Research & Reviews: Journal of of Statistics and Mathematical Sciences

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1 Reseach & Reviews: Jounal of of Saisics and Mahemaical Sciences Complexiy in he Sochasic Kaldo-Kalecki Model of Business Cycle wih Noise Lin Zeng Xuehan Xu and Zaiang Huang* School of Mahemaical Sciences Guangxi Teaches Educaion UnivesiyNanning 5 P R China RESEARCH ARTICLE Received dae: //5 Acceped dae: //5 Published dae: 6//5 *Fo Coespondence Zaiang Huang School of Mahemaical Sciences Guangxi Teaches Educaion UnivesiyNanning 5 P R China ABSTRACT In he pape he sochasic Kaldo-Kalecki model of business cycle wih noise is invesigaed By analyzing he Lyapunov exponen invaian measue and singula bounday heoy some new cieia ensuing sochasic sabiliy P-bifucaion and pichfok bifucaion fo sochasic Kaldo-Kalecki model ae obained especively Numeical simulaion esuls ae given o suppo he heoeical pedicions zaianghuang@6com Keywods: Sochasic bifucaion and sabiliy; Lyapunov exponen; Invaian measue; Kaldo-Kalecki model INTRODUCTION Kaldo [] poposed a odinay diffeenial sysem o model business cycle in which he goss invesmen depends on he level of oupu and capial sock Theeafe his model was ofen discussed see [-] and efeences hee in The Kalecki business model [5] was a few yeas ealie han he Kaldo one Kalecki assumed ha he saved pa of pofi is invesed and he capial gowh is due o pas invesmen decisions Thee is a gesaion peiod o a ime lag afe which capial equipmen is available fo poducion In 999 Kawiec and Szydlowski [6] have fomulaed he Kaldo-Kalecki business cycle model based on he muliplie dynamics which is he coe of boh he Kaldo (afe Keynes) and Kalecki appoach Howeve hey employed Kaleckis appoach o invesmen and of a ime lag beween invesmen decisions and implemenaion The model is as following fom: Y ( ) = [ IY ( ( ) X ( )) SY ( ( ) K ( ))] K () ()= IY ( () K ()) qk () Clealy inoducing noise and ime delays ino he business model is moe easonable On he model Kawiec and Szydlowski [-] have sudied he sabiliy and exisence of Hopf bifucaions by analyzing he chaaceisic equaion associaed wih he model he mehod canno be applied o he sochasic model In geneal sochasic delay-diffeenial equaions exhibi much moe complicaed dynamics han he esponding odinay diffeenial equaions since a ime delay o noise could cause he change of sabiliy of an equilibium and hence Hopf bifucaion occus I is ineesed o invesigae he noise o ime delay how o affec he dynamics of a sysem and i is impoan o deemine he diecion of he Hopf bifucaion and he sabiliy of he bifucaing peiodic soluions when a Hopf bifucaion occus Taking hese facos ino accoun we inoduce andomness ino he model by eplacing he paamees β and q by β β ξ() and q q ξ () This is only a fis sep in inoducing sochasiciy ino he model Ideally we would also like o inoduce sochasic envionmenal vaiaion ino he ohe paamees such as he ansmission coefficien and γ he oal ae of poducion of healhy cells pe uni ime bu o do his would make he analysis much oo difficul In his pape we conside he Kaldo-Kalecki model of business cycle wih noise as following RRJSMS Volume Issue Decembe 5

