Hopf Bifurcation and Stability Analysis of a Business Cycle Model With Time-Delayed Feedback

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1 Hopf Bifurcaion and Sabiliy Analysis of a Business Cycle Model Wih ime-delayed Feedback Lei Peng #, Yanhui Zhai # Suden, School of Science, ianjin Polyechnic Universiy, ianjin 3387, China Professor, School of Science, ianjin Polyechnic Universiy, ianjin 3387, China Absrac In his paper, a business cycle model wih ime-delayed feedback is invesigaed Firsly, we add a ime-delayed feedback conroller o he business cycle model and propose a new model Secondly, he linear sabiliy of he model and he local Hopf bifurcaion are sudied and we derive he condiions for he sabiliy and he exisence of Hopf bifurcaion a he equilibrium of he sysem Besides, he direcion of Hopf bifurcaion and he sabiliy of bifurcaion periodic soluions are sudied by adoping he cener manifold heorem and he normal form heory A las, some numerical simulaion resuls are presened o confirm ha he conroller can effecively increase he sabiliy region of he business cycle model Keywords Business cycle model,ime-delayed feedback, Sabiliy, Hopf bifurcaion, Numerical simulaion I INRODUCION In recen years, wih he differenial equaions have been widely applied o biology, economics and oher fields, many scholars have esablished some models ha can reflec he characerisics of he dynamical sysems of differenial equaions [-3] Business cycle is also called economic cycle I refers o he phenomenon ha economic expansion and economic conracion occur alernaely and repeaedly in economic operaion In he heory of Macroeconomics, he business cycle is characeried by flucuaion in macroeconomic variables, which is caused by he insabiliy of he business sysems[4-6] he dynamics propery of a Kaldor-Kalecki business model are sudied in [7-9] In[], he auhors invesigaed a business model based on Keynesian s heory and firs sudied Hopf bifurcaion for his model wih delay In[], Jinchen Yu, Mingshu Peng and Caiyan Zhang discuss he sabiliy and bifurcaion of he equilibrium by applying he mehod of muliple scales However, here few resuls on he business cycle model wih ime-delayed feedback conrol In his paper, we invesigaed a business clcle model wih ime-delayed feedback and analysis he sabiliy and Hopf bifurcaion of he model [] described by he following nonlinear differenial equaions: x ax qx vx vx ux 3 3 () ( ) () () () () () where represens he delay parameer, x is gross naional income, he do is derivaive wih respec o ime, a denoes he marginal propensiy o consume, u and /u is he Keynesian muliplier, q is a fixed ineres rae, v is capial-oupu raio, also known as he acceleraor In[], le x() y(), hen he Eq () can be changed ino he following form: x() y(), y() ax( ) uy() f ( x, y) () 3 3 where f ( x, y) qx ( ) vy ( ) vy ( ) and he oher parameers are definiion of his model are he same as model () In his paper, based on he above model (), we add a ime-delayed feedback conroller ISSN: hp://wwwijejournalorg Page 38

2 k( x() x( )) o i, Hence, we propose a new model as follows: x() y(), y() ax( ) uy() k( x() x( )) f ( x, y) (3) where he parameer k is he feedback gain he res of he paper is arranged as follows, he linear sabiliy of he model and he local Hopf bifurcaion are sudied and he condiions for he sabiliy and he exisence of Hopf bifurcaion a he equilibrium are derived in Secion In Secion 3, according o he mehod of heory and applicaions of Hopf bifurcaion by Hassard, we analysis he direcion and sabiliy of bifurcaing periodic soluions In Secion 4, he correcness of heoreical analysis are confirmed by some numerical simulaion resuls A las, some conlusions are obained in Secion 5 II SABILIY AND LOCAL HOPF BIFURCAION ANALYSIS In his secion, we focus on he problems of he Hopf bifurcaion and sabiliy for he sysem(3) We also derive he sufficien condiions for he he sabiliy and he exisence of Hopf bifurcaion a he equilibrium poin Obviously, sysem (3) has he unique equilibrium poin (,) he linearaion of sysem (3) a (,) is x() y(), y() kx() ( k a) x( ) uy() (4) Obviously, he correspoding characerisic equaion of model (3) a he equilibrium poin is as follows u ( k a) e k (5) Lemma When is saisfied, he equilibrium poin (,) of model (3) is locally asympoically sable Proof When is me, Eq (5) becomes u a (6) hen we have he following condiions: D u, D a (7) According o he Rouh-Hurwi crieria, all roos of characerisic equaion (6) have negaive real pars Hence, when hold, he equilibrium poin (,) of sysem (3) is locally asympoically sable Lemma Assume ha ka a, namely a k is me hen Eq(5) has a pair of purely i, where imaginary roos when, ( k u ) ( k u ) 4( ka a ) arccos( ) k k a Proof Le i ( ) is a soluion of he characerisic equaion (5), hen iu k a i k ( )(cos sin ) he separaion of he real and imaginary pars, i follows ( k a)cos k u ( k a)sin (8) From (8) we obain 4 ( k u ) ( ka a ) (9) Hence, ( k u ) ( k u ) 4( ka a ) k j j arccos( ) j,,, k a ISSN: hp://wwwijejournalorg Page 39

