁amplitude ▁of ▁ ▁grows ▁expon entially ▁unless ▁it ▁is ▁initially ▁zero . ▁ ▁Ex pressed ▁in ▁Fourier ▁space , ▁the ▁multiplication ▁ ▁is ▁a ▁convolution ▁of ▁their ▁Fourier ▁transform s ▁ ▁and ▁. ▁The ▁positive ▁feedback ▁ar ises ▁because ▁the ▁ ▁component ▁of ▁ ▁converts ▁the ▁ ▁component ▁of ▁ ▁into ▁a ▁driving ▁signal ▁at ▁, ▁and ▁vice ▁vers a ▁( reverse ▁the ▁signs ). ▁This ▁explains ▁why ▁the ▁p ump ing ▁frequency ▁must ▁be ▁near ▁, ▁twice ▁the ▁natural ▁frequency ▁of ▁the ▁oscill ator . ▁P ump ing ▁at ▁a ▁gross ly ▁different ▁frequency ▁would ▁not ▁couple ▁( i . e ., ▁provide ▁mut ual ▁positive ▁feedback ) ▁between ▁the ▁ ▁and ▁ ▁components ▁of ▁. ▁ ▁Param et ric ▁reson ance ▁Param et ric ▁reson ance ▁is ▁the ▁paramet rical ▁reson ance ▁phenomen on ▁of ▁mechanical ▁perturb ation ▁and ▁oscill ation ▁at ▁certain ▁frequencies ▁( and ▁the ▁associated ▁harm on ics ). ▁This ▁effect ▁is ▁different ▁from ▁regular ▁reson ance ▁because ▁it ▁exhib its ▁the ▁inst ability ▁phenomen on . ▁ ▁Param et ric ▁reson ance ▁occurs ▁in ▁a ▁mechanical ▁system ▁when ▁a ▁system ▁is ▁param etr ically ▁excited ▁and ▁oscill ates ▁at ▁one ▁of ▁its ▁reson ant ▁frequencies . ▁Param et ric ▁reson ance ▁takes ▁place ▁when ▁the ▁external ▁exc itation ▁frequency ▁equals ▁twice ▁the ▁natural ▁frequency ▁of ▁the ▁system . ▁Param et ric ▁exc itation ▁diff ers ▁from ▁forcing ▁since ▁the ▁action ▁appears ▁as ▁a ▁time ▁varying