Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local Bifurcation Thor: Continuous Tim Sstms Considr th sstm: whr is a paramtr. Th quilibrium point of this sstm is givn b solving. Not: th quilibrium point a stationar point dpnds on th valu of. At crtain valus of th charactristics of th sstm changs somtims quit dramaticall. B th implicit function thorm th quilibria ar continuousl diffrntiabl functions of : Som dfinitions:. a If at an quilibrium point th Jacobian is zro and svral branchs of quilibria com togthr on sas that is a point of bifurcation. b Altrnativ dfinition: lt N dnot th numbr of quilibrium valus of th sstm whn th paramtr is qual to thn if for an intrval ε ε N is not constant is calld a bifurcation valu and th sstm is said to undrgo a bifurcation as passs through. c Anothr dfinition: a valu of quation for which th solution of is not structurall stabl is a bifurcation valu of. d An quilibrium point at which no bifurcation occurs is calld a hprbolic fixd point. Th bifurcation diagram is a diagram in which th branchs of quilibria ar shown in spac.
Not: in th cas of codimnsion on bifurcations th conditions on th Jacobian and its roots can b rplacd with singl partial drivativs.. Saddl-nod fold bifurcation prototp function: Considr th on paramtr first-ordr diffrntial quation f 4 and assum that whn thr is an quilibrium for which th following assumptions ar satisfid: f [ f has a stationar point with rspct to at ] 5.a f [ is an xtrmum] 5.b f. [ f is not stationar with rspct to at ] 5.c Thn dpnding on th signs of th xprssion 5.b and 5.c thr ar i no quilibria nar whn < > ; ii two quilibria nar for ach paramtr valu > <. Ths quilibria ar Exampl: hprbolic; on of thm is stabl and th othr unstabl.. Transcritical bifurcation prototp function: Considr th on paramtr first-ordr diffrntial quation f 6 and assum that whn thr is an quilibrium for which th following assumptions On-dimnsional sstms.
ar satisfid: f [ f has a stationar point with rspct to at ] 7.a f [ is an xtrmum] 7.b f. [Chang in shifts th phas curv] 7.c Thn dpnding on th signs of th xprssion 7.b and 7.c i th quilibrium is stabl unstabl whn < > ; ii th quilibrium bcoms unstabl stabl for ach paramtr valu > < and a branch of additional stabl unstabl quilibria mrgs. Exampl:. Pitchfork bifurcation prototp function: Considr th on paramtr first-ordr diffrntial quation f 8 and assum that whn thr is an quilibrium for which th following assumptions ar satisfid: f [ f has a stationar point with rspct to at ] 9.a f [Excluding th prsnc of a horizontal inflction at ] 9.b f. [Shift of th phas curv] 9.c Thn dpnding on th signs of th xprssion 9.b and 9.c iii th quilibrium is stabl unstabl whn < > ;
4 iv th quilibrium bcoms unstabl stabl for ach paramtr valu > < and two branchs of additional stabl unstabl quilibria mrg. Exampl: Th Hopf bifurcations in continuous tim Not: th Hopf bifurcation rquirs at last a sstm to appar. Considr th sstm of first-ordr diffrnc quations and assum that for ach in th rlvant rang this sstm has an isolatd quilibrium point obtaind b solving th sstm. Th solution to will giv as continuousl diffrntiabl functions of th paramtr naml if th following Jacobian matrix of is non singular at th quilibrium point J Th cas in which th two additional quilibria wr stabl is calld suprcritical pitchfork. Th cas in which th two additional quilibria wr unstabl is calld subcritical pitchfork.
Hopf bifurcation thorm Assum that Jacobian matrix valuatd at has th following proprtis: i it posssss a pair of simpl complx conjugat ignvalus θ ± iω that bcom pur imaginar at th critical valu of th paramtr i.. θ whil ω ; dθ ii ; d THEN sstm has a famil of priodic solutions. Not: th critical valu is calld Hopf bifurcation point of sstm. Whn a stabl ccl mrgs w hav suprcritical Hopf bifurcation s fig. i othrwis it is subcritical s fig. ii. Not: Conditions for dtrmining th tp of Hopf bifurcation suprcritical or subcritical do xist but involv th cofficints of third-ordr approximation to th nonlinar trms which ar tpicall undtrmind in conomic modls Prko L.: Diffrntial Equation and Dnamical Sstms. Springr-Vrlag Brlin 99. Chaptr 4. Conditions i and ii can also b applid to mor gnral n n sstms of diffrntial quations. In th cas n th Hopf bifurcation thorm rquirs all rmaining roots xcpt for th pair of complx conjugat roots undr invstigation to hav a ngativ ral part as othrwis th sstm would b unstabl anwa. 5
Local Bifurcation Thor: Discrt Tim Sstms Not: in discrt tim sstms th root with unit modulus taks th plac of th zro-ral-part root of th Jacobian matrix.. Saddl-nod fold bifurcation prototp functions: Considr th on paramtr first-ordr diffrnc quation t t. t t t f t t and assum that whn thr is an quilibrium. W hav fold bifurcation if th following hpothss ar satisfid: f 4.a f f 4.b. 4.c. Transcritical bifurcation prototp functions: Considr th on paramtr first-ordr diffrnc quation t t t t t t t. t f t 5 and assum that whn thr is an quilibrium. W hav a transcritical bifurcation if th following hpothss ar satisfid: f 6.a f 6.b f. 6.c 6
. Pitchfork bifurcation prototp functions: t t t Considr th on paramtr first-ordr diffrnc quation t t t t. t f t 7 and assum that whn thr is an quilibrium. W hav pitchfork bifurcation if th following hpothss ar satisfid: f 8.a f 8.b f. 8.c 4. Flip priod doubling bifurcation prototp functions: t t t t t t t. Not: flip bifurcation can onl aris in discrt dnamical sstms. Considr th on paramtr first-ordr diffrnc quation t f t 9 and assum that whn thr is an quilibrium for which th following hpothss ar satisfid: f.a f f f.b f f a..c Thn dpnding of th signs of th xprssions.b and.c i th quilibrium is stabl unstabl whn < > ; ii th quilibrium bcoms unstabl stabl for ach paramtr valu > 7
< and a branch of additional stabl unstabl quilibria of ordr mrgs two-ccl. Not: an quilibrium point or fixd point of ordr is an quilibrium point of th following diffrnc quitation : f f f f. t t t t Whn a in.c is positiv ngativ th mrging quilibrium points of ordr ar stabl unstabl and th flip bifurcation is said to b suprcritical subcritical rspctivl. Hopf bifurcation thorm for discrt-tim sstms Not: unlik with th continuous tim cas th Hopf bifurcation thorm xists onl for discrt tim sstms. Considr a non-linar diffrnc sstm with on paramtr t t and suppos that for ach it has a smooth famil of quilibrium points at which th ignvalus ar complx conjugat λ θ ± iω. If thr is a critical valu of th paramtr such that i λ θ ω λ j for j 4 ; d λ ii ; d THEN thr is an invariant closd curv bifurcating from. An quilibrium point of ordr is a point that rpats itslf vr two priods i.. a constant-amplitud altrnation. 8