3 Obviously, le j, hen () () ( k u ) ( k u ) 4( ka a ) arccos( ) k k a As a resul, when, he equaion (5) have a pair of purely imaginary roos Lemma 3 Le ( ) ( ) i( ) be he roo of (6) wih ( ) and ( ) When ka a and u k hold, hen he ransversaliy condiion d Re( ) d is saisfied Proof By differeniaing boh sides of Eq (5) wih regard o and applying he implici funcion heorem, we have d ( k a) e d u k e hen ( ) d Re d ( u k) ( u ( k a) cos ) ( ( k a) sin ) ( ) Lemma 4 For Eq (5), when, all of his roos have negaive real pars he equilibrium (,) is locally asympoically sable, and sysem (3) produces a Hopf bifurcaion a he equilibrium (, ) when By applying he Hopf bifurcaion heorem for ime-delayed differenial equaion and he above four lemmas [], we have he following consequences heorem conclusions hold: a) If a k and For sysem (3), he following u k hold, he equilibrium poin (,) is asympoically sable for [, ) b) If,model (3) exhibis a Hopf bifurcaion a he equilibrium poin (,) c) If, hen he equilibrium poin of sysem III (4) is unsable DIRECION AND SABILIY OF HE HOPF BIFURCAION In his secion, by using he normal form heory and he cener manifold heorem inroduced in [3-5], we discuss he direcion of Hopf bifurcaion and he sabiliy of he bifurcaing periodic soluions ( k a) sin i( k a) cos when ( u ( k a) cos ) i( ( k a) sin ) For noaional convenience, le, u( ) ( x ( ), x ( )) and u ( ) u( ), clearly, is Hopf for, bifurcaion value for (3) For iniial condiion ( ) ( ( ), ( )) C,, he sysem (3) is equivalen o he following Funcional Differenial Equaion (FDE) sysem u( ) Lu F( u, ) (3) wih L B B ( ) () ( ) Since d d Re( ) u k, hus he proof is compleed (4) and F(, ) 3 3 q () v () v () ISSN: hp://wwwijejournalorg Page 4

4 (5) where L is he one family of bounded linear operaor in C, and o B k, B u ( ka) By he Ries represenaion heorem[6], here exiss a bounded variaion funcion (, ) for [,], such ha L d(, ) ( ), C (6) we can choose B B, ( ) ( ) (7) where is a Dela funcion For C([,]), he operaors A and R are defined as follow d( ( )), [,), A( ) ( ) d d( (, ) ( )), (8), [,), R( ) ( ) F(, ), (9) Hence, he Eq (3) can be wrien as he following form: u A( ) u R( ) u () Since du d du d, hen Eq() can be wrien as du d () du, [,), d Lu F( u, ), For C, operaor A ( ) A( ) ( ) (), we define he adjoin of A( ) as d () s, s (, ], ds s d s For ( ) C[,) ( (,) ( )),, and C, bilinear inner produc, (3), define a () () ( )[ d( )] ( ) d where ( ) (,) Le ; o deermine he normal form of operaor A, we need o calculae he eigenvecors q () and q (s) of A and A corresponding o i and i, respecively we can obain A() q( ) iq( ) A () q ( s) iq ( s) (4) i Assume ha q( ) Ve is eigenvecor of A () i s corresponding o i and q () s DV e is eigenvecor of A () corresponding o i By direc calculae, we ge q( ) Ve ( v, v ) e (5) i i i,, i e i e q ( s) DV e D( v, v ) e is is (6),, i s is D e D u i e Now, we verify ha q, q and ISSN: hp://wwwijejournalorg Page 4