2 Y ()= [ IY ( ()) βk () γy ()] Y () ξ() βη() K IY K qk K () ()= ( ()) β () () () ξ() βη () whee Y is he goss poduc and K he capial poduc of he business cycle; > measue he eacion of he sysem o he diffeence beween invesmen and saving; q () is he depeciaion ae of capial sock; IS : R R R ae invesmen and saving funcion of Y and K especively; ξ () is he muliplicaive andom exciaion and η () is he exenal andom exciaion diecly(namely addiive andom) ξ () and η () ae independen in possession of zeo mean value and sandad vaiance Gauss whie noises ie [ ξ( )]= [ η( )]= [ ξ( ) ξ( τ)]= δτ ( ) [ η( ) η( τ)]= δτ ( ) [ ξ( ) η( τ)]= And ( β ) is he inensiies of he whie noise The heoy of andom dynamical sysem povides a vey poweful mahemaical ool fo undesanding he limiing behavio of sochasic sysem Recenly i has been applied o economics and finance o help in undesanding he sochasic naue of financial model wih andom peubaions [-] In paicula he sudy of he limiing disibuing of vaious sochasic models in economics and finance give a good descipion of saionay business cycle Thee seems o have been no applicaion of i o Kaldo-Kalecki model of business cycle Ou pupose in his pape is o invesigae he sochasic bifucaion and sabiliy fo () by applying he singula bounday heoy Lyapunov exponen and he invaian measue heoy he diecion of he Hopf bifucaion and he sabiliy of bifucaing peiodic soluions ae also deemined We also give numeical example o simulae he esuls found by using he Malab and Mahemaica sofwae The sucue of he pape is as follows In Secion we fis ouline he exended model of Kaldo-Kalecki model of business cycle In Secion and he sochasic dynamical behavio is analyzed fom he viewpoin of saionay measues and invaian measue especively The pape is hen concluded in Secion 5 PRELIMINARY In he secion we pesen some peliminay esuls o be used in a subsequen secion o esablish he sochasic sabiliy and sochasic bifucaion Befoe poving he main heoem we give some lemmas and definiions Thoughou he es of his pape we assume ha β > q γ () and ha Is () is C * * C Le ( Y K ) be an equilibium poin * * * * * * of sysem () I = IY ( ) and x= YY y= K K fs ( )= Is ( Y) I Then sysem () can be ansfomed as x ()= [ f( x ()) βy () γx ()] x () ξ() βη () y ()= I( x ()) βy () qy () y () ξ() βη () Le he Taylo expansion of f a Then we can ewie () as he following equivalen sysem x ()= ax () ax () ax () β y () γ x () x () ξ() βη () y ()= bx () bx () bx () βy () qy () y () ξ() βη () a = f () a = f () a = f () b = f () b = f () b = f ()!!!! whee Le Y=[ xy ] Τ hen by subsiuing he coesponding vaiables in Eq () Y = AY f( Y ξ( ) η( )) (5) So discussing he sabiliy of sysem () a equilibium poin Q is equivalen o discussing he sabiliy of sysem (6) a equilibium poin O() Le Y = PX X = [ x( ) y( )] P = Τ whee T T i i () () b b T = T = q β γ a ( q β γ ) ( q β γ ) a a βb q β γ a ( q β γ ) ( q β γ ) a a βb Then by subsiuing he coesponding vaiable in he equaions we obain X P APX P f PX ie = ( ξ( ) η( )) x () = cx bx bxy by bx b5x y b6xy b y ( k x k y) ξ() η() y () = c y b x b xy b y b x b x y b xy b y ( kx k y) ξ() η() 5 6 RRJSMS Volume Issue Decembe 5 (6)

