Bifurcation Theory. , a stationary point, depends on the value of α. At certain values

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1 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.

2 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.

3 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 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.

5 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

6 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

7 . 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

8 < 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