Goodwin Accelerator Model Revisited with Piecewise Linear Delay Investment

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1 Advnes in Pure Mthemtis SSN Online: 6-8 SSN Print: 6-68 Goodwin Aelertor Model Revisited with Pieewise Liner Dely nvestment Aio Mtsumoto Keio Nym Feren Szidrovszy Deprtment of Eonomis nterntionl Center for Further Development of Dynmi Eonomi Reserh Chuo University Hhioji Jpn Deprtment of Eonomis Chuyo University Ngoy Jpn Deprtment of Applied Mthemtis University of Pés Pés Hungry How to ite this pper: Mtsumoto A Nym K nd Szidrovszy F (8 Goodwin Aelertor Model Revisited with Pieewise Liner Dely nvestment Advnes in Pure Mthemtis Reeived: Jnury 8 Aepted: Ferury 5 8 Pulished: Ferury 8 8 Astrt t is well-nown tht Goodwin s nonliner dely elertor model n generte diverse osilltions (ie smooth nd swtooth osilltions t is however less-nown wht onditions re needed for these dynmis to emerge n this study using pieewise liner investment funtion we solve the governing dely differentil eqution nd otin the expliit forms of the time trjetories n doing so we detet onditions for persistent osilltions nd lso onditions for the irth of suh yli dynmis Copyright 8 y uthors nd Sientifi Reserh Pulishing n This wor is liensed under the Cretive Commons Attriution nterntionl Liense (CC BY Open Aess Keywords Nonliner Aelertor nvestment Dely Suessive ntegrtion Smooth nd Kined Solutions Numeril Anlysis ntrodution t hs een well-nown tht Goodwin s usiness yle model with delyed nonliner elertor [] n generte multiple solutions Depending on speified forms of the initil funtions nd speified prmeter vlues it gives rise to smooth yli osilltions or swtooth (ie slow-rpid osilltions This pper ims to nlytilly nd numerilly investigte these yli properties of Goodwin s model y solving the time dely equtions nd performing simultions [] presents five different versions of the nonliner elertor-multiplier model with investment dely The first version hs the simplest form ssuming pieewise liner funtion with three levels of investment nd ims to exhiit how non-linerities give rise to endogenous yles without relying on struturlly un- DO: 6/pm88 Fe Advnes in Pure Mthemtis

2 A Mtsumoto et l stle prmeters exogenous shos et The seond version reples the pieewise liner investment funtion with smooth nonliner investment funtion Although persistent ylil osilltions re shown to exist the seond version inludes unfvorle phenomen tht is disontinuous investment jumps whih re not oserved in the rel eonomi world n order to ome lose to relity ([] p the third version introdues n investment dely However no nlytil onsidertions re given to this version The existene of n endogenous usiness yle is onfirmed in the fourth version whih is liner pproximtion of the third version with respet to the investment dely Finlly lterntion of utonomous expenditure over time is ten into ount in the fifth version whih eomes fored osilltion system This pper reonstruts the third version hving pieewise liner investment funtion with fixed time dely t is omplement to [] in whih the effets used y investment dely s well s onsumption dely re onsidered t is lso n extension of [] in whih the dynmis of Goodwin s model is exmined under ontinuously distriuted time delys nd the existene of the multiple limit yles is nlytilly nd numerilly shown Following the method of suessive integrtion provided y [] we derive expliit forms of the solutions nd otin onditions under whih the smooth or swtooth osilltions emerge With the sme spirit [5] exmines Goodwin s model Their fous is minly put on the relxtion (ie swtooth osilltions We step forwrd nd investigte periodi properties of the smooth osilltions whih will e lled Goodwin osilltions heneforth Our min onerns in this pper re on the role of the fixed dely for the irth of yli mro dynmis The pper is orgnized s follows n Setion the Goodwin model without dely is onsidered to see how nonlinerities of the model ontriute emergene of yli dynmis n Setion n investment dely is introdued to onstrut Effets of investment dely on the smooth osilltions re onsidered in Setion nd those on the swtooth osilltions re done in Setion 5 Setion 6 ontins some onluding remrs Bsi Model To find out how nonlinerity wors to generte endogenous yles we review the seond version of Goodwin s model whih we ll the si model ( t ( t ( α y( t ( ( t ϕ ( t Here is the pitl sto y the ntionl inome α the mrginl propensity to onsume whih is positive nd less thn unity nd the reiprol of is positive djustment oeffiient Sine the dot over vriles mens time differentition ( t nd ( t re the rtes of hnge in pitl (ie investment nd ntionl inome The first eqution of ( defines n djustment proess of the ntionl inome in whih ntionl inome rises or flls ording to whether DO: 6/pm88 79 Advnes in Pure Mthemtis

3 A Mtsumoto et l investment is lrger or smller thn svings The seond eqution determines the indued investment sed on the elertion priniple with whih investment depends on the rte of hnges in the ntionl inome in the following wy n n if ( t U ν n n ϕ ( t ν( t if ( t M ν ν n n if ( t L ν ( ν > nd n > This funtion is pieewise liner nd hs three distint regions Aordingly there re two threshold vlues of ( t denoted s n ν nd n ν nd the investment is proportionl to the hnge in the ntionl inome in the middle region M ut eomes perfetly inflexile (ie inelsti in the upper region U or the lower region L These vlues re thought to e eiling nd floor of investment the eiling is ssumed to e three time higher thn the floor s it ws the se in Goodwin s model nserting the seond eqution of ( into the first one nd moving the terms on the right hnd side to the left give n impliit form of the dynmi eqution for ntionl inome y ( ( ( ( ( t ϕ t + α y t ( the sttionry point of the si model is y( t y( t Eqution ( it is redued to either ( ν y( t ( y( t if ( t is in the middle region or ( ( α ( for ll t With ( t y t + d (5 d n t is in the upper region nd d n if in the lower region Equtions ( nd (5 re liner nd thus solvle We see grphilly nd then nlytilly how dynmis proeeds if ( Phse Plot Solving ( nd (5 for y( t presents n lterntive expression of dynmi eqution y( t f ( t n ( t if ( t U ( ν ( t f ( t if ( t M (6 n + ( t if ( t One the initil vlue is given the whole evolution of ntionl inome is de- L DO: 6/pm88 8 Advnes in Pure Mthemtis