3 whee he coefficien ae denoed as following: c = ( qβ γ ak) c = ( qβ γ a k) qa βa γ a a a βb a k k = = = b b b qa βa γ a aa βb ak b = b = a5 = a6 = b k qa βa γ a aa βb ak b = b = a5 = a6 = b k k b β k = k b ( q β γ a k) β ( q β γ a k) = kq ( β γ a k) k b ( q β γ a k) β ( q β γ a k) = kq ( β γ a k) k b β k = k = ( q β γ ) ( q β γ ) a a βb bβ bβ = ( β β) = q β γ a k q β γ a k Se he coodinae ansfomaion x= cos θ y = sin θ and by subsiuing he vaiable in (6) we obain ( ) = c ( cos θ csin θ) ( bcos θ ( b a) cos θsinθ ( b a)cos θsin θ bsin θ) ( bcos θ ( b5 b) cos θsinθ ( b6 b5) cos θsin θ ( b b6) cos θsin θ bsin θ) k ( cos θ ( k k)cosθsinθ ksin θξ ) () ( cos θ sin θη ) ( ) θ( ) = ( c c )cosθsin θ b [ cos θ ( b b) cos θsinθ ( b b)cos θsin θ bsin θ] [( bcos θ ( b5 b) cos θsinθ ( b6 b5) cos θsin θ ( b b6) cos θsin θ bsin θ)] ( kcos θ ( k k)cosθsin θksin θξ ) ( ) ( cos θ sin θη ) ( ) I is difficul o calculae he exac soluion fo sysem () oday Accoding o he Khasminskii limi heoem when he inensiies of he whie noises ( β β ) is small enough he esponse pocess { ( ) θ ( )} weakly conveged o he wodimensional Makov diffusion pocess [-] Though he sochasic aveaging mehod we obained he I ô sochasic diffeenial equaion () he pocess saisfied d = md σdw σdwθ dθ = mθd σdw σdwθ whee W () and W θ ae he independen and sandad Wiene pocesses As fo he wo-dimensional diffusion pocess i is necessay o calculae is wo-dimensional ansiion pobabiliy densiy Thee is no geneal and igh mehod fo he calculaion As fo he concee sysem we could finish he calculaion wih some special echniques Unde he condiion σ = σ sysem () is ewien as follows 5 d = d dw ( ) dwθ 6 dθ = d ( 5) dw dw θ whee () () (9) = ( ) =5 5 6 c c k k k k k k k k = ( ) = k k k k k k k k 5 = ( )( ) 6 = k k k k k k k k k k k k RRJSMS Volume Issue Decembe 5

4 =b b b b =b b b b Fom he diffusion maix we can find ha he aveaging ampliude () is a one-dimensional Makov diffusing pocess when σ = σ = ie k k = o k = k Thus we have he equaion as following d = d dw This is an efficien mehod o obain he ciical poin of sochasic bifucaion hough analyzing he change of sabiliy of he aveaging ampliude () in he meaning of pobabiliy STOCHASTIC STABILITY In ode o deec he local sochasic sabiliy of he sochasic aveaging sysem he mehod ha we ofen used is o calculae he maximum Lyapunov exponen Theoem If = = (i) When hen he sochasic sysem () is sochasically sable < 6 > 6 (ii) When hen he sochasic sysem () is sochasically unsable Poof When = = Then sysem () becomes () d = d dw Using he soluion of linea Iô sochasic diffeenial equaion we obain he soluion of sysem () as follow () σ () ( ) = ()exp m() ds σ () dw ( ) s whee m() = σ() = Using he heoem of qusi-non-inegable Hamilonian sysem hee we define a new nom: ( ) = ( ) hus he appoximaion of Lyapunov exponen of he linea Iô sochasic diffeenial equaion is: σ () m() λ = ln ( ) = = lim 6 Thus we have: When < ha is λ < hus he ivial soluion of he linea Iô sochasic diffeenial equaion = is sable in 6 he meaning of pobabiliy ie he sochasic sysem is sable a he equilibium poin Q In addiion he linea Iô sochasic diffeenial equaion have obusness ie he ivial soluion = of he nonlinea Iô sochasic diffeenial equaion () is sable in he meaning of pobabiliy This demonsaes ha he deeminisic sysem is seady a is equilibium poin i may also be seady in he meaning of pobabiliy a is equilibium poin unde andom exciaions When > ha is λ > Thus he ivial soluion of he linea Iô sochasic diffeenial equaion = is unsable in 6 he meaning of pobabiliy ie he sochasic sysem is unsable a he equilibium poin Q This demonsaes ha alhough he deeminisic sysem is seady a is equilibium poin he sochasic sysem may be unsable in he meaning of pobabiliy a is equilibium unde andom exciaions When = ha is λ = Whehe = o no can be egaded as he ciical condiion of bifucaion a he 6 6 equilibium poin And whehe he Hopf bifucaion could occu o no ae wha we will discuss in he nex secion The max Lyapunov exponen based on Oseledec muliplicaive egodic heoy can only be used o judge he local sabiliy hee we judge he global sabiliy by he singula bounday heoy In he secion accoding o he singula bounday heoy we will obain sabiliy of he sochasic aveaging sysem Theoem Le = < and < Poof When = he sysem () can be ewien as follows: d = d dw Then he sochasic sysem () is sochasically sable Thus = is he fis kind of singula bounday of sysem () When = we can find m = ; hus = is he second kind of singula bounday of sysem () () () RRJSMS Volume Issue Decembe 5

