▁equations ▁ ▁These ▁equations ▁may ▁be ▁dec ou ple d ▁and ▁solved ▁by ▁making ▁another ▁change ▁of ▁variables ▁ ▁which ▁yields ▁the ▁equations ▁ ▁where ▁for ▁bre vity ▁the ▁following ▁are ▁defined ▁ ▁and ▁the ▁det uning ▁. ▁ ▁The ▁ ▁equation ▁does ▁not ▁depend ▁on ▁, ▁and ▁linear ization ▁near ▁its ▁equilibrium ▁position ▁ ▁shows ▁that ▁ ▁dec ays ▁expon entially ▁to ▁its ▁equilibrium ▁ ▁where ▁the ▁decay ▁constant ▁. ▁ ▁In ▁other ▁words , ▁the ▁paramet ric ▁oscill ator ▁phase - lock s ▁to ▁the ▁p ump ing ▁signal ▁. ▁ ▁T aking ▁ ▁( i . e ., ▁assuming ▁that ▁the ▁phase ▁has ▁locked ), ▁the ▁ ▁equation ▁becomes ▁ ▁whose ▁solution ▁is ▁; ▁the ▁amplitude ▁of ▁the ▁ ▁oscill ation ▁diver ges ▁expon entially . ▁However , ▁the ▁corresponding ▁amplitude ▁ ▁of ▁the ▁un transform ed ▁variable ▁ ▁need ▁not ▁diver ge ▁ ▁The ▁amplitude ▁ ▁diver ges , ▁dec ays , ▁or ▁stays ▁constant , ▁depending ▁on ▁whether ▁ ▁is ▁greater ▁than , ▁less ▁than , ▁or ▁equal ▁to ▁, ▁respectively . ▁ ▁The ▁maximum ▁growth ▁rate ▁of ▁the ▁amplitude ▁occurs ▁when ▁. ▁At ▁that ▁frequency , ▁the ▁equilibrium ▁phase ▁ ▁is ▁zero , ▁imp lying ▁that ▁ ▁and ▁. ▁As ▁ ▁is ▁varied ▁from ▁, ▁ ▁moves ▁away ▁from ▁zero ▁and ▁, ▁i . e ., ▁the ▁amplitude ▁grows ▁more ▁slowly . ▁For ▁sufficiently ▁large ▁devi ations ▁of ▁, ▁the ▁decay ▁constant ▁ ▁can ▁become ▁purely ▁imag inary ▁since ▁. ▁ ▁If ▁the ▁det uning ▁ ▁exceed