4 A Mtsumoto et l termined The phse digrm with < ν is shown in Figure in whih y f ( is desried y mirror-imged N-shped urve in the ( yy plne The sttionry point is t the origin denoted y E The lous of y f ( is the positive sloping line in the middle region while it is the negtive sloping upper or lower line in the upper or lower region For eh vlue of there is unique orresponding y vlue determined to me point ( yy stisfy Eqution (6 nd it is lso determined whether y is inresing or deresing t tht point So the diretion of the trjetory is given in ll points of the phse digrm The diretions re shown y rrows Let A denote the lol mximum point of the urve with positive y nd lwys nd let C e the lol minimum point with negtive y nd Point B nd D hve the sme y vlues s t points A nd C respetively Notie tht the diretion of the dynmi evolution goes from to C from the origin to C from the origin to A nd lso from + to A Seleting the initil point denoted s S on the positive-sloping line the evolution strts t this point nd moves upwrd until point A s indited y rrow Then it nnot ontinue on the ontinuous urve fter point A sine the diretion of evolution hnges Therefore it jumps to point B nd ontinues long the sme diretion until point C the sme prolem ours so nother jump ours to point D nd evolution ontinues until point A t whih the next round repets itself Thus the differentil Eqution ( with the pieewise liner investment funtion ( n give rise to losed orit ABCD onstituting self-sustining slow-rpid osilltion The sttionry point is unstle if < ν however the osilltion is stle in sense tht the lous ( yy sooner or lter onverges to the sme osilltion regrdless of Figure Slow-rpid osilltion long y f ( with ν > Mthemti version is used to perform simultions nd illustrte this nd the following figures The olor versions of the figures re found in DP78 t DO: 6/pm88 8 Advnes in Pure Mthemtis

5 A Mtsumoto et l seletion of the initil point This is simple exhiition of emerging stle endogenous yle of ntionl inome The vertex of the losed osilltion in Figure re n n A y B y y C y nd D y y ν ν ( ( mx m mx min M min the mximum nd minimum vlues of long the yle re ( ν n( ν n M nd m ν while the mximum nd minimum vlues of y long the yle re y mx ( ( ν ( ( n ν n ν nd ymin ν α ν α t should e notied tht the instility nd the nonlinerity is ruil soures for the irth of persistent osilltions sine the instility of the sttionry point prevents trjetories from onverging nd the nonlinerities suh s the eiling nd floor prevent trjetories from diverging Jumping ehvior leds to the ined time trjetory of y( t nd the disontinuous time trjetory of ( t tht re shown s the lue nd red urves in Figure A B holds t the upper ined point of the lue urve nd C D t the lower ined point The red trjetory from to t desries the movement of ( t from point S to A At time t when the left-most red urve defined on intervl [ t ] rrives t the upper horizontl dotted line the red urve jumps strightly down to the strting point of the lower red urve defined on intervl [ t t ] the point of whih orrespond to point B At time t the red urve rosses the lower dotted line from elow nd the intersetion orresponds to point C t whih the red urve jumps strightly Figure Time trjetories of y( t (the ined lue urve nd y( t (the disontinuous red urves DO: 6/pm88 8 Advnes in Pure Mthemtis

6 A Mtsumoto et l up to the downwrd-sloping red urve defined on intervl [ ] t t tht then rosses the upper horizontl dotted line nd the ross-point orrespond to point A Figure illustrtes the sme dynmis of Figure from different view point Expliit Solutions Seleting n initil point we n determine n expliit form of the orresponding time trjetory nd its rte of hnge n prtiulr we te n initil point on the positive sloping prt of the y( f ( urve suh s n < y( onstnt nd ( y( < ν ν ymin < y( < ymx f y( t expliit forms of the solution ( is in the middle region Eqution ( yields y t nd its time derivtive t t α ν ν ( e nd ( e with ( y t K t K K y (7 ν nd if ( t enters the upper or lower region Eqution (5 yields the following forms of the solution y( t nd its time derivtive ( α t ( α d i α i i i i y t e K + nd y t e K for i (8 di n if is even nd di n if is odd Sine ( < n ν nd y ( t presents n rrivl time t t ν n ( ν t log ν ( K Sustituting t into y ( t nd y ( t implying tht point ( y( t y( t is inresing in t solving ( yields n y ( t ymx nd ( t ν t n ν for t orresponds to point A in Figure Solutions in (7 desrie the movement from point S to point A At t t the dynmi system ( is swithed to Eqution (5 with d n Equtions in (8 with i presents expliit forms of the solution nd its time derivtive ( α ( α n α y t e K nd y t e K (9 Sine the time trjetory y( t is ontinuous in t solving y ( t y ( t gives the vlue of K K n ν ( ν ω ( e With this K we hve t nd y t y ( ( m mx DO: 6/pm88 8 Advnes in Pure Mthemtis

7 A Mtsumoto et l Thus point ( y( t y( t orresponds to point B to whih point A jumps This rpid hnge is desried y the vertil movement of the red urve long the vertil line t t t in Figure Sine ( t < n ν nd ( t inreses in t solving ( t n ν gives neessry time to rrive t n ν ω ν t t+ log n the sme wy ove it is possile to show tht n ( ( t y ( t y ν min whih is point C The movement from point B to C is desried y solutions in (9 At time t t the dynmi system with d n is swithed to the dynmi system with d n nd (8 with i presents expliit forms of the solutions n α ( + ( y t e K nd y t e K Solving y ( t y ( t presents the vlue of K n( ν K e ν ( t is lso le to e shown tht point ( y( t y( t D implying jump to point D from point C Solving ( time when ( t rrives t n ν ν t t + log ( is identil with point t n ν presents At t t we n onfine tht the trjetory omes to point A t whih the following holds n ( ( t y( t ym ν A new round strts s time goes further nd the sme proedure is pplied to otin expliit forms of the solutions for t t Time segments of y( t nd ( t tht onstitute one yle of ntionl inome re now given y (9 nd ( Sine the length of one yle is mesured y the time period etween one upper (or lower ined point nd next upper (or lower ined point it is given y ( ν ( ν t log Further the length of the reession period long segment BC in whih ntionl inome is deresing is ω ν t log while the length of the reovery period long segment DA in whih ntionl in- DO: 6/pm88 8 Advnes in Pure Mthemtis

8 A Mtsumoto et l ome is inresing is ν t log n wht follows we will perform numeril simultions with the set of the prmeter vlues given elow whih re the sme prmeter vlues used in [] nd [] Needless to sy these prtiulr vlues of the prmeters re seleted only for nlytil simpliity nd do not ffet qulittive spets of the results to e otined Assumption α 6 5 ν nd n n prtiulr Figure nd Figure re illustrted under Assumption nd the initil vlues of y ( nd y ( 8 5 the pir ( ( y( orresponds to point S in Figure The ritil times t whih the system swithing ours re given t 7998 t t 6 t 6 t 7998 nd t 6 y solving 5 6 n n i( t if iis odd or i( t if iis even ν ν The length of one yle given y t i+ t is out 58 i yers n the sme wy the reession period from one pe to trough of the yle is given y ti i 6 yers for i eing even while the reovery period from one trough to pe y ti i yers for i eing odd The onstnt K i solves y t y t ( ( i i i+ i with K nd the numeril results re s follows K 65 K 967 K K nd K Dely Model 5 6 We now investigte how the investment dely ffets time pths of ntionl inome Oserving the ft tht in rel eonomy plns nd their reliztions need time to te effets [] introdues the investment dely > etween deisions to invest nd the orresponding outlys in order first to ome loser to relity nd seond to eliminte unrelisti disontinuous jumps Consequently the investment funtion (6 is modified s follows ( ( ( ( n if y t in U ϕ ( t ν t if t in M n if y t in M With this modifition the dynmi Eqution ( turns to e ( ( ( ( ( ( t ϕ t + α y t ( n [] it is ssumed tht is one yer dely Under the speified vlues of the prmeters t 58 i+ i implying the length is onsidered to e 58 yers DO: 6/pm88 85 Advnes in Pure Mthemtis

