Stability and Analytical Approximation of Limit. Cycles in Hopf Bifurcations of Four-dimensional. Economic Models

Transcription

1 Appled Mathematcal Scences, Vol. 8, 4, no. 8, HIKARI Ltd, Stablty and Analytcal Approxmaton of Lmt Cycles n Hopf Bfurcatons of Four-dmensonal Economc Models M. P. Markaks and P. S. Dours Department of Electrcal and Computer Engneerng Unversty of Patras, GR 654, Greece Copyrght 4 M.P. Markaks and P.S. Dours. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Abstract Paper constructs approxmate analytcal expressons of perodc dsequlbrum fluctuatons - busness cycles - occurrng n connecton wth Hopf bfurcatons n nonlnear problems of economc nteractons descrbed by four-dmensonal contnuous tme dynamcal systems. Two nonlnear macrodynamc models are employed as tests models. The regon of equlbrum stablty n parameter space s obtaned n each case and a Hopf bfurcaton curve s dentfed as a boundary of the regon. Valdty of the analytcal approxmatons obtaned for the cycles generated by the loss of equlbrum stablty on ths curve s confrmed by comparson to numercally determned cycles. Explct analytcal descrpton of such lmt cycles s of partcular nterest n the case of subcrtcal bfurcaton, due to the dffculty of the numercal determnaton of the generated unstable cycles. Mathematcs Subject Classfcaton: 4A4, 4C7, 4C, 7G Keywords: Stablty; Hopf bfurcatons; Lmt cycles; Analytcal expressons

2 968 M. P. Markaks and P. S. Dours. Introducton The man purpose of ths paper s to apply a modfed verson of the engneerng method of harmonc balancng to address the problem of constructng approxmate analytcal expressons of perodc dsequlbrum fluctuatons occurrng n connecton wth Hopf bfurcatons n economc dynamcs. Such fluctuatons are wdely consdered as stylzed mathematcal representatons of busness cycles n contnuous tme models of economc nteractons. Busness cycle theory has been further developed recently by Lorenz [], Gandolfo [5], Aglar and Dec [] and others. Nonlneartes of the models are consdered to be responsble for the occurrence of cycles as an endogenous feature of the economy. The Harmonc Balance (HB) Method, not wdely known among economsts, s a classcal technque for the study of lmt cycles n nonlnear dynamcal systems that has been successfully appled to engneerng problems, ether n ts classc form (wth the use of crcular functons; see e.g. [6], [4]) or n ts generalzed form (use of ellptc functons; see e.g. [5]). In fact the HB method s the Descrbng Functon (DF) Technque ntroduced by Krylov and Bogolybov [9] (see also [6], []) and has been wdely used n classcal Nonlnear Feedback Control Theory (see for example [], [7] and []). In the feld of Economc Dynamcs, the DF method has also been formed as a useful tool n the nvestgaton of relevant nonlnear problems. See for example the applcaton to the classc nonlnear busness cycle model of Goodwn, by Bothwell [4] (the Method of Equvalent Lnearzaton). The DF method n general ams to establsh the exstence of oscllatons of the system, as well as to obtan approxmate analytcal expressons for the frequency and the ampltude of these oscllatons. The verson of the HB method used heren, yelds explct approxmate analytcal expressons of the famly of perodc solutons emergng at a Hopf bfurcaton, wthout requrng a pror nformaton about the behavor of the system (see [8]). It proceeds along lnes smlar to the center manfold procedure and respectve egen-analyss (see []), followed by a perturbaton technque constructng the soluton n seres form step by step. The procedure also yelds the requred nformaton for the determnaton of the nature of the Hopf bfurcaton occurrng, supercrtcal or subcrtcal (equvalently the stablty of the occurrng cycles), n terms of the sgn of the second order term n the resultng expanson of the bfurcaton parameter wth respect to a small perturbaton parameter.