5 Accoding o he singula bounday heoy we can calculae he diffusion exponen difing exponen and chaaceisic value of bounday = and he esuls ae as follows: = β = ( ) ( ) β m ( ) ( ) = lim = = lim σ c So () if c > ie > he bounday = is exclusively naual If c < ie < he bounday = is aacively naual If c = ie = he bounday = is sicly naual We can also calculae he diffusion exponen difing exponen and chaaceisic value of bounday = and he esuls ae as follows: = β = ( ) ( ) β m ( ) c = lim = = lim (5) σ So if c > ie < he bounday = is exclusively naual If c < ie > he bounday = is aacively naual If c = ie = he bounday = is sicly naual As we know if he singula bounday = is aacively naual bounday and = is enance bounday his siuaion is all he solve cuves ene he inne sysem fom he igh bounday and is aaced by he lef bounday he equilibium poin is global sable Fom he analysis above we can daw a conclusion ha he equilibium poin is global sable when he singula bounday = is aacively naual bounday and = is enance bounday Combine he condiion of local sabiliy he equilibium poin = is sable when < and < Theoem Le Then he sochasic sysem () is no sochasically sable Poof When he sysem () can be ewien as follows: d = d dw One can find σ a = so = is a nonsingula bounday of sysem (6) Though some calculaions we can find ha = is a egula bounday(eachable) The ohe esul is m = when = so = is second singula bounday of (6) The deails ae pesened as follows: = β = ( ) ( ) β m () c = lim = = lim σ ( ) So if c > ie If c < ie < > he bounday = is exclusively naual he bounday = is aacively naual If c = ie = he bounday = is sicly naual Thus we can daw he conclusion ha he ivial soluion = is unsable ie he sochasic sysem is unsable a he equilibium poin Q no mae whehe he deeminisic sysem is sable a equilibium poin Q o no STOCHASTIC BIFURCATION In he secion We will see how he inoducion of andomness change he sochasic behavio significanly fom boh he dynamical and phenomenological poins of view [] (6) () RRJSMS Volume Issue Decembe 5 5

6 Theoem (D-bifucaion) Le = = Then sysem () undegoes sochasic D-bifucaion Poof When = = Then sysem () becomes d = d dw When = equaion () is a deeminae sysem and hee is no bifucaion phenomenon Hee we discuss he siuaion le σ m ( )= ( )= 6 The coninuous andom dynamic sysem geneae by () is ϕ() x = x m( ϕ()) s x ds σϕ ( ()) s x dw whee dw is he diffeenial a he meaning of Saonovich i is he unique song soluion of () wih iniial value x And m() = σ () = so is a fixed poin of ϕ Since m () is bounded and fo any i saisfy he ellipiciy condiion: σ ( ) ; i assue ha hee is a mos one saionay pobabiliy densiy Accoding o he Io ˆ equaion of ampliude () we obain is FPK equaion coesponding o () as follows p = p p p Le = hen we obain he soluion of sysem (9) mu ( ) p()= c σ () exp du () σ ( u) The above dynamical sysem (9) has wo kinds of equilibium sae: fixed poin and non-saionay moion The invaian measue of he fome is δ and i s pobabiliy densiy is δ x The invaian measue of he lae is ν and i s pobabiliy densiy is () In he following we calculae he lyapunov exponen of he wo invaian measues Using he soluion of linea Iô sochasic diffeenial equaion we obain he soluion of sysem () m ds dw σ() σ () ( ) = ()exp () σ () () The lyapunov exponen wih egad o of dynamic sysem ϕ is defined as: λ ϕ ( ) = lim ln ( ) subsiuing () ino () noe ha ( m ds σ dw s ) λϕ ( δ) = lim ln () () () ( ) W () = m () σ ()lim = m () 6 = σ() = σ () = we obain he lyapunov exponen of he fixed poin: () Fo he invaian measue which egad () as is densiy we obain he lyapunov exponen: λ ()= lim ϕ ν m () () () ds σ σ σ() σ () = m () p() d R m () = p() d R σ () 6 = m ( ) exp m ( ) = exp ( ) Le = We can obain ha he invaian measue of he fixed poin is sable when < bu he invaian 6 () (9) () () RRJSMS Volume Issue Decembe 5 6

