Notes on the stability of dynamic systems and the use of Eigen Values.

Transcription

1 Noes on he sabl of dnamc ssems and he use of Egen Values. Source: Macro II course noes, Dr. Davd Bessler s Tme Seres course noes, zarads (999) Ineremporal Macroeconomcs chaper 4 & Techncal ppend, and Hamlon (994) Tme Seres nalss chaper & Mahemacal Revew ppend. Prepared b rel Vale 3 Le he frs-order lnear non-lnear non-homogeneous ssem of dfference equaons be n general form: z () z f where s a n n mar of me nvaran coeffcens and z and f are n vecor of daed varables. The varable z s pcall he sae vecor of he ssem a me ; f s a vecor of (possbl me-dependen) forcng erms ofen hough of as eogenous. For llusrave purposes we wll work wh a ssem of he form: a c b d () or, a c b d. (3) We can wre he general soluon o an lnear ssem lke n () n he form of a complemenar homogeneous ssem plus he parcular soluon, or: z z z g C P (4) Ths resul (4), allows us o solve for he equlbrum of he ssem n () n wo sages: we sar wh he homogeneous ssem z z, and complee he ask b fndng he parcular soluon o he full ssem. Parcular soluons are eas o fnd when he lernavel he analss can be done n erms of dfferenal equaons, changng noaon (). Recall b Kolmogorov s connu heorem, ha for he dscree { } process and T > f here es posve consans α β [ s ] D s ; s, T E α, β, D s.., hen here ess a connuous verson of, ().

2 forcng erms are me nvaran because n he sead sae and. Thus, I I where d c b a (5) regroupng erms, ( ) Mar Iden I where (6) and, ( ) I (7) ha s ff ( s nverble,.e. non-sngular. Pluggng (6) no (3): ) (8) ( ) I Regroupng erms, we ge he homogeneous lnear ssem of frs-order dfference equaons: (9) ssume we know he nal condon, hen. () Hence, he ssem converges o he sead sae ff. Ths can be deermned hrough a specral or egenvalue represenaon of. To do hs we need o use Jordan decomposon, whch requres ha mar has lm s n lnearl ndependen egenvecors,.e. dsnc egenvalues (could have some bu never all egenvalues repeaed). The Jordan decomposon of s: () Λ e e

3 where e s he mar of egenvecors and Λ s he mar of egenvalues, a dagonal mar wh he egenvalues along he prncpal dagonal and zeros elsewhere. For he ssem: λ Λ λ () and because Λ s dagonal, mus be: Λ λ, where λ (3) λ λ The egenvalues are obaned as follows. Whou loss of general drop he me superscrp n (), hen mus be ha: e Λ e defne λ as he vecor of egenvalues and I an (4) n n den mar, hen Λ λ I, and ( λ I ) e λ I e e (5) Thus, ff e s non-sngular, hen mus be ha: ( I ) de λ (6) (6) s known as he characersc equaon from where we oban he egenvalues. Noce, ha for he ssem: a b λ de c d λ ( λ I ) de a λ b de (7) c d λ ( a λ)( d λ) bc ad aλ λ dλ bc λ ( a d ) λ ( ad bc) ( ) T where a d Trace( ) and ( ad bc) de( ) D. Then, he characersc equaon can be re-wren n erms of he T and D as follows: λ Tλ D (8)

4 Solvng he quadrac equaon for he egenvalues: T T 4D T T 4D λ, and λ (9) λ s he larges egenvalue and λ s he smalles egenvalue. lso, λ λ T and λ λ λ D. Recall ha, Λ, hen we can assess he local sabl properes λ of he planar ssem from he dscrmnan T 4D (a parabola) of he characersc equaon: ) Iff T > 4D hen boh egenvalues are real and dsnc. ) Iff T 4D hen boh egenvalues are equal and mar canno be dagonalzable and we need anoher mehod o fnd he egenvalues. 3) Iff T < 4D, hen boh egenvalues are dsnc and comple conjugaes: T 4D T T 4D T λ, and λ () lso: ) Iff λ <, he modulus of he egenvalues le whn he un crcle (SINK),.e. he sead sae s sable as λ as. B) Iff λ >, he modulus of he egenvalues le ousde he un crcle (SOURCE),.e. he sead sae s unsable as λ ± as. C) Iff s.. λ >, here ess some egenvalue whch modulus les ousde he un crcle (bu no all) (SDDLE), hen he sead sae s neher sable nor unsable. Onl a ver specfc se of nal condons wll ake he ssem o he sead sae along he saddle pah (sable manfold) drven b he sable egenvalues. Fnall facorzng he characersc equaon p, we oban he sragh lnes p() p( ), whch jonl wh he dscrmnan parabola dvdes he plane n each regons for he characerzaon of he sabl properes of he ssem n erms of he egenvalues (roos), or equvalenl from he race (T) and he deermnan (D) of he coeffcen mar (see graph below):

5 D T < - T > D > Comple Roos D > D > p() p(-) Comple Roos D < 6 7a 7c - - T (,) & D (, ) T Real Roos Real Roos T < 7b Real Roos T > 3 - Real Roos D < - Real egenvalues zones: Regon : Saddle, Regon : Source, oscllaor dvergence, Regon 3: Saddle, Regon 4: Source, monoone dvergence, Regon 7: Snk, 7a & 7c monoone convergence, 7b oscllaor convergence, Regon 8: Source, monoone dvergence, Comple egenvalues (conjugae pars) zones: Regon 5: Source, oscllaor dvergence, Regon : Snk, oscllaor convergence.

6 FINL REMRK: For he nonlnear case, we lnearzed he ssem usng Talor s epanson and fndng he Jacoban (J) evaluaed a he sead sae. Then we can proceed wh he prevous analss changng for J.