3 Stablty and analytcal approxmaton of lmt cycles 969 Two four-dmensonal models are employed as the test models. The Extended Schnas open economy model proposed by Makovínyová and Zmca [4] and Strežovská and Zmca [], and the Puu s model of nterregonal macrodynamcs [8]. The present work s an extenson to four dmensonal systems of the work presented n Asada et al. []. We explore the test models wth regard to the appearance of busness cycles through Hopf bfurcatons. We consder the effects of the changes of the parameter values on the stablty of equlbrum and determne the stablty regon n a subspace of the parameter space. In ths subspace we further dentfy the Hopf bfurcaton curves and seek to determne the type of Hopf bfurcatons and to obtan an analytcal descrpton of the dsequlbrum fluctuatons occurrng. Although Asada and Yoshda [] have provded a complete mathematcal characterzaton of the four-dmensonal Hopf bfurcaton (referred to as Asada- Yoshda theorem n [5] (p. 48), to the author s knowledge the problem of explct analytcal descrpton of the cycles generated by the change of equlbrum stablty at Hopf bfurcatons has not been addressed suffcently n the lterature of economc dynamcs ([], remark 5). Such descrpton of the occurrng cycles s of nterest especally n the case of subcrtcal bfurcaton, when the generated cycles are unstable and dffcult to obtan numercally, and may prove useful n testng for subsequent bfurcaton of the cycles. The paper proceeds as follows. In secton we present the macrodynamc models under consderaton, and n sectons and 4 we outlne the basc features of the method employed for the determnaton of the type of Hopf bfurcaton and the analytcal approxmaton of the perodc cycles occurrng. In secton 5 we determne the regons of stablty of equlbrum n parameter space, apply the above procedure to the models under consderaton and test the valdty of the analytcal approxmatons obtaned, by comparson to numercally determned cycles. Secton 6 concludes.. The models Two four-dmensonal models are treated n the present work: Model. Extended Schnas s open economy model.

4 97 M. P. Markaks and P. S. Dours Ths model s structured as an extenson to an open economy of the Schnas s model [9], descrbng the development of output, nterest rate and money supply n a closed economy. Thus by consderng also the exchange rate, the followng system of equatons [4] s obtaned: and Y I( Y, R) S Y T Y, R D Y F( Y, ),, (.) FY (, ) C R,, (.) R L( Y, R) L S,, (.) L D Y F( Y, ) C R, (.4) S, IY, R I Y R S Y T Y, R S Y T Y, R,,,, Y R Y R DY DY GTY ( ),, Y F Y, FY (, ) FY (, ) XMY (, ),,, Y C R CR ( ) CmRCxR,, R (.5) (.6) (.7) (.8) LYR, LY (, R),, (.9) Y R where dot denotes dfferentaton wth respect to the tme perod t and Yt () output, () t exchange rate, R() t nterest rate, L ( t) money supply, I nvestments, S savngs, G government expendtures (fxed), T tax collectons, X export, M mport, Cx captal export and Cm captal mport. The system (.) - (.4) has been studed wth regard to equlbrum stablty and the exstence of busness cycles (see for example []). In [4], the exstence of the cycles s more thoroughly nvestgated and formulae for the calculaton of bfurcaton coeffcents are obtaned when the functons I and S are nonlnear n Y. The sgn of these coeffcents determne the exstence and the stablty of the lmt cycles. Let us now adopt the followng functonal form for I, T, S, D, F, C, L [4]: S

5 Stablty and analytcal approxmaton of lmt cycles 97 I( Y, R) Y R, T Y t ty, (.) S Y T Y, R s s Y T Y s R, (.) DY ddy, FY (, ) f fy f, (.) C R c c R, L( Y, R) l ly l R, (.) where the coeffcents,, t, s, s, d, f, f, c, l, l are postve. In ths case the dynamcal system (.) - (.4) s descrbed by the followng equatons: where Y c Y cy cy f cr, (.4), c fy f c R (.5) R l ly l RL S, (.6) L c c Y f c R (.7) S 4 4, c s st d f, c t s t d f, (.8) c s t, c s, (.9) c f c, c d f c, c d f. (.) 4 4 Model. Puu s two-regon busness cycle model. Ths model has been suggested by Puu [8] and t s a typcal example of nonoptmzng four-dmensonal nonlnear macrodynamc model wth contnuous tme. The model conssts of the followng system of equatons: X M m Y my X M, (.) j j Y I X M sy, (.) I v Y Y I, (.) where the subscrpts, j,( j) denote the economc regons (or the countres) and Y ( t) regonal ncome, X ( t) regonal export, M ( t) regonal mport, I ( t) regonal nvestment expendture, m margnal propensty to