7 measue of he non-saionay moion is sable when > so = D = is a poin of D -bifucaion Then sysem () undegoes sochasic D-bifucaion Theoem Le = = Then sysem () dose no undego sochasic P-bifucaion Poof Simplify Eq () we can obain ( ) ps ( )= c whee c is a nomalizaion consan hus we have p ( ) = o ( v ) (6) s ( ) whee v = Obviously when v < ha is < ps () is a δ funcion when < v < ha is > 6 6 = is a maximum poin of ps () in he sae space hus he sysem occu D-bifucaion when v = ha is is he = 6 ciical condiion of D-bifucaion a he equilibium poin When v > hee is no poin ha make ps () have maximum value hus he sysem does no occu P-bifucaion Theoem (P-bifucaion) Le = < hen sysem () undego a P-bifucaion a he paamee value Poof When = hen Eq () can ewie as following d = d dw Le φ = = σ = dφ φ φ d σφ dw = hen we conside he sysem () becomes () which is solved by φ exp W φ ψ ( ωφ ) = / φ exp Ws ds We now deemine he domain D ( ω ) whee D ( ω):={ φ R:( ωφ ) D}( D= R Ω X) is he (in geneal possibly empy) se of iniial values φ R fo which he ajecoies sill exis a ime and he ange R ( ω ) of ψ( ω): D( ω) R( ω) We have R D ( ω)= ( d ( ω ) d ( ω )) < whee d ( ω) = > exp W ( ) s ω ds and P = (5) () (9) () ( ( ω) ( ω)) > R( ω)= D( ϑ() ω)= R whee () exp W ( ω)= d( ϑ() ω)= > exp W ( ) s ω ds We can now deemine E := D ( ω) R and obain RRJSMS Volume Issue Decembe 5

8 E = ( d( ω) d( ω )) > {} () whee ± < d ( ω)= < ± exp Ws ds The egodic invaian measues of sysem () ae º () i Fo he only invaian measues is ω = δ ( ii ) Fo > we have he hee invaian fowad Makov measues ω = δ and υ± ω = δ± k whee k ( ω ):= exp W ds We have k = Solving he fowad Fokkwe-planck equaion * Lp = φ φ φ P( φ) ( φ P( φ) ) = yield () p = δ fo all i ( ii ) fo p > 6φ q ( )= Nφ exp φ > φ φ and q ( φ) = q ( φ) whee N = Γ Naually he invaian measues υ± ω = δ± k ae hose coesponding o he saionay measues q Hence all invaian measues ae Makov measues The wo families of densiies ( q ) > clealy undego a P-bifucaion a he paamee value P = Then sysem () undego a P-bifucaion Theoem Le q β γ k = a o q β γ = a k = < Then sysem () undegoes sochasic pichfok bifucaion Poof We deemine all invaian measues(necessaily Diac measue) of local RDS χ geneaed by he SDE = dφ φ φ d φ dw on he sae space R () = R and σ = The lineaized RDS χ = Dϒ ( ωφ ) χ saisfies he lineaized SDE = ( ϒ ( )) dχ ωφ χ d χ dw hence Dϒ W ϒ s ds ( ωφ ) χ= χ exp ( ( ωφ )) Thus if νω = δ φ is a ϒ - invaian measue is Lyapunov exponen is λ ( ) = lim log Dϒ( ωφ ) χ We now calculae he Lyapunov exponen fo each of hese measue RRJSMS Volume Issue Decembe 5