5 q, q From(3), we obain q, q q q() q ( ) d( ) q( ) d ( ) q, q,( A() R()) i( ) i [ ( ) } D V V V e d Ve d [ i [ ( )] ] D V V V d e V i D[ V V e V B V ] (3) (7) i Le D [ V V e V B V ], we can ge q, q By, A A,, we obain real, herefore we only discuss real soluions Since, i is easy o find ha (3) Le q, A q, R i q f (, ) ' ( ) i g(, ), where g (, ) g (33) g g from () and (3), we have, i q, q q, Aq A q, q (8) i q, q i q, q herfore q, q he proof is compleed Using he same noaions as in Hassard e al [3], we firs compue he coordinaes o describe he cener manifold C a Define ( ) q, u, W (, ) u ( ) Re{ ( ) q( )} (9) On he cener manifold C,we have W(, ) W( ( ), ( ), ) (3) Where W (,, ) W( ) W( ) W( ) and are local coordinaes for cener manifold C in C in he direcion of q and q, respecively Noe ha W is real if u is W u q q AW q f q AW q f q f (34) Re () (, ) ( ), [,], Re{ () (, ) ( )} (, ), Which can be rewrien as W AW H(,, ) (35) where H(,, ) H (36) ( ) H On he oher hand, on C, W W (37) W ( ) H Using (3) and (3) o replace ( ) W and and heir conjugaes by heir power series expansions, we obain ISSN: hp://wwwijejournalorg Page 4

6 W i W ( ) i W ( ) (38) Comparing he coefficiens of he above equaion wih hose of (35) and (38), we ge ( A i ) W ( ) H( ), AW ( ) H( ), ( A i ) W ( ) H( ) (39) Noice ha u u( ) W( ( ), ( ), ) q q and q( ) (, ) i e, we ge where K K 3 v, K v, v, K 3 q v[ W () W () 3 ] () () 4 When q () D(, ), we can ge ha g(, ) q () f (, ) D(, ) K K K3 K4 3 4 D ( K K K K ) x ( ) u x( ) () W (,, ) i i e e () W (,, ) () () () () W () W () W () () () () () W () W () W () W ( ), we obain H (,, ) Re[ q () f(, ) q( )] () () () [ W () W ()] ( g( ) g g ) q( ) () () () [ () ()] W W In order o ge he values of g, g, g and g Comparing he cofficiens of he above equaion wih hose in (33), we ge g D K, g D K, g D K, g D K 3 4 (4) In order o deermine he value of g, we also need o compue he values of W ( ) and ( g ( ) g g ) q( ) 3 () 3 () 3 3 From he (3 ) and (33), we obain f (, ) K K K3 K4 (4) Comparing he coefficiens wih (36), we gives ha H H ( ) g ( ) g q( ) g q( ) g (4) When, we have q( ), q( ) ISSN: hp://wwwijejournalorg Page 43

7 H (,,) Re[ q () f (, ) q()] f (, ) ( g g g ) q() ( g ( ) g g ) q() K K K3 K4 Comparing he coefficiens wih (4), we have H() gq() g q(), K H() gq() gq() K (43) Using (39), (4), we obain ig ig W ( ) q() e q() e E e, i i i 3 ig ig W ( ) q() e q() e E i i where E ( E, E ), E ( E, E ) () () () () From he definiion of A () and (39), we have d ( ) W ( ) i W () H(), d( ) W ( ) H() Noice ha i ( i I e d( )) q() i ( i I e d( )) q() Hence, we can ge i ( ii e d( )) E K ( d ( )) E herefore, we have K () i E i () ( k a) e i u E K () E () a ue K (44) hen we can ge E () i ( k a) e k 4 iu E i E () () (45) Similarly, we have K E a () E () (46) K Based on he above analysis, we have he following parameers [7-9]: i g C () ( g g g g ), 3 Re{ C()} ', Re{ ()} Re{ C ()}, ' Im{ C()} (Im{ ()}) (47) which deermine he quaniies of bifurcaing periodic soluion in he manifold a he criical value, now we have he following heorem for he sysem (3) [-] heorem a) he direcion of he Hopf bifurcaion is deermined by he parameer If, he Hopf bifurcaion is supercriical If, he Hopf bifurcaion is subcriical b) deermines he sabiliy of he bifurcaing periodic soluion If, he bifurcaing periodic soluions is sable; if bifurcaing periodic soluions is unsable, he ISSN: hp://wwwijejournalorg Page 44