6 97 M. P. Markaks and P. S. Dours mport, m, s propensty to save, s and v accelerator coeffcent of nvestment, v. Moreover, Puu [8], by dfferentatng Eqn. (.) and then substtutng Eqns. (.) - (.), derves the followng four-dmensonal system of equatons: Y (.4) W, Y (.5) W, v W smy my v sw W, (.6) v W my s my v sw W, (.7). State transformaton and egen-analyss We consder a nonlnear autonomous system d z Zz,, (.) dt where z ( z, z, z, z4) s the four-dmensonal state vector, s an ndependent real scalar parameter and Z ( Z, Z, Z, Z4) s a vector functon. The nonlnear functons Z( zk, ),, k,,,4 are assumed to be real analytc n the state varables z k and and span R 5 n a regon of nterest. We further assume that (.) has a sngle valued smooth equlbrum soluton z ( ) n R. By consderng now the equlbrum Jacoban of (.) Z J DzZz, Z, jzk,,, j, k,...,4 Z, j, z (.) z z we suppose that J has a par of complex conjugate egenvalues, denoted as r( ) ( ), where r, are smooth functons of, and the remanng two egenvalues are ether a complex par or two dstnct real ones, denoted as r( ) ( ) and r ( ) ( ), where r r n the frst case, whle n the second case. If now the followng holds true: S : The real part of the one par of complex egenvalues, r( ), crosses the axs at a crtcal value wth nonzero speed, whle the real parts of the remanng egenvalues are negatve, at that crtcal value, j

7 Stablty and analytcal approxmaton of lmt cycles 97 then accordng to the Hopf bfurcaton theorem (the smple case) [see [7], Secton.] an oscllatory type of nstablty occurs close to, leadng to a famly of lmt cycles. Moreover, n order to nvestgate such a bfurcaton phenomenon, by usng the crteron of Lu [], f we denote the characterstc polynomal of J ( ) by n n p ; p p... p n, (.) where every p ( ) s a smooth functon of, we have that the suffcent condtons ( S ) of the Hopf theorem are equvalent to the followng condtons on the coeffcents of p( ; ) : C p D D D :,,...,,, C dd n n : / d, n where Dk( ) det( Lk( )), k,..., n, wth L k the k k matrx: L k p p p p, pk pk where p ( ) f or n. Especally for n 4, condtons ( C ) and ( C ) become p, p, p, (.4) p p p p p p, (.5) d p p p p p p d. (.6) Addtonally, for the study of the famly of perodc orbts, the consdered method proceeds n the transformaton of the state varables: T x x x x z z x, x,,, 4. (.7) The matrx T s chosen such that the new Jacoban evaluated at the crtcal value has the block-dagonal form:

8 974 M. P. Markaks and P. S. Dours, j k,. x r J X x r (.8) Here and r, r represent the pure magnary and the complex or real (dependng on the case) egenvalues of J( ), whle X ( x, ),, k,,,4 are the new rght-hand sdes of the system equatons: k d x Xx, T Z T, X, X, X, X4. dt z x (.9) To obtan the block-dagonal form (.8) the matrx T s chosen as T cdpq,,,, (.) where c and d denote the real and the magnary parts, respectvely, of the egenvector correspondng to the complex egenvalue r( ) ( ) of J ( ), whle pq, are ether the real and the magnary parts, respectvely, of the egenvector assocated to the complex egenvalue r( ) ( ), or the egenvectors correspondng to the real egenvalues r, r. Moreover, due to transformaton (.7) we have the followng propertes of the vector functon X : X, X, X,..., (.) where the subscrpt denotes dfferentaton wth respect to the parameter. 4. The perodc solutons As mentoned above, n the case where condtons ( C),( C ) (Eqns. (.4)-(.6) when n 4 ) are satsfed then a famly of perodc solutons of (.) bfurcates from the equlbrum path x at the crtcal value. By ntroducng a perturbaton parameter, we assume that the perodc solutons are n parametrc form: t t t t xx ;, x ; x ;, (4.),, (4.),, (4.)

9 Stablty and analytcal approxmaton of lmt cycles 975 where t s the perod. Furthermore we ntroduce Fourer seres expanson: x t; p p cos mtr sn mt, (4.4) m m m whch by means of the tme scalng t, (4.5) becomes -perodc: x ; p p cosm r sn m. (4.6) m m m Wth ths tme scalng, system (.9) s transformed to x, ; X xk ;,,, k,,,4 (4.7) where the subscrpt denotes dfferentaton wth respect to ths scaled tme varable. In addton (4.6) yelds and x, ; mpmsn m mrm cosm (4.8) p m m r, (4.9) m for all m, snce corresponds to the crtcal pont (, ) n the stateparameter space. Dfferentatng now successvely (4.7) wth respect to and usng (.) and (4.9), we obtan perturbaton equatons of the respectve order (the k th order s generated by k tmes dfferentaton of (4.7)). The equatons up to the second order are: X X X X x x x x x,,..., 4 (4.),, x x x x x x X x X x X x,,, ( ) ( ) ( ) x x x X X X X X X x x x x x x x x x xx xx xx x x x