9 A Mtsumoto et l tht we ll the dely model Eqution ( is redued to liner dely differentil eqution of neutrl type if the delyed rte of hnge in ntionl inome stys in the middle region ( ( ( ( t ν t + α y t ( nd it remins to e liner ordinry differentil Eqution (5 if the delyed rte is in the upper or lower region To solve the dely eqution we need n initil funtion tht determines ehvior of y prior to time zero ( Φ( for y t t t Although [] does not nlyze dely dynmis generted y the third version [] in ddition to numeril nlysis derive the expliit forms of the pieewise ontinuous solutions of y( t under the pieewise liner investment funtion ( We follow their method of suessive integrtion to solve the dely eqution nd derive the expliit forms of time trjetories of y( t nd ( t Sine yli osilltion hs een shown to exist in the si model our min onern here is to see how the presene of the investment dely nd the seletion of the initil funtion ffet hrteristis of suh swtooth osilltion otined in the si model t hs een exmined y [] tht the irth of osilltions in the Goodwin model re used y seleted form of the initil funtion nd the length of dely For the se of nlytil simpliity we ssume the onstnt initil funtion in the following numeril simultions Assumption Φ ( t y nd Φ ( t for t ( Lol Stility t is well nown tht if the hrteristi polynomil of liner neutrl eqution hs roots only with negtive rel prts then the sttionry point is lolly symptotilly stle The norml proedure for solving this eqution is to try n exponentil form of the solution Sustituting y( t y e λt into ( nd rerrnging terms we otin the orresponding hrteristi eqution: ( λ λ νλe + α To he stility we determine onditions under whih ll roots of this hrteristi eqution lie in the left or right hlf of the omplex plne Dividing oth sides of the hrteristi eqution y nd introduing the new vriles ν > nd > (5 we rewrite the hrteristi eqution s λ λe λ + (6 [6] derive expliit onditions for stility/instility of the n-th order liner slr neutrl dely differentil eqution with single dely Sine ( is spe- Goodwin ssumes smooth form of the nonliner investment funtion in his third version DO: 6/pm88 86 Advnes in Pure Mthemtis

10 A Mtsumoto et l il se of the n-th order eqution pplying their result (ie Theorem leds to the following: the rel prts of the solutions of Eqution (6 re positive for ll > if > The first result on the fixed dely model is summrized s follows: Lemm f ν > then the zero solution of the fixed dely model ( is lolly unstle for ll > On the other hnd if v or hrteristi Eqution (6 hs t most finitely mny eigenvlues with positive rel prts The eigenvlue is rel nd negtive when The roots of the hrteristi eqution re funtions of the dely Although it is expeted tht ll roots hve negtive rel prts for smll vlues of the rel prts of some roots my hnge their signs to positive from negtive s the lengths of the dely inreses The stility of the zero solution my hnge Suh phenomen re often referred to s stility swithes We will next prove tht stility swithing however nnot te ple in the delyed model Lemm f v then the zero solution of the fixed dely model ( is lolly stle for ll > Proof t n e heed tht λ is not solution of (6 euse sustituting λ yields tht ontrdits > f the stility swithes t then (6 must hve pir of pure onjugte imginry roots with Thus to find the ritil vlue of we ssume tht λ iω with ω > is root of (6 for > Sustituting λ iω into (6 we hve ω sinω nd ω ω osω Moving nd ω to the right hnd sides nd dding the squres of the resultnt equtions we otin ( + ω Sine > nd > s < is ssumed there is no ω tht stisfies the lst eqution n other words there re no roots of (6 rossing the imginry xis when inreses No stility swith ours nd thus the zero solution is lolly symptotilly stle for ny > n se of ν in whih the hrteristi eqution eomes λ ( λ e + (7 t is ler tht λ is not solution of (7 sine > Thus we n ssume tht root of (7 hs non-negtive rel prt λ u + iv with u for some > From (7 we hve the lst inequlity is due to u ( u+ + v ( u + v ( u + v e u e for u nd > Hene u + DO: 6/pm88 87 Advnes in Pure Mthemtis

11 A Mtsumoto et l the diretion of inequlity ontrdits the ssumption tht u nd > Hene it is impossile for the hrteristi eqution to hve roots with nonnegtive rel prts Aordingly ll roots of (7 must hve negtive rel prts for ll > Lemms nd imply the following theorem onerning lol stility of the dely model ( Theorem For ny > the zero solution of the dely model ( is lolly symptotilly stle if ν nd unstle if ν > We ll y in Assumption n initil vlue for onveniene Fixing the length of dely t we illustrte ifurtion digrm with respet to the initil vlue in Figure For given vlue of y the dynmi system runs for t T 5 The solution for t 5 re disrded to eliminte the initil disturnes nd the mximum nd minimum vlues of the resultnt solutions for 5 t 5 re plotted ginst y The ifurtion prmeter y inreses from to 6 with inrement of nd for eh vlue of y the sme lultion proedure is repeted As is seen in Figure nd lredy pointed out y [5] the dely dynmi system with the onstnt initil funtion hs the two threshold initil vlues mx min y nd y suh tht the swtooth osilltions min mx rise for y y y nd so do the Goodwinin osilltions otherwise These vlues depend on the length of the dely nd re numerilly determined mx min s y 9 nd y 68 under n the following we first set G y y 5 nd onsider Goodwinin osilltions in Setion nd then exmine swtooth osilltions with y y in Setion S 5 Goodwinin Osilltions G Given Assumptions nd with y y the time trjetories of y( t nd ( t re illustrted y the lue nd red urves respetively in Figure in Figure Bifurtion digrm with respet to y DO: 6/pm88 88 Advnes in Pure Mthemtis

12 A Mtsumoto et l Figure Time trjetories of y( t nd y( t whih we n see tht dely time trjetories show shrp differenes from non-dely time trjetories depited in Figure The intervl inluding the whole prts of one yle y( t strts t point S nd ends t point E is divided into eight suintervls eh of whih is distinguished y hevy or light gry olor Solving non-dely dynmi Eqution (5 or dely dynmi Eqution ( we will derive the expliit forms of these trjetories in eh suintervl the detiled derivtions re presented in Appendix Time Trjetories We omit onsidertion in intervl [ t S ] with ts strongly depends on hoie of initil point n the first intervl [ t t ] s ehvior there S t t + nd t n ν the lue nd red trjetories re ontrolled y Eqution (5 with d n ( G- t solves the eqution ( n y ( t e K for t α ( t e K K 9 At t t t rosses the lower horizontl dotted line t n ν nd the rossing point is denoted y the left most green dot in Figure The oundry vlues of this intervl re y t t 76 nd y t 666 t 67 the red trjetory ( ( ( ( ( S S As seen in Figure the lue trjetory is ined nd the red urve jumps downwrd t t ts This disontinuity is shown s follows Let y ( t nd ( t e solution nd its derivtive in intervl [ t S ] Then onstnt K is The expliit forms re given in Appendix DO: 6/pm88 89 Advnes in Pure Mthemtis

