▁network ▁with ▁four ▁eigen ve ctors ▁with ▁nodes ▁at ▁the ▁connections . ▁This ▁can ▁be ▁implemented ▁as ▁a ▁trans imp ed ance ▁ampl ifier , ▁a ▁travel ing - wave ▁ampl ifier ▁or ▁by ▁means ▁of ▁a ▁circul ator . ▁ ▁Mathemat ical ▁equation ▁The ▁paramet ric ▁oscill ator ▁equation ▁can ▁be ▁extended ▁by ▁adding ▁an ▁external ▁driving ▁force ▁: ▁. ▁ ▁We ▁assume ▁that ▁the ▁d amp ing ▁ ▁is ▁sufficiently ▁strong ▁that , ▁in ▁the ▁absence ▁of ▁the ▁driving ▁force ▁, ▁the ▁amplitude ▁of ▁the ▁paramet ric ▁oscill ations ▁does ▁not ▁diver ge , ▁i . e ., ▁that ▁. ▁In ▁this ▁situation , ▁the ▁paramet ric ▁p ump ing ▁acts ▁to ▁lower ▁the ▁effective ▁d amp ing ▁in ▁the ▁system . ▁For ▁illustr ation , ▁let ▁the ▁d amp ing ▁be ▁constant ▁ ▁and ▁assume ▁that ▁the ▁external ▁driving ▁force ▁is ▁at ▁the ▁mean ▁reson ance ▁frequency ▁, ▁i . e ., ▁. ▁The ▁equation ▁becomes ▁ ▁whose ▁solution ▁is ▁roughly ▁. ▁ ▁As ▁ ▁approaches ▁the ▁threshold ▁, ▁the ▁amplitude ▁diver ges . ▁When ▁, ▁the ▁system ▁enters ▁paramet ric ▁reson ance ▁and ▁the ▁amplitude ▁begins ▁to ▁grow ▁expon entially , ▁even ▁in ▁the ▁absence ▁of ▁a ▁driving ▁force ▁. ▁ ▁Adv antages ▁It ▁is ▁highly ▁sensitive ▁low ▁noise ▁level ▁ampl ifier ▁for ▁ult ra ▁high ▁frequency ▁and ▁mic row ave ▁radio ▁signal ▁The ▁unique ▁cap ability ▁to ▁operate ▁as ▁a ▁wireless ▁power ed ▁ampl ifier ▁that ▁doesn ' t ▁require ▁internal ▁power ▁source ▁ ▁Other ▁relevant ▁mathematical ▁results