10 976 M. P. Markaks and P. S. Dours X X X X X x x x,,..., 4 x x x (4.) where prmes denote dervatves wth respect to, evaluated at ( x, ). Note that the thrd order equatons [see [8], Eqn. (4), not lsted here due to ther length] are requred to obtan the second order coeffcent of the expanson of ( ) (Eqn. (4.4) below) whch determnes the type of Hopf bfurcaton and the stablty of the resultng cycles. Thus by evaluatng the dervatves of X at the crtcal pont (, ) and substtutng for k k x ( ;) and x, ( ;) the k -dervatves of the expressons (4.6) and (4.8), respectvely, and then equatng the m -harmonc coeffcents appearng n the perturbaton equatons, we extract the dervatves of pm( ), r ( ) m, ( ) and ( ) at. Wth ths procedure, and puttng p, r, (4.) ordered approxmatons are constructed n the form: x x ; x ; ;..., (4.)..., (4.4)... (4.5) We note that the sgn of () determnes the stablty of the lmt cycles bfurcatng from the crtcal pont (, ). Indeed, when r( ) becomes zero at by crossng the axs wth postve slope, then the followng two stuatons are possble: f () we have unstable lmt cycles coexstng wth the stable equlbrum pont. The Hopf bfurcaton at s subcrtcal; f () stable lmt cycles are generated when equlbrum losses stablty. The Hopf bfurcaton at s supercrtcal.

11 Stablty and analytcal approxmaton of lmt cycles 977 The stablty property for the lmt cycles and the equlbrum path s reversed when r( ) crosses the axs wth negatve slope. In that case () mples supercrtcal and () subcrtcal Hopf bfurcaton. 5. Regons of stablty, Hopf bfurcatons and cycles 5.. Extended Schnas s open economy model. By adoptng now the same values for the coeffcents nvolved n the functonal forms (.) - (.), as n the numercal example presented by Makovínyová and Zmca [4], that s.8,.6,., t., t., (5.) s.8, s., s., (5.) d.7, d., f.6, f.5, f., (5.) c.4, c., l, l.8, l., (5.4) the dynamcal system (.4) (.7) takes the form: Y Y R, (5.5) 7 Y R, 5 5 (5.6) 4 R Y RL S, 5 5 (5.7) 89 L S Y R (5.8) The equlbrum values are: Y,, R, L S 9,,, 9,.547, , (5.9) and the Jacoban at equlbrum s

12 978 M. P. Markaks and P. S. Dours J ;, (5.) The characterstc equaton of J ( ;, ) s p p p p (5.) 4, wth coeffcents p p ( ;, ),,...,: p 9 ;,, (5.) 5 p ;, 49 47, 5 (5.) p ;, , (5.4) p ;, 4. (5.5) By (5.), (5.) and (5.5) we conclude that the condtons (.4) are satsfed, whatever (postve) values the parameters,, may take. Thus the crtcal value s found by fxng (, ) (, ) and solvng Eqn. (.5).

13 Stablty and analytcal approxmaton of lmt cycles 979 Fg.. The stablty regon n parameter plane (, ) ( ). Equlbrum s stable n the rght sde of the boundary curve ; p p p p p p. On the other hand, wth the curve ( ;, ) n the parameter plane (, ) represents the boundary of the stablty regon, as shown n Fg. ( ), snce along ths curve the characterstc equaton has a par of pure magnary roots, whle due to (.4), the real parts of the remanng egenvalues are negatve. Let us now fx (, ) (,) and consder the Hopf bfurcaton occurrng wth as the bfurcaton parameter. In ths case ( ), gven by (.5), becomes ppp pp p (5.6) and ts postve root gves the crtcal value of the parameter, namely (5.7) As regards the knd of the roots of the characterstc equaton (5.), we fnd that f les nsde the nterval.9,.46, then J possesses a par of complex egenvalues together wth two dstnct real ones. For we have, r, r, , (5.8) r , r