13 A Mtsumoto et l determined so s to stisfy y ( ts y ( ts tht is the end point of ( S oinided with the strting point of ( S of y( t t t t Seondly the solutions of y ( t nd y ( S the following dynmi equtions respetively ( ts + ( α y( ts t + α y t n y t is y t Hene it first implies the ontinuity ( ( ( S S Sutrting the seond eqution from the first presents t t n t t ( ( > or ( > ( S S S S the lst inequlity implies disontinuity of the derivtive t t should stisfy t t n the seond intervl [ t t ] with t t + 5 pplying suessive integrtion for dynmi eqution ( t + ( α y( t ν ( t gives expliit forms of the solutions ( G- The integrl onstnt y t t y t t ( e ( α + α ( e ( β + β ( β α α 567 for t α ν α e K 59 β α 7 α 9 oundry vlues for the end points of intervl α is otined y solving y ( t y ( t re ( ( ( ( S The y t 666 y t 67 nd y t 569 y t 85 t n e numerilly s well s grphilly heed tht y ( t y ( t ( t ( t n onsequene y ( t nd y ( (ie ontinuous nd differentile On the other hnd y( t n ν indues system hnge t t t leding to tht ( t nd y ( t for t t re onneted with in (ie ontinuous nd non-differentile For t in the third intervl [ t t ] n ν > ( t > n ν ( with Φ y ( t νy ( t nd t re smoothly onneted with t t + 6 we hve Hene suessive integrtion implies tht Eqution yields the trjetories desried y ( G- y t t t y t t t ( e ( α + α + α ( e ( β + β + β for t DO: 6/pm88 9 Advnes in Pure Mthemtis

14 A Mtsumoto et l ν β α e 57 β α 9 9 ν ( α e β β 5558 β α α 5568 α 6 β α α 679 Sine y ( t y ( t nd y ( t y ( t hold the lue nd red tr- t t The oundry vlues for the right endpoint of jetories re ontinuous t intervl re ( ( y t 55 nd y t 698 rosses the upper horizontl dotted line from elow t 59 t n ν for t < t < t This rossing point is lso denoted y the green dot n the fourth intervl [ t t ] with t t + 69 Eqution ( with Φ ( t ν ( t determines the trjetories y ( t e ( α t + α t + α t+ α ( G- for t y ( t e ( β t + β t + β t+ β As is seen in Figure the red urve y ( t t nd ( < ν β α e 7576 β α 6 ν β β α e 87 β α α 655 α β β + β 87 β α α 8799 α 6796 β α α Sine intervl is very nrrow the right end point of is lelled t the upper prt of Figure to void the nottionl ongestion Due to the ontinuity of the lue nd red trjetories t t t y ( t y ( t nd ( t ( t hold The oundry vlues for the right end point of intervl re lulted s ( ( y t 9 nd y t 6 Sine ( t > n ν holds in the fifth intervl [ t t ] nd t 778 solves ( t n ν y ( t n t t Eqution ( with Φ gives the following forms of the solution nd its time derivtive ( G- n y ( t e K + α ( t e K for t DO: 6/pm88 9 Advnes in Pure Mthemtis

15 A Mtsumoto et l K 895 The rossing point of the red urve with the upper horizontl dotted line t t is denoted y the green dot The oundry vlues for the right end point of this intervl re ( ( y t 997 nd y t de- Sine y ( t < n ν holds for t of the sixth intervl [ t t ] t t dynmi Eqution ( with ϕ y ( t νy ( t termines the following evolution of y( t nd ( t : ( G- α β α y t t y t t ( e ( α + α ( e ( β + β ( α ν e K 9 α β α α 89 The oundry vlues for the right end point of intervl ( ( with for t y t 76 nd y t 556 re t is seen tht the red urve rosses the lower horizontl dotted line from ove t td 95 n the seventh intervl [ t t ] with t t + 5 pplying suessive integrtion to dynmi Eqution ( with ( ( ϕ t ν t yields the forms of the solutions y ( t e ( α t + α t+ α ( G- for t y ( t e ( β t + β t+ β ν β e α 7997 β α ν B e ( α β β 88 β α α 8678 α 7986 β α α 7668 The oundry vlues for the right end point of intervl ( ( y t 8 nd y t 76 Finlly in the eighth intervl [ t te ] ( < t t n ν for d re with t t + 5 implies dynmi Eqution (5 with E DO: 6/pm88 9 Advnes in Pure Mthemtis

16 A Mtsumoto et l ( ϕ t n ontrols the trjetories n y ( t e K ( G- α ( t e K for t 669 y te y ts te ts hold The length of the period is out yers 5 ery roughly speing the reovery period ould e pproximtely 7 yers from t to t nd then the reession period is 5 yers The sme yle repets itself for t > t K Notie tht ( ( nd ( ( Phse Plot Clulting the oundry vlues of eh intervl i we hve the following set of t y t in the phse digrm of Figure 5 points ( ( ( ( S ( 76 ( ( ( ( ( ( 7 55 ( 5 ( 6 9 ( 6 ( 997 ( 7 ( ( 8 ( 768 ( E ( 76 Point denoted y (S nd (E is the strting point nd the ending point of the yli osilltion oth of whih re identil Eqution (5 with d n governs deresing movements from (S to ( nd from (8 to (E long the lower red line s Eqution (5 with d n ontrols upwrd movements E Figure 5 Phse digrm of Goodwin Eqution ( 5 With the sme prmeter vlues ut different initil vlues [] nlytilly otined 9 yers yle nd [] numerilly got n 8 yer yle DO: 6/pm88 9 Advnes in Pure Mthemtis

