s ▁, ▁ ▁becomes ▁purely ▁imag inary ▁and ▁ ▁var ies ▁sin us oid ally . ▁Using ▁the ▁definition ▁of ▁the ▁det uning ▁, ▁the ▁p ump ing ▁frequency ▁ ▁must ▁lie ▁between ▁ ▁and ▁ ▁in ▁order ▁to ▁achieve ▁exponential ▁growth ▁in ▁. ▁Exp and ing ▁the ▁square ▁roots ▁in ▁a ▁bin omial ▁series ▁shows ▁that ▁the ▁spread ▁in ▁p ump ing ▁frequencies ▁that ▁result ▁in ▁expon entially ▁growing ▁ ▁is ▁approximately ▁. ▁ ▁Int uit ive ▁deriv ation ▁of ▁paramet ric ▁exc itation ▁The ▁above ▁deriv ation ▁may ▁seem ▁like ▁a ▁mathematical ▁sle ight - of - hand , ▁so ▁it ▁may ▁be ▁helpful ▁to ▁give ▁an ▁intuit ive ▁deriv ation . ▁The ▁ ▁equation ▁may ▁be ▁written ▁in ▁the ▁form ▁ ▁which ▁represents ▁a ▁simple ▁harm onic ▁oscill ator ▁( or , ▁altern atively , ▁a ▁band pass ▁filter ) ▁being ▁driven ▁by ▁a ▁signal ▁ ▁that ▁is ▁proportional ▁to ▁its ▁response ▁. ▁ ▁Assume ▁that ▁ ▁already ▁has ▁an ▁oscill ation ▁at ▁frequency ▁ ▁and ▁that ▁the ▁p ump ing ▁ ▁has ▁double ▁the ▁frequency ▁and ▁a ▁small ▁amplitude ▁. ▁App lying ▁a ▁trig on ometric ▁identity ▁for ▁products ▁of ▁sin us oid s , ▁their ▁product ▁ ▁produces ▁two ▁driving ▁signals , ▁one ▁at ▁frequency ▁ ▁and ▁the ▁other ▁at ▁frequency ▁▁ ▁Being ▁off - res on ance , ▁the ▁ ▁signal ▁is ▁at ten u ated ▁and ▁can ▁be ▁neglect ed ▁initially . ▁By ▁contrast , ▁the ▁ ▁signal ▁is ▁on ▁reson ance , ▁serves ▁to ▁ampl