14 98 M. P. Markaks and P. S. Dours Fg.. The real part of the complex egenvalues as a functon of the bfurcaton parameter for,,. It becomes zero at by crossng the axs wth negatve slope. Moreover, Fg. demonstrates the fact that the par of complex egenvalues, r( ) ( ), cross the magnary axs wth non-zero speed at the crtcal value (t follows that the equvalent condton (.6) s satsfed). Consequently, the suffcent condtons ( S ) of the Hopf bfurcaton theorem are satsfed. Thus takng nto account that the nequaltes (.4) (concernng the negatve sgn of the remanng egenvalues) are always satsfed, and assumng that r( ) crosses the axs wth non zero speed at every crtcal trplet (,, ), we conclude that the boundary curve of the stablty regon represents a bfurcaton curve, along whch smple Hopf bfurcatons occur. Fnally, we note that the slope of the crossng s negatve. Proceedng now to apply the analyss of sectons and 4 to the present system, frst the change of varables s carred out by means of the 4 4 matrx T of the assocated egenvectors (Eqns. (.7) and (.)): Y Y c d p q x R R c d p q x L S L c S 4 d4 p4 q4 x4 x, (5.9) where c, c, c4, d, d, d 4 are functons of r( ) and ( ), whle p, p, p 4 and q, q, q 4 are functons of r ( ) and r ( ). Moreover the expanson (4.4) (up to the second order) of the bfurcaton parameter gves

15 Stablty and analytcal approxmaton of lmt cycles 98 ( ) ( ), (5.) wth () (5.) Snce r( ) crosses the axs wth negatve slope (see Fg.), as noted at the end of secton 4, the postve sgn of () marks the Hopf bfurcaton as subcrtcal. The occurrng cycles are unstable and coexst wth the stable equlbrum. For the assocated bfurcaton parameter we have that ( ), whle at these unstable cycles collapse on the equlbrum pont causng t to lose stablty. The method outlned n secton 4 provdes analytcal approxmatons of the cycles, gven by Eqn. (4.). Thus, up to second order terms we obtan x( ) cos g( ) O( ), (5.) x( ) sn g( ) O( ), (5.) x( ) g( ) O( ), (5.4) x4( ) g4( ) O( ), (5.5) wth g ( ) q c cos s sn,,,,4, (5.6) where,,, r, r and q c s are numercal coeffcents dependng on the crtcal values. For the specfc values of the parameters n the case treated heren, we have q c s q c s q c s q4 c4 s

16 98 M. P. Markaks and P. S. Dours A comparson of the second order approxmaton wth the numercally determned cycles s shown n Fg. for two values of the bfurcaton parameter. For.68 the analytcal approxmaton s almost ndstngushable from the numercally determned perodc orbt n ths scale. Fg.. Behavour of orbts near the equlbrum pont (large dot n the centre) projected n the Y R(left) and Y LS (rght) planes, for,,.68,, (the nner orbts) and,,.7,,(the outer orbts). The sold closed curves represent the perodc orbts determned by numercal experments. Ther analytcal approxmatons are shown as dashed curves.

17 Stablty and analytcal approxmaton of lmt cycles Puu s two-regon busness cycle model. In ths case, the equlbrum pont of the system of Eqns. (.4) (.7) s located at (,,,) and settng for the parameters s, m,, the values: s., m., s., m., (5.7) the Jacoban of the system at equlbrum s 6 Jv; v v, 5 5 v 5 5 (5.8) whle the coeffcents of the characterstc equaton 4 p p p p of J are: 9 pv; v, (5.9) 5 pv; v 5vv, (5.) 5 pv; v 65v6v 5vv 8, (5.) 5 5 pv; v v v. (5.) By (5.9), (5.) and (5.) we conclude that the condtons (.4) are satsfed f and only f the followng nequaltes hold true: 5 vv and vv (5.) 5 Thus by fxng v v, n the case where (5.) as well as (.6) (wth v) are satsfed, then the soluton(s) of Eqn. (.5) (we set v) represents a crtcal value(s) v (for smple Hopf). On the other hand, as shown n Fg.4, n the parameter plane ( v, v ), the curves ( v; v) (along whch a par of pure magnary roots of the characterstc equaton s obtaned) do not only represent the boundares of the stablty regon, but they also trace paths nsde the