17 A Mtsumoto et l from (5 to (6 long the upper red line On the other hnd movements from ( to (5 nd from (6 to (8 long the dotted urves etween these two lines re desried y Eqution ( The swithing of dynmi equtions ours t the following points: Point ( t whih Eqution (5 with d n is hnged to Eqution (; Point (5 t whih Eqution ( to Eqution (5 with d n ; Point (6 t whih Eqution (5 with d n to Eqution (; Point (8 t whih Eqution ( to Eqution (5 with d n ; Points ( ( nd (7 t whih Equtions ( hve t different forms of Φ ( t We n verify the following Theorem The yli time trjetories of y( t nd ( t re ontinuous t these swithing points Proof At point ( with t t y ( t is onneted to y ( t nd so is ( t to ( t ntegrl onstnt α of y ( t is determined so s to solve y ( t y ( t Further y ( t nd y ( t should stisfy the dynmi equtions t t t ( t + ( α y ( t n t + α y t ν t ( ( ( ( ( t ( t holds t t t + nd ( t oth of whih led to ν ( t n t n ν y definition of Therefore the ove two dynmi equtions re identil nd thus two solutions of these dynmi equtions te the sme vlues t t t nmely y ( t y ( t nd ( t ( t At point (5 with t t y ( t is onneted to y ( t nd so is ( t to ( t ntegrl onstnt K of y ( t is determined so s to solve y ( t y ( t Further y ( t nd y ( t should stisfy the dynmi equtions t t t ( t + ( α y ( t ( t t + α y t n ( ( ( ( t ( t holds t t t + nd ( t oth of whih led to ( t n t n ν y definition of Therefore the ove two dynmi equtions re identil nd thus two solutions of these dynmi equtions te the sme vlues t t t nmely y ( t y ( t nd ( t ( t The sme proedure pplies for points (6 nd (8 At point ( with t t y ( t nd y ( t stisfy the dynmi equtions ( t + ( α y ( t ( t t + α y t t ( ( ( ( ( t ( t nd ( t ( t s lredy hve y ( t y ( t solves y ( t y ( t t t + From (i we Further the integrl onstnt α of y ( t Then sustituting the seond eqution from the first DO: 6/pm88 9 Advnes in Pure Mthemtis

18 A Mtsumoto et l The sme proedure pplies for Points ( nd (7 This theorem onfirms no jumps of the derivtives t the swithing points of the dynmi system implying the smooth time trjetory of ntionl inome just lie oserved usiness yle This is wht [] ims to otin So we summrize this results s follows: Theorem 5 f the initil vlue y of the initil funtion nd the length of dely re seleted suh s y min < y ( or y mx > y ( then the dely model n generte smooth osilltions of ntionl inome eqution presents y ( t y ( t 5 Swtooth Osilltions S Under Assumptions nd with y o Figure 6 illustrtes trjetories of y( t (lue urve nd ( t (red urve for t [ 5] The lue trjetory hs ins nd the red trjetory jumps t ti n These re initil prts of the trjetories tht eventully onverge to swtooth osilltions The shpes of these trjetories re different from those in Figure nd Figure t hs een pointed out y [] tht the dely model lso gives rise to swtooth-lie osilltions 6 Our min im of this setion is to nlytilly reprodue these numeril results to understnd why trjetory y( t hs ins nd its derivtive ( t mes jumps To this end we strt to divide the intervl [ 5] into five suintervls with respet to the length of dely [ t t ] for i 5 i i i t i i nd ti ti + with Detiled derivtions of the forms t in eh intervl re presented in Appendix of y( t nd ( Figure 6 Time trjetories of y( t (lue nd ( y' t (red for t 5 6 More preisely [] found t lest twenty five other limit yles were lso solution to the dely model with the sme prmeter vlues Further it ws indited tht there were n infinite numer of dditionl solutions DO: 6/pm88 95 Advnes in Pure Mthemtis

19 A Mtsumoto et l 5 Time Trjetories The onstnt initil funtion Φ( t for t [ t t] dynmi Eqution ( with ( solution nd its derivtive ( S- is seleted The ϕ Φ t yields the following forms of the y( t e K with K α ( t e K The oundry vlues for the end points of intervl re ( ( ( ( y t y t 6 nd y t 897 y t 79 n the seond intervl [ t t] y ( t y ( t solving ( with ϕ ν y suessive integrtion yields the following forms y( t e ( α t+ α ( S- y ( t e ( β t+ β α ( K ( K e β e 95 α 6 β ntegrl onstnt e 78 K+ K α is determined so s to stisfy y ( t y ( t implying the ontinuity of the lue urve t t The disontinuity of the red urve t tht point n e shown in the sme wy s in the se of Goodwin yle The solutions y( t nd ( t t y t stisfy the orresponding dynmi equtions t ( + ( α ( ( + ( ( ( t y t t α y t νy t t t + nd > the first eqution from the seond eqution presents t implying tht ( ( ( t ( t ν ( > or ( t > ( t The lst inequlity onfirms the disontinuity of the red urve t t t The oundry vlues for the end points of intervl re ( ( ( ( y t 897 y t 79 nd y t 7 y t 898 t is then numerilly onfirmed tht DO: 6/pm88 96 Advnes in Pure Mthemtis

20 ( ( ( ( y t y t ontinuity of the lue urve t t t t y t disontinuity of the red urve t t t A Mtsumoto et l As shown in the Appendix t is the vlue t whih the red urve rosses the upper horizontl dotted line one from ove nd divides the intervl [ t t] into two suintervls t t nd t t t t + So we derive solution of the differentil eqution in eh intervl n intervl Eqution (5 with d n ( t : ( S- solving y ( t y ( t presents the forms of ( n y ( t e K + t α y ( t Ke y t nd gives onstnt vlue K 99 The oundry vlues for the end points of intervl re y t 7 t 6 nd y t 6565 t 78 ( ( ( ( the ontinuity of the lue urve nd the disontinuity of the red urve t t t re lso numerilly onfirmed ( ( ( ( y t y t nd y t y t On the other hnd in intervl Eqution ( with ϕ ( t ν ( t yields the solution nd its derivtive y ( t e ( α t + α t+ α ( S- y ( t e ( β t + β t+ β β α e 579 β α 79 ( α e β β 98 β α α 5597 α 85 β α α 5597 The oundry vlues for the end points of intervl re y t 6565 t 78 nd y t 86 t 9 ( ( ( ( the lue nd red urves re onfirmed to e ontinuous t ( ( ( ( y t y t nd y t y t t Due to the vlues of t nd t otined in the Appendix intervl [ t t] is divided into three suintervls t t t t nd t t y t t + nd t t + Sine ϕ ( t ϕ ( t n for t Eqution (5 with d n yields t DO: 6/pm88 97 Advnes in Pure Mthemtis

21 A Mtsumoto et l the solution of the differentil eqution nd its derivtive ( S- n y ( t e K + with K 599 α α ( t e K with K 95 The oundry vlues for the end points of intervl re ( ( ( ( y t 86 y t nd y t 56 y t 76 the lue urve is ontinuous nd the red urve jumps t t t n ( ( ( ( y t y t nd y t y t Eqution ( with ϕ y ( t νy ( t its derivtive ( S- gives the solution nd y t t t t y t t t t ( ( e { α + α + α + α } ( e { β + β + β + β } β α e 86 β α 96 β β α e 76 β α α 99 α e β β + β 95 β α α α 5678 β α α 7 The oundry vlues for the end points of intervl re ( ( ( ( y t 56 y t 76 nd y t 677 y t 5 the lue nd red urves re ontinuous t solution nd its derivtive Sine ϕ ( t t ( ( ( ( y t y t nd y t y t t n for t ( S- Eqution (5 with d n yields the n y( t e K7 with K7 59 α α ( t e K7 with K7 7 The oundry vlues for the end points of intervl re ( ( ( ( y t 677 y t 5 nd y t 9 y t 56 the lue nd red urves re ontinuous t t t DO: 6/pm88 98 Advnes in Pure Mthemtis