9 = lim ( ϒ( s ωφ )) = φ ds povided he IC φ L ( ) is saisfied () i Fo R he IC fo ν ω = δ is ivially saisfied and we obain ( )= λν ν So is sable fo < and unsable fo > ( ii ) Fo > ν ω = δ is measuable hence he densiy p of ρ = ν saisfies he Fokke-Planck equaion d ω * = ( ) L p ( p ( ) ) = ν φ φ φ φ φ φ which has he unique pobabiliy densiy soluion φ P ( φ) = Nφ exp φ > Since φ = ( d ) = φ p ( φ ) d φ < ν he IC is saisfied The calculaion of he Lyapunov exponen is accomplished by obseving ha d exp W Ψ () ( ϑω ) = = Ψ exp s Ws ds whee Ψ ( ) = exp s Ws ds Hence by he egodic hoeem lim Ψ ( d ) = log ( ) = finally ( ) = < λν ( iii ) Fo > ν = δ ω d ω ( d ) = ( ) = hus d ( ) = < is measuable Since ( d )= ( d ) λν Fom Theoem he wo families of densiies ( q ) > clealy undego a P-bifucaion a he paamee value P = - which is he same value as he ansciical case Hence we have a D-bifucaion of he ivial efeence measue δ a D = and a P-bifucaion of P = Then sysem () undegoes sochasic pichfok bifucaion NUMERICAL SIMULATIONS In his secion we give some examples o veify he hepeical esuls obained in and Se = β = γ = 565 q = 9 hen Sys() becomes Y ( ) = IY ( ( )) K ( ) 565 Y ( ) Y ( ) ξ( ) βη ( ) K ()= IY ( ()) K () K () ξ() βη () Fo simpliciy we assume ha ( ) is ivial equilibium poin of Sys() Choose Is ( ) = anh(65 s ) and hen I () = 65 I () = I () = 99 () is asympoically sabiliy (Figues -) () RRJSMS Volume Issue Decembe 5 9

10 5 5 Xemp = 5 = β = β = Figue Red Repesen The Simulaion Of Deemine Sysem Blue Repesen The Simulaion Of Sochasic Sysem 5 5 Xemp = 5 = β = β = Figue Red Repesen The Simulaion Of Deemine Sysem Blue Repesen The Simulaion Of Sochasic Sysem 5 5 Xemp Xemp = 5 = β = β = Figue The Simulaion Of Sochasic Sysem RRJSMS Volume Issue Decembe 5

11 x 5 9 Xemp = 5 = β = β = Figue Red Repesen The Simulaion Of Deemine Sysem Blue Repesen The Simulaion Of Sochasic Sysem x 5 9 Xemp = 5 = β = β = Figue 5 Red Repesen The Simulaion Of Deemine Sysem Blue Repesen The Simulaion Of Sochasic Sysem x 5 9 Xemp x 5 = 5 = β = β = Figue 6 The simulaion of sochasic sysem RRJSMS Volume Issue Decembe 5