18 984 M. P. Markaks and P. S. Dours nstablty regon. Ths means that by fxng ether of v, v and crossng, f addtonally (.6) s satsfed (wth the unsettled parameter), then a smple Hopf bfurcaton arses when (the equlbrum pont) changes stablty. Fg.4. The stablty regon n parameter plane v, v. Axes orgn s stable n the dark v ; v are also shown. regon. The paths of the curves Βy fxng now v and consderng v as the bfurcaton parameter, the crtcal value s derved as the postve real soluton of Eqn. (.5) ( v), that s v (5.4) In ths case, wth regard to the knd of the egenvalues of J, we have that for v takng values nsde the nterval.8,.7, the characterstc equaton has two pars of complex roots. For v v we have, r, r,.5496, (5.5) r, r , ,4 Furthermore, as n the numercal example of the frst model, Fg.5 shows that the roots rv ( ) ( v) cross the magnary axs wth non-zero speed at the crtcal value v. Thus takng nto account that the nequaltes (5.) are satsfed along

19 Stablty and analytcal approxmaton of lmt cycles 985 the borders of the stablty area, and assumng that rv ( ) crosses the v axs wth non zero speed at every crtcal par ( v, v ), then smple Hopf bfurcatons occur along the parts of the curves whch represent the boundares of the stablty regon. Here the slope of the crossng s postve. Fg.5. The real part of the complex egenvalues as a functon of the bfurcaton parameter v for v. It becomes zero at v by crossng the v axs wth postve slope. As n the frst model, the harmonc balancng analyss used n ths work, frst carres out the change of varables: Y x Y c d p q x, W c d p q x W c d p q x (5.6) where c, c, c4, d, d, d 4 are functons of rv ( ) and ( v), whle p, p, p 4 and q, q, q 4 are functons of r ( v ) and ( v). Moreover the expanson (4.4) (up to the second order ( v () )) of the bfurcaton parameter gves v v ( ) v v( ) v, v (5.7)

20 986 M. P. Markaks and P. S. Dours wth v ().884. (5.8) Here the postve sgn of v () marks the Hopf bfurcaton as supercrtcal, snce rv ( ) crosses the v axs wth postve slope (see Fg.5). We have v ( ) v and the bfurcatng cycles are stable, coexstng wth the unstable equlbrum. Fnally analytcal approxmatons of the cycles, gven by Eqn. (4.), are obtaned here as follows by keepng up to thrd order terms: 4 x( ) cos g( ) g( ) O( ), (5.9) 6 4 x( ) sn g( ) g( ) O( ), (5.4) 6 4 x( ) g( ) g( ) O( ), (5.4) 6 4 x4( ) g4( ) g4( ) O( ), 6 (5.4) wth g ( ),,,, 4, (5.4) g ( ) c cos s sn c cos s sn,,,, 4, (5.44) where,,, c s c s are numercal coeffcents dependng on the crtcal values, r, as well as v. Wth the present numercal values of the parameters these coeffcents are c s c s c s c s c s c s c4 s4 c4 s As n the prevous model, a comparson of the thrd order approxmaton wth the numercally determned cycles s shown n Fg.6 for two values of the bfurcaton parameter.

21 Stablty and analytcal approxmaton of lmt cycles 987 Fg.6. Behavour of orbts near the orgn projected n the Y Y(left) and Y Y (rght) planes, for v, v.4,(the nner orbts) and,.55, v v (the outer orbts). The sold closed curves represent the perodc orbts determned by numercal experments. Ther analytcal approxmatons are shown as dashed curves. 6. Concludng remarks We explored two contnuous tme nonlnear economc models as test models of four-dmensonal contnuous dynamcal systems governng problems of economc nteractons, where Hopf bfurcatons occur. In partcular the correspondng bfurcatons curves have been traced n a subspace of the parameter space, where the crossng of the boundary of the stablty regon generates smple Hopf bfurcatons of lmt cycles.