22 ( ( ( ( y t y t nd y t y t A Mtsumoto et l Due to the rossing vlues t d nd t e otined in the Appendix intervl 5 [ t t5] is divided into three suintervls 5 t t d nd 5 td t e nd 5 te t 5 y td td + nd te te + Sine ϕ ( t ϕ ( t n for t 5 Eqution (5 with d n implies the solution nd its derivtive ( S- n y5 ( t e K8 + with K8 88 α α 5 ( t e K8 with K8 567 The oundry vlues for the end points of intervl 5 re y t 9 t 6 nd y t 5 t 668 ( ( ( d ( d y t is ontinuous ut ined t t t nd or- the lue urve of ( dingly the red urve of ( t jumps t t t y ( t y ( t nd ( t ( t n 5 nd its derivtive 5 5 Eqution ( with ϕ y ( t νy ( t ( S- yields the solution y t t t t t y t t t t t 5 ( e { α + α + α + α + α } 5 ( e { β + β + β + β + β } β α e 69 β α 795 β β α e β α α 975 β β + β α e 755 β α α ( α e β β β β 98 β α α α K β α α The oundry vlues for the end points of intervl 5 re y t 5 t 668 nd y t 79 t 7 ( d ( d ( e ( e DO: 6/pm88 99 Advnes in Pure Mthemtis

23 A Mtsumoto et l Sine ϕ ( f t n for t 5 solution nd its derivtive ( S- Eqution (5 with d n yields the n y5 ( t e K with K 8658 α α 5 ( t e K with K 6886 The oundry vlues for the end points of intervl 5 re ( e ( e ( ( y t 79 y t 7 nd y t 867 y t the lue nd red urves re ontinuous t t t e ( e ( e ( e ( e y t y t nd y t y t Notie tht the red urves in intervls nd 5 interset the upper nd lower horizontl dotted urves nd differene etween t-vlues is getting smller t t 8 t t 6 nd t t 5 d e f g As seen ove the dely differentil Eqution ( desries dynmi ehvior of y( t for determines the form of y( t for t for 5 while the liner ordinry Eqution (5 t or For 6 the sme types of the solutions re otined nd the size of y t pprohes the sw- reses the resultnt shpe of the solution form of ( tooth shpe t whose verties ( t jumps 5 Phse Plot shrins implying tht s in- We now turn ttention to the phse digrm in the ( y( t y( t plne The oundry vlues of eh trjetory tht hve een otined re summrized in the following tle nd plotted in Figure 7 The red urves re the lous of y( t f ( t nd the green prllelogrm is swtooth limit yle Bl dotted urve onnets the oundry vlues The following points re shown in Figure 7: ( ( 6 ( ( 7 9 ( ( 7 9 ( ( 9 7 ( 5 ( 6 ( 6 ( ( 7 ( 86 ( 8 ( 86 ( 9 ( 76 ( ( 568 ( ( 5 9 ( ( 6 9 ( ( 6685 ( ( 779 ( 5 ( 687 Point ( is the strting point of intervl t t nd the dely Eqution ( trnsports it to point ( t t At the eginning of the seond intervl t jumps whih mes the horizontl move of point ( to point ( tht ( rrives t point ( t the end of At the eginning of the third intervl ( t jumps gin nd point ( horizontlly shifts to point (5 t whih the dynmi system is hnged to the liner Eqution (5 with d n ming move long the upper downwrd line to point (6 s t proeeds from t t to DO: 6/pm88 Advnes in Pure Mthemtis

24 A Mtsumoto et l Figure 7 Phse digrm of swtooth osilltion t t The dynmi system is hnged to the dely eqution t t t nd then the dotted trjetory leves the upper red urve heding to point (7 This is euse for t t t the dely eqution with ( t M ontrols dynmi ehvior On the wy the ( t urve rosses the upper nd lower horizontl dotted urves s seen in Figure 6 The move rehes point (7 t the end of nd jumps to point (8 t the eginning of intervl in whih we see signs of swtooth osilltions Two intersetions otined in intervl uses two hnges of the dynmi system; the liner Eqution (5 with d n governs the movement from point (8 to point (9 nd the dely Eqution ( ontrols the movement from point (9 to point ( nd then the system is hnged to the liner Eqution (5 with d n mnging the movement long the lower downwrd line from point ( to point ( At the eginning of intervl 5 jump from point ( to point ( ours nd the further movement long the upper red line to point ( is ontrolled y the liner eqution with d n point ( to point ( y the dely eqution nd point ( to point (5 y the liner eqution with d n Point (5 jumps to point on the upper red line nd the dynmi system hnge ours s well in for integer 6 s in 5 By doing so the trjetory grdully pprohes to the green swtooth limit yle s time goes on t is notied tht jump ours t the lol mximum or minimum point in the non-dely model s even t the middle of these oundry vlues in the dely model 6 Conluding Remrs This pper presented Goodwin s nonliner elertor model ugmented with DO: 6/pm88 Advnes in Pure Mthemtis

25 A Mtsumoto et l investment dely in ontinuous time sles Assuming pieewise liner investment funtion nd speifying the vlues of the model s prmeters expliit forms of swtooth osilltions were derived when the initil vlue of the onstnt initil funtion ws seleted in the neighorhood of the stedy stte Otherwise the sme ws done for Goodwin osilltion With these numeril results the pper exhiited vlule insights into the mro dynmis of mret eonomies: the dely nonliner elertor-multiplier mehnism n e soure of vrious types of usiness yles; eonomies strting in the neighorhood of the stedy stte ould hieve regulr ups nd downs while eonomies strting wy from the stedy stte presented persistent nd irregulr yles Referenes [] Goodwin R (95 The Nonliner Aelertor nd the Persistene of Business Cyles Eonometri [] Mtsumoto A nd Szidrovszy F (8 Goodwin Aelertor Model Revisited with Fixed Time Delys Communitions in Noliner Siene nd Numeril Simultion [] Mtsumoto A (9 Note on Goodwin's Nonliner Aelertion Model with n nvestment Dely Journl of Eonomi Dynmis nd Control [] Strotz R MAnulty J nd Nines J (95 Goodwin s Non Liner Theory of the Business Cyle: An Eletro-Anlog Solution Eonometri 9- [5] Antonov A Rezni S nd Todorv M ( Relxtion Osilltion Properties in Goodwin's Business Cyle Model nterntionl Journl of Computtionl Eonomis nd Eonometris 9- [6] Freedmn H nd Kung Y (99 Stility Swithes in Liner Slr Neutrl Dely Equtions Funilj Evioj 87-9 DO: 6/pm88 Advnes in Pure Mthemtis