12 5 5 5 Xemp Xemp = = β = β = Figue The simulaion of sochasic sysem CONCLUSION In his pape we have consideed a Kaldo-Kalecki model of business cycle wih noise Alhough hee ae los of papes on he sabiliy and Hopf bifucaion of Kaldo-Kalecki model of business cycle wih delays he mehod canno be applied o he pesen model By using he he singula bounday heoy Lyapunov exponen and he invaian measue heoy we have sudied he a geneal hid degee polynomial sochasic diffeenial equaion Applying he obained esuls o sysem () we have found ha unde ceain condiions when o vaies he zeo soluion loses is sabiliy and Hopf bifucaion occus ha is a family of peiodic soluions bifucae fom he zeo soluion when o passes a ciical In addiion o povide a complee picue of he equilibium behavio of he model as a paamee capuing he behavio changes of Kaldo-Kalecki model of business cycle we conduc ou analysis fom he viewpoins of boh dynamical and phenomenological bifucaions Thee numeical simulaion esuls ae given o suppo he heoeical pedicions Acknowledgemens This eseach was suppoed by he Naional Naual Science Foundaion of China (No 9) and and (No9) Guangxi Naual Science Foundaion(No GXNSFAA9) and (No GXNSFBA96) REFERENCES Kaldo N A model of he ade cycle Econom J 9; : -9 Kalecki N A macodynamic heoy of business cycles Ecconomeica 95; : - Chang W and Smyh D The exisence and pesisence of cycles in nonlinea model: Kaldois 9 model e-examined Rev Econ Sud 9; : - Gasman J and Wenzel J Co-exisence of a limi cycle and an equilibium in Kaldos business cycle model and is consequences J Econ Behav Oganizaion 99; : 69-5 Ichimua S Towad a geneal nonlinea macodynamics heoy of economic flucuaions In: Kuihaa K edio Pos- Keynesian economics Ruges Univesiy Pess; 95; Kawiec A and Szydlowski M The Kaldo-Kalecki business cycle model Ann Ope Res 999; 9: 9- Szydlowski M e al Nonlinea oscillaions in business cycle model wih ime lags Chaos Solions Facals ; : 55-5 Zhang C and Wei J Sabiliy and bifucaion analysis in a kind of business cycle model wih delay Chaos Solions Facals ; : Szydlowski M and Kawiec A The Kaldo-Kalecki model of business cycle as a wo-dimensional dynamical sysem J Nonlinea Mah Phys ; : 66- Ma J and Gao Q Sabiliy and Hopf bifucaions in a business cycle model wih delay Appl Mah Compu 9; 5: 9- Qin X e al Muli-paamee bifucaions of he Kaldo-Kalecki model of business cycles wih delay Nonlinea Analysis ; : 69- Beja A and Coldman M On he dynamic behaviou of pices in disequilibium J Finace 9; 5: 5- RRJSMS Volume Issue Decembe 5

13 Bohm V and Chiaella C Maen vaiance pefeences expecaions fomaion and he dynamics of andom asse pices Mah Finance 5; 5: 6-9 Chiaella C e al The sochasic pice dynamics of speculaive behaviou Woking PapeQFRC Univesiy of Technology Sydney 5 Follme H e al Equilibium in financial makes wih heeogeneous agens: A pobabilisic pespecive J Mah Econom 5; : Rheinlaende T and Seinkamp M Asochasic vesion of Zeeman s make model Sud Nonlinea Dynam Economeics ; : - Huang Z e al Sochasic sabiliy and bifucaion fo he chonic sae in Machuk s model wih noise Applied Mahemaical Modelling ; 5: Zhu W Nonlinea Sochasic Dynamics and Conol in Hamilonian Fomulaion Beijing: Science Pess 9 Khasminskii R On he pinciple of aveaging fo Io sochasic diffeenial equaions KybeneikaPague 96; : 6-9 Lin Y and Cai G Pobabilisic Sucual Dynamics Mcgaw-hill Pofessional Publishing Anold L Random Dynamical Sysems Spinge New yok 99 Namachchivaya N Sochasic bifucaion Applied Mahemaics and Compuaion 99; : -59 RRJSMS Volume Issue Decembe 5

7 Wave Equation in Higher Dimensions

7 Wave Equation in Higher Dimensions 7 Wave Equaion in Highe Dimensions We now conside he iniial-value poblem fo he wave equaion in n dimensions, u c u x R n u(x, φ(x u (x, ψ(x whee u n i u x i x i. (7. 7. Mehod of Spheical Means Ref: Evans,