22 988 M. P. Markaks and P. S. Dours Further, by applcaton of a modfed verson of the analytcal method of harmonc balance, approxmate analytcal expressons of the emergng famly of perodc solutons have been constructed n trgonometrc seres form. By the followed procedure we also determne the nature of the Hopf bfurcaton occurrng, equvalently the stablty of the emergng cycles. Explct analytcal formulae of such dsequlbrum fluctuatons generated by the change of stablty at Hopf bfurcatons s of partcular nterest n the case of subcrtcal bfurcaton, as n the frst model treated heren, due to the dffculty of the numercal determnaton of the bfurcatng unstable cycles. Such analytcal descrpton may n future work prove useful n testng for subsequent bfurcaton of the famly of cycles. References [] A. Aglar and R. Dec, Coexstence of attractors and homoclnc loops n a Kaldor-lke busness cycle model, n Busness Cycle Dynamcs. Models and Tools, Sprnger-Verlag, 6. [] T. Asada and H. Yoshda, Coeffcent Crteron for four-dmensonal Hopf Bfurcatons: A Complete Characterzaton and Applcaton to Economc Dynamcs, Chaos Solton Fract., 8 () (), [] T. Asada, V. Kalantons, M. Markaks and P. Markellos, Analytcal Expressons of Perodc Dsequlbrum Fluctuatons Generated by Hopf Bfurcatons n Economc Dynamcs, Appl. Math. Comput., 8 (), [4] F.E. Bothwell, The Method of Equvalent Lnearzaton, Econometrca, () (95), [5] G. Gandolfo, Economc Dynamcs, 4 th ed., Sprnger, Berln, 9. [6] A. Gelb, and W.E. Vander Velde, Multple Input Descrbng Functons and Nonlnear System Desgn, McGraw-Hll Book Co., New York, 968 [7] J. Hale and H. Koçak, Dynamcs and Bfurcatons, Texts n Appled Mathematcs, Sprnger-Verlag, New York, 99.

23 Stablty and analytcal approxmaton of lmt cycles 989 [8] K. Huseyn and A.S. Atadan, On the analyss of Hopf bfurcatons, Int. J. Eng. Sc., (98), [9] N. Krylov and N. Bogolyubov, Introducton to Nonlnear Dynamcs, Prnceton Unversty Press, 947. [] Y.A. Kuznetsov, Elements of Appled Bfurcaton Theory, rd ed., Sprnger, New York, 4. [] G.A. Leonov and N.V. Kuznetsov, Algorthms for Searchng for Hdden Oscllatons n the Azerman and Kalman Problems, Dokl. Math., 84 () (), [] W.M. Lu, Crteron of Hopf Bfurcatons wthout Usng Egenvalues, J. Math. Anal. Appl., 8 (994), [] H.W. Lorenz, Nonlnear Dynamcal Economcs and Chaotc Moton, Sprnger-Verlag, 99. [4] K. Makovínyová and R. Zmca, On the Exstence of Hopf Bfurcaton n an Open Economy Model, n Equadff-, Proceedngs of Ιnt. Conf. on Dfferental Equatons, July 5-9, Bratslava, Slovaka, 5, pp [5] J.G. Margallo, J.D. Bejarano, and S.B. Yuste, Generalzed Fourer seres for the study of lmt cycles, J. Sound Vb., 5 (988), -. [6] R.E. Mckens and K. Oyedej, Constructon of approxmate analytcal solutons to a new class of nonlnear oscllator equatons, J. Sound Vb., (985), [7] P.W.J.M. Nuj, O.H. Bosgra and M. Stenbuch, Hgher Order Snusodal Input Descrbng Functons for the Analyss of Nonlnear Systems wth Harmonc Responses, Mech. Syst. Sgnal Pr., (8) (6), [8] T. Puu, Nonlnear Economc Dynamcs, 4 th ed., Sprnger-Verlag, Berln and New York, 997. [9] J.G. Schnas, Fluctuatons n a Dynamc, Intermedate-Run IS-LM Model: Applcatons of the Poncaré Bendxon Theorem, J. Econ. Theory, 8 (98), [] L. Strežovská and R. Zmca, Cycles n an Open Economy Model under Fxed Exchange Rate Regme wth Pure Money Fnancng, Oper. Res. Dec. Quarterly, -4, Wroclaw, (), -4.

24 99 M. P. Markaks and P. S. Dours [] J.H. Taylor, A General Lmt Cycle Analyss Method for Multvarable Systems, n Holmes P.J. (Ed.), New Approaches to Nonlnear Problems n Dynamcs, SIAM, 98, pp [] J.H. Taylor, Descrbng Functons n Electrcal Engneerng Encyclopeda, John Wlley & Sons Inc., New York, 999. [] K.V. Veluplla, A dsequlbrum macrodynamc model of fluctuatons, Journal of Macrodynamcs, 8 (6), [4] S.B. Yuste and J.D. Bejarano, Constructon of approxmate analytcal solutons to a new class of nonlnear oscllator equatons, J. Sound Vb., (986), Receved: Aprl, 4