26 A Mtsumoto et l Appendix n this Appendix we provide mthemtil underpinnings for Goodwin osilltions Sine the investment dely ould me y( t ined nd ( t disontinuous t t n n for integer n the time intervl for t is reonstruted s the union of unit intervls n [ tn t n + ] for n nd then dynmi eqution defined over intervl n is solved to otin expliit forms of time trjetory nd its derivtive Dynmi eqution is solved with suessive integrtion in whih n initil point or funtion is the solution of dynmi eqution defined in the proeeding suintervl ntervl : [ t t] t nd t The initil funtion Φ( t determines dynmis for t Sine ϕ Φ ( t y Assumption solving ( t + ( α y ( t presents expliit forms of the solution nd its derivtive n α ( ( y t e K nd y t e K with K ν nd s n e seen in Figure the red urve is elow the lower dotted line or n ( t < for t (A- ν Derivtion of (G- ntervl : [ t t] t (A- implies ϕ ( t n for t nd then (8 with d n re n t α y( t e K+ nd ( t e K solving y ( t y ( t gives nd the following holds ntervls nd : [ ] n K K + e 9 α n ( t < for t (A- ν i ti t i + ti+ ti + for i n the sme wy s in intervl (8 with d y t nd i ( t for t i s the ones defined in y t y t nd t t with K K forms of ( i ( ( ( ( i i i n leds to the identil Notie tht the red urve rosses the lower dotted horizontl line from elow in intervl Solving ( t n ν for t gives t with whih the following inequlities hold n n n ( t < for t < t nd < ( t < for t < t t (A- ν ν ν ntervl : [ t t ] t nd t DO: 6/pm88 Advnes in Pure Mthemtis

27 A Mtsumoto et l Due to (A- t t + divides intervl into two suintervls [ t t ] nd [ t t5] First ( with ϕ ( t n for t presents the sme forms ( ( ( ( y t y t nd y t y t with K K So fr we hve seen tht in [ ts t] trjetory of y( t is desried y with ts t time t n y ( t e K + y ( t y ( t nd K K A form of ( t is otined y time-differentiting y ( t nd is identil with ( t y ( t nd ( t onstrut system (G- defined in Setion Derivtion of (G- On the other hnd for t the investment is delyed nd ( with ϕ ( t ν ( t is written s ( t α ν α ( t + y( t e K Multiplying oth sides y the term e t nd rrnging the terms present ( ( α α d α α t ν t y t e e K e e dt ntegrting oth sides yields Thus the form of the solution is ( ( α α t ν y t e K e t+ K α The integrl onstnt ν The derivtive of y( t is t n e heed tht iv iv ( e ( α + α y t t ( α iv iv K α K e nd K is otined y solving y ( t y ( t K 98 iv iv ( e ( β + β t t iv iv iv iv iv β α nd β α α n n y( t for t ν < < ν (A- DO: 6/pm88 Advnes in Pure Mthemtis

28 A Mtsumoto et l ntervl 5: [ t t ] ( ( ( ( t t for t nd t t for t t 6 6 As in intervl the threshold vlue t t + 5 divides intervl 5 into two suintervls 5 [ t5 t ] nd 5 [ t t6] For t 5 Eqution ( with ϕ ( t ν ( t leds to the solution nd its derivtive tht re the sme s the ones otined in ( ( ( ( y t y t nd y t y t with K K Therefore time trjetories for t 5 [ t t ] y ( t y ( t nd ( t ( t re desried y oth of whih form (G- defined in Setion j re written s α j nd Derivtion of (G- β j Dynmi Eqution ( with ϕ y ( t νy ( t solution Solving y ( t y ( t 5 5 Differentiting y ( Sine the ( α iv j nd for t 5 v v v 5 ( e ( α + α + α y t t t v ν α e iv v ν iv iv e v 5 β ( α β β α K presents 5 t gives K5 66 v v v 5 ( e ( β + β + β t t t v v v v v v β α β α α β α α 5 t urve rosses the upper horizontl line from elow t t 5995 iv β j for hs we then hve n n < 5 ( t < for t5 t t ν ν n n < 5 ( t < for t t t ν ν n 5 ( t > for t t t6 ν (A-5 DO: 6/pm88 5 Advnes in Pure Mthemtis

29 A Mtsumoto et l ntervl 6: [ t t ] t Due to the two vlues t nd t we define two threshold vlues t t + 6 nd t t + 69 oth of whih then divide intervl into three suintervls [ t t ] [ t t ] nd [ t t ] Aordingly onditions in (A-5 determine the indued investment s ( ( 6 7 ϕ 5 t νy5 t for t 6 ϕ ( t ϕ 5 ( t ν5 ( t for t 6 ϕ 5 ( t n for t 6 n onsequene the form of the solution nd its derivtive in 6 re y t y t nd t t ( ( ( ( Therefore lue nd red trjetories for t [ t t ] re desried y ( ( nd ( ( y t y t y t y t oth of whih form (G- defined in Setion j re written s α j nd Derivtion of (G- β j For t 6 suessive integrl leds to the solution Solving y ( t y ( t 6 6 Differentiting y ( written s vi vi vi vi 6 ( e ( α + α + α + α y t t t t v v v vi ν β vi ν β β α α e e α β β + β α K vi v v v vi 6 yields 6 K α v j nd v β j for t with respet to t is fter rrnging the terms it n e vi vi vi vi 6 ( e ( β + β + β + β t t t t vi vi β α vi vi β α α vi vi β α α vi vi β α α Therefore lue nd red trjetories in 6 re desried y ( ( nd ( ( y t y t y t y t 6 6 DO: 6/pm88 6 Advnes in Pure Mthemtis

30 A Mtsumoto et l oth of whih form (G- defined in Setion j re written s α j nd β j Derivtion of (G- For t 6 Eqution (9 implies form of the solution n y6( t e K6 + α α 6( t e K6 y6 t y6 t gives K6 895 solving ( ( ntervl 7: 7 [ t7 t8] t 8 8 n Sine Eqution ( implies ϕ ( Eqution (9 implies tht Notie tht the y ( t α vi j nd vi β j for 6 t n for n nd t 7 ( ( ( ( y7 t y6 t t y t urve intersets the horizontl dotted line t n ν from ove t the following point t 778 Therey n 7( t > for t7 t < t ν n n < 7( t < for t < t t8 ν ν (A-7 ntervl 8: 8 [ t8 t9] t 9 9 The threshold vlue in intervl 7 defines new threshold vlue t t in intervl 8 tht divides intervl 8 into two suintervls 8 [ t8 t ] nd 8 [ t t9] Sine ϕ 7 ( t n for t 8 (9 implies y t y t nd t t 8 ( 7( 8 ( 7( Therefore trjetories in [ t t ] re desried y y ( t y ( t y ( t y ( t ( t ( t ( t ( t y ( t nd y ( t nd onstrut (G- defined in Setion 6 K is repled with K Derivtion of (G- On the other hnd ϕ 7( t ν 7( t for t 8 suessive integr- y t hs the form tion implies tht the solution of ( viii viii 8 ( e ( α + α y t t α α ( α ν K e K 5588 viii 8 viii 8 DO: 6/pm88 7 Advnes in Pure Mthemtis