8 c) he period of he bifurcaing periodic soluion is decided by he parameer If ( ), he period increases(decreases) IV NUMERICAL SIMULAION In his secion, some numerical resuls are presened o confirm he analyical predicions obained in he previous secion We ake he parameers a 5, u 6, q, v 8, k B y simply compuing, we obained ha , From he above arihmeic in secion, If we choose 5, he equilibrium poin (,) of he sysem (3) is asympoically sable proved by numerical simulaions (see Figs -3) If he delay value passes hrough he criical value, he he equilibrium poin (,) loses is sabiliy and a Hopf bifurcaion occurs, namely, here are periodic soluions bifurcaing ou from he equilibrium poin (,) (see Figs4-6) For convenien comparison, we can choose he parameers a 5, u 6, q, v 8, k, namely for he unconrolled model (), we obained ha 598, 3357 When, he equilibrium poin (,) of he sysem () is asympoically sable (,) (see Figs7-9) When 6>, and he periodic soluions occur from he equilibrium (,) (see Figs 9-) Figure Sae plo of x () wih 5 Figure 3 Sae plo of y () wih 5 Figure 4 Phase plo of x ( ) y() wih 3 Figure Phase plo of x ( ) y() wih 5 ISSN: hp://wwwijejournalorg Page 45

9 Figure 5 Sae plo of x () wih 3 Figure 8 Sae plo of x () wih Figure 6 Sae plo of x () wih 3 Figure 9 Sae plo of y () wih Figure 7 Phase plo of x ( ) y() wih Figure Phase plo of x ( ) y() wih 6 ISSN: hp://wwwijejournalorg Page 46

10 Figure Sae plo of x () wih 6 Figure Sae plo of y () wih 6 V CONCLUSION Based on he he conrol and bifurcaion heory, We discussed he effec of he feedbak delay on he sysem Unil now, here are few resuls abou a business cycle model wih feedback delay and we provide an insigh o unexplored aspecs of hem Firs, we inroduce a ime-delayed feedback conroller o his model which aim is o conrol he bifurcaion Second, we derived he condiions for he sabiliy and he exisence of Hopf bifurcaion a he equilibrium of he sysem Moreover, by employing he he cener manifold heorem and he normal form heory, we obained he he direcion of Hopf bifurcaion and he sabiliy of bifurcaion periodic soluions A las, Some compuer simulaion resuls have been presened o illusrae he validiy of he heoreical analysis he research of his paper furher enriches and develops he sudies on business cycle models ACKNOWLEDGMEN he auhors are graeful o he referees for heir helpful commens and consrucive suggesions REFERENCES [] CH Zhang, XP Yan, GH Cui () Hopf bifurcaions in a predaor prey sysem wih a discree delay and a disribued delay Nonlinear Analysis Real World Applicaions, (5), [] J Li, W Xu, W Xie, Z Ren (8) Research on nonlinear sochasic dynamical price model Chaos Solions & Fracals, 37(5), [3] D ao, X Liao, Huang (3) Dynamics of a congesion conrol model in a wireless access nework Nonlinear Analysis: Real World Applicaions, 4(), [4] Puu, Irina Sushko (4) A business cycle model wih cubic nonlineariy Chaos, Solions and Fracals, 9(3), [5] Chuirui Zhang, Junjie Wei (4) Sabiliy and bifurcaion analysis in a kind of business cycle model wih delay Chaos, Solions and Fracals, (4), [6] XD Liu, WL Cai, JJ Lu, e al (5) Sabiliy and Hopf bifurcaion for a business cycle model wih expecaion and delay Communicaions in Nonlinear Science & Numerical Simulaion, 5(-3), 49-6 [7] M Sydlowski, A Krawiec (5) he sabiliy problem in he Kaldor-Kalecki business cycle model Chaos Solions & Fracals 5(), [8] A Kaddar, H alibi Alaoui (8) Hopf bifurcaion analysis in a delayed Kaldor-kalecki model of business cycle Nonlinear Analysis Modelling & Conrol, 3(4), [9] J Yu, M Peng (6) Sabiliy and bifurcaion analysis for he Kaldor-kalecki model wih a discree delay and disribued delay Physica A Saisical Mechanics & Is Applicaions, 46, [] Junhai Ma, Qin Gao (9) sabiliy and Hopf bifurcaions in a business cycle model wih delay Applied Mahemaics and Compuaion, 5(), [] Jinchen Yu, Mingshu Peng, Caiyan Zhang (3) Hopf bifurcaion of a business cycle model wih ime dealy Journal of Beijing Jiaoong Universiy, 37(3), 39-4 [] J Hale (977) heory of Funcional Differenial Equaions, Springer ISSN: hp://wwwijejournalorg Page 47

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