Is the largest Lyapunov exponent preserved in embedded dynamics?

Transcription

1 3 Ocober Physics Leers A 76 ) Is he larges Lyapunov exponen preserved in embedded dynamics? W. Davis Decher a, Ramazan Gençay b,c, a Deparmen of Economics, Universiy of Houson, USA b Deparmen of Economics, Mahemaics and Saisics, Universiy of Windsor, Canada c Deparmen of Economics, Bilken Universiy, Bilken, Ankara, Turkey Received July ; acceped 4 Sepember Communicaed by C.R. Doering Absrac The mehod of reconsrucion for an n-dimensional sysem from observaions is o form vecors of m consecuive observaions, which for m>n, is generically an embedding. This is Takens resul. Our analyical examples show ha i is possible o obain spurious Lyapunov exponens ha are even larger han he larges Lyapunov exponen of he original sysem. Therefore, we presen examples where he larges Lyapunov exponen may no be preserved under Takens embedding heorem. Elsevier Science B.V. All righs reserved.. Inroducion Lyapunov exponens measure he rae of divergence or convergence of wo nearby iniial poins of a dynamical sysem. A posiive Lyapunov exponen measures he average exponenial divergence of wo nearby rajecories whereas a negaive Lyapunov exponen measures exponenial convergence of wo nearby rajecories. If a discree nonlinear sysem is dissipaive, a posiive Lyapunov exponen quanifies a measure of chaos. The inroducion of Lyapunov exponens o economics was in []. Brock and Sayers [] noe ha he Wolf [3] algorihm is sensiive o he number of ob- * Corresponding auhor. Ramazan Gençay hanks he Naural Sciences and Engineering Research Council of Canada and he Social Sciences and Humaniies Research Council of Canada for financial suppor. address: gencay@uwindsor.ca R. Gençay). servaions as well as o he degree of measuremen or sysem noise in he observaions. This observaion sared a search for new algorihmic designs wih improved finie sample properies. The search for an algorihm o calculae Lyapunov exponens wih desirable finie sample properies has gained momenum in he las few years. Abarbanel e al. [4 6], Ellner e al. [7], McCaffrey e al. [8], Gençay and Decher [9] and Decher and Gençay [] came up wih improved algorihms for he calculaion of he Lyapunov exponens from observed daa. Gençay [] worked on he calculaion of he Lyapunov exponens wih noisy daa when feedforward neworks were used as he esimaion echnique. The main algorihmic design in all papers above is o embed he observaions in an m-dimensional space, hen by heorems of Mañé [] and Takens [3] he observaions are used o reconsruc he dynamics on he aracor. The Jacobian of he reconsruced dynamics as demonsraed in [4,5] is hen used o calculae he //$ see fron maer Elsevier Science B.V. All righs reserved. PII: S375-96)657-5