31 A Mtsumoto et l Time differentition of y ( t n e heed tht ntervl 9: [ t t ] 8 t is viii viii 8 ( e ( β + β t t viii viii viii viii viii β α β α α n n < 8 ( t < for t 8 ν ν n n < 8 ( t < for t 8 ν ν (A t t9 + The threshold vlue t t + divides intervl 9 into two suintervls 9 [ t9 t ] nd 9 [ t t9] Sine the first eqution of (A-8 implies ϕ 8( t ν 8 ( t for t 9 the solution of ( is ( ( nd ( ( y t y t y t y t Therefore trjetories in 8 9 [ t t ] y ( t y ( t y ( t nd ( t ( t ( t re desried y oth of whih form (G- defined in Setion j re written s α j nd Derivtion of (G- β j 8 8 for t 9 On the other hnd ϕ y ( t νy ( t following form of the solution nd ix ix ix 9 ( e ( α + α + α y t t t ix ν α viii β e ix ν viii viii ix 9 ( α e β β 6 α K 75 ix ix ix 9 ( e ( β + β + β t t t ix ix β α ix X ix β α α ix X ix β α α α viii j nd implies the viii β j for DO: 6/pm88 8 Advnes in Pure Mthemtis

32 A Mtsumoto et l t is to e notied tht the y ( t n ν from ove t with whih nd ntervl : [ t t ] urve intersets the horizontl line t 6 t 9967 d n 9 ( t > for t9 t < t ν n y9 t y9 t < for td < t t9 ν ( ( ( E te 579 The threshold vlue t td + divides intervl into two suintervls [ t t ] nd [ t te ] Sine ϕ 9 ( t ν 9 ( t for t we hve y t y t nd t t ( ( ( ( 9 9 Therefore time trjetories in 9 [ t t ] y ( t y ( t y ( t nd ( t ( t ( t d re desried y 9 9 oth of whih form (G- defined in Setion j re written s α j nd β j α ix j nd ix β j for Derivtion of (G- On the other hnd ϕ 9 ( t ϕ 9 ( t n for t implies tht nd n ( + y t e K with K α ( t e K The end point e y t y t s for t Therefore time trjetories in re given y y t y t nd t t y ( t nd y ( t t is otined y solving ( ( ( ( ( ( form (G- defined in Setion K is denoted y K A yle strts t t ts y ( ts nd finishes t t te with y ( te The length of this yle is equl to t E S tht is out 6 Appendix Our min im of this ppendix is to nlytilly reprodue these numeril results of swtooth osilltions to understnd why trjetory y( t hs ins (lterntively its derivtive ( t mes jumps To this end we strt to divide the whole intervl [ 5] into five suintervls with respet to the length of DO: 6/pm88 9 Advnes in Pure Mthemtis

33 A Mtsumoto et l dely [ ] t t for i 5 i i i with ti i nd ti ti + Derivtive of (S- t t t nd t ntervl : [ ] Eqution ( with the onstnt initil funtion y ( t ( t the solution of the form ( y t e K with K nd differentition gives its derivtive form α ( t e K Φ yields t 6 stys in the middle region Both of whih form (S- defined in Setion 5 Sine ( ( t nd y ( t > t y ( t 79 Derivtive of (S- ntervl : [ t t ] Due to ( nd (A-9 for n n y( t for t (A-9 t t+ whih is sustituted into ( to otin ( ( ϕ t ν t ( α t ( t + y( t Q( t with Q( t K e e Sine this eqution n e written s d y t dt t t ( e Q( t ntegrting oth sides nd rrnging the terms present the solution of y( t denoted s y ( t ( ( α α y t e K e t+ K Sine the trjetory of y ( t is pieewise ontinuous solving y( t y( t presents The form of y ( with t is rewritten s K 6 ( e ( α + α y t t e DO: 6/pm88 Advnes in Pure Mthemtis

34 A Mtsumoto et l α ( α K e nd α K A time derivtive of y ( with These y ( t nd y ( t t is ( e ( β + β t t ( α β e K 95 nd β e K+ K 78 form (S- defined in Setion 5 Under Assumption we lulte the oundry vlues of intervl y t 899 t 79 nd y t 7 nd t 898 ( ( ( ( The red urve rosses the upper horizontl dotted line one from ove t point t 86 with whih the following inequlities hold s is seen in in Figure n ( t > for t t < t ν n n < ( t < for t < t t ν ν (A- Derivtions of (G- (G- nd (G- ntervl : [ t t ] t t + Due to the vlue of t the intervl is divided into two suintervls t t nd ( t t t t + Dely investment is differently determined ording to onditions in (A- n if t ϕ ( t ( t if t (8 different dynmi systems re defined on different suintervls So we derive the solution of the differentil eqution in eh suintervl ntervl -: t t < t n this suintervl Eqution (8 with the first eqution of (A- yields the solution nd its derivtive ( e n y t K + α ( e t K DO: 6/pm88 Advnes in Pure Mthemtis

35 A Mtsumoto et l solving y ( t y ( t gives K 99 These two funtions form (S- y ( t nd y ( t y ( t nd ( t Boundry vlues of this intervl re ( ( ( ( re denoted y y t 7 y t 6 nd y t 6565 y t 78 ntervl -: t < t t n this intervl we hve Eqution ( with the seond eqution of (A- tht is rewritten s t y t Q t ( + ( ( ( t β ( β Q( t ( t e { t + } Rewriting the dynmi eqution s d y t dt t t ( e Q( t nd integrting oth sides give the following form of solution β y( t e e t + ( β β t + K solving y( t y( t Then the form of y ( nd A derivtive of y ( gives K 85 t is rewritten s ( e ( α + α + α y t t t α β e 579 ( α e β β 98 t is α K ( e ( β + β + β t t t e β α 79 DO: 6/pm88 Advnes in Pure Mthemtis

36 A Mtsumoto et l nd nd Boundry vlues re β α α 5597 β α α 5597 ( ( y t 6565 y t 78 ( ( y t 86 y t 9 re de- As is seen in Figure 6 the red urve rosses the horizontl dotted lines t n nd n one t points These y ( t nd ( t form (S- in whih y ( t nd y ( t noted s y ( t nd ( t t is pprent from Figure 6 tht ( t 55 with t n n t 8 with ( t n ( t > for t t < t n ( t > for t t < t n n > ( t > for t < t < t n > ( t for t < t t (A- Derivtion of (S- (S- nd (S- ntervl : [ t t] t t + Due to the properties desried in (A- intervl is divided into three suintervls y t t + nd t t + in whih ϕ ( t ϕ ( t n for t t < t (A- nd ( ( ϕ t t for t < t < t (A- ( ϕ t n for t < t t (A-5 ntervl -: t t < t Eqution ( with (A- implies tht the solution of the differentil eqution is given y ( ( α ( t + y t n n 5( 5 5 y t e K + with K 599 DO: 6/pm88 Advnes in Pure Mthemtis