2 6 W.D. Decher, R. Gençay / Physics Leers A 76 ) Lyapunov exponens of he unknown dynamics. The mehod of reconsrucion for a n-dimensional sysem from observaions is o form vecors of m consecuive observaions, which for m>n is generically an embedding. The Jacobian mehods for Lyapunov exponens uilize a funcion of m variables o model he daa and he Jacobian marix is consruced a each poin in he orbi of he daa. When embedding occurs a dimension m = n, hen he Lyapunov exponens of he reconsruced dynamics are he Lyapunov exponens of he original dynamics. However, if embedding only occurs for an m>n, hen he Jacobian mehod yields m Lyapunov exponens, only n of which are he Lyapunov exponens of he original sysem. The problem is ha as currenly used, he Jacobian mehod is applied o he full m-dimensional space of he reconsrucion, and no jus o he n-dimensional manifold ha is he image of he embedding map. Our examples show ha i is possible o ge spurious Lyapunov exponens ha are even larger han he larges Lyapunov exponen of he original sysem.. The Jacobian algorihm The Lyapunov exponens for a dynamical sysem, f : R n R n, wih he rajecory, x + = fx ), =,,,..., are measures of he average rae of divergence or convergence of a ypical rajecory. For an n-dimensional sysem as above, here are n exponens which are cusomarily ranked from larges o smalles λ λ λ n. I is a consequence of Oseledec s [6] heorem ha he Lyapunov exponens exis for a broad class of funcions. The addiional properies of Lyapunov exponens and a formal definiion are given in []. In pracice one rarely has he advanage of observing he sae of he sysem, x, le alone knowing he acual funcional form f which generaes he dynamics. The model which is widely used is ha associaed wih he dynamical sysem here is an observer funcion h : R n R which generaes he observaions, y = hx ). I is assumed ha all ha is available o he researcher is he sequence {y }. For noaional purposes, le y m = y,y +,...,y +m ). If he se U is compac manifold hen for m n + J m x) = hx), h fx) ),...,h f m x) )) ) ) generically is an embedding. 3 For m n + here exiss a funcion g : R m R m such ha y m + = gym ) where y m + = y +,y +,...,y +m ). Bu noice ha y m + = J m x + ) = J m fx ) ). 3) Hence from Eqs. ) and 3) J m f x )) = gj m x )). The funcion g is opologically conjugae o f.this implies ha g inheris he dynamical properies of f. Decher and Gençay [] prove he following heorem o show ha n of he Lyapunov exponens of g are he Lyapunov exponens of f. Theorem. Decher and Gençay []). Assume ha M is a smooh manifold dimension n, f : M M and h : M R are a leas) C. Define J m : M R m by J m x) = hx), hf x)),..., hf m x))). Le µ x) x) µ n x) be he eigenvalues of he symmeric marix DJ m ) x DJ m ) x, and suppose ha inf x M µ n x) >, sup x M µ x) <. Le λ f λf λf n be he Lyapunov exponens of f and λ g λg λg m be he Lyapunov exponens of g, whereg : J m M) J m M) and J m f x)) = gj m x)) on M. Then generically {λ f i } {λg i }. By Theorem., n of he Lyapunov exponens of g are he Lyapunov exponens of f. The approach of Gençay and Decher [9] is o esimae he funcion g based on he daa sequence {J m x )}, and o calculae he Lyapunov exponens of g. The rajecory is also wrien in erms of he ieraes of f. Wih he convenion ha f is he ideniy map, and f + = f f,hen we also wrie, x = f x ). A rajecory is also called an orbi in he dynamical sysem lieraure. Also see [7 9] for precise condiions and proofs of he heorem. 3 By generic is mean ha in every neighborhood of f and h here are funcions f and h so ha he funcion J m corresponding o hese funcions is an embedding of he aracor of f and he image of he image of he aracor under J m.heren + ishe wors-case upper limi.

3 W.D. Decher, R. Gençay / Physics Leers A 76 ) From Eq. ) he mappingg which is o be esimaed may be aken 4 o be y y + y + g : 4). y +. y +m vy,y +,...,y +m ) and his reduces o esimaing y +m = vy,y +,..., y +m ).Herev is an unknown map. Linearizaion of he map g yields y+ m = Dg) y m y m. The soluion can be wrien as y m = Dg ) y m y m. The Lyapunov exponens can be calculaed from he eigenvalues of he marix Dg ) y m using QR decomposiion. This mehod is discussed in [4,5,] and a modified version is presened in [6]. 3. An example If x is a fixed poin, hen he subspaces V j = V j do no depend upon. Le us consider he mapping fx) a he fixed poin x. Choose V = R, V = span{, )} and V 3 ={}.For µ > consider 5 [ ] µ Df x) =. 5) This will saisfy pars ) and ) of Definiion in [] and we will have λ = lim ln µ v + µ v ) = ln µ for v V \ V, λ = lim ln µ v + µ v ) = ln for v V \ V 3. This definiion mainly generalizes he idea of eigenvalues o give average linearized conracion and expansion raes on a rajecory. An aracor is a se of poins owards which he rajecories of f converge. More precisely, Λ is an aracor if here is an open se U R n wih Λ U, fu) U and Λ = f U) where U is he closure of U. The aracor Λ is said o be indecomposable if here is no proper subse of Λ which is also an aracor. An 4 Here, he ime sep is assumed o be equal o he delay ime. 5 This example is from Guckenheimer and Holmes []. aracor can be chaoic or ordinary or nonchaoic). There is more han one definiion of a chaoic aracor in he lieraure. In pracice he presence of a posiive Lyapunov exponen is aken as a signal ha he aracor is chaoic. Now, suppose ha he observaions come from he following: y = hx) = x + x, 6) where h : R R. Le us consider a 3-embedding hisory generaed from hx) so ha, J 3 x) = µ x and 7) µ J 3 fx)= x. 8) µ 3 µ 3 Le gy) = y µ µ + for y R 3.Then µ g J 3 x) = x = J 3 fx). µ 3 µ 3 Therefore, he condiion for conjugacy is saisfied. Also, Dg) y =. 9) µ µ + Le W = R 3, W = span{,, ),,, )}, W 3 = span{,, )} and W 4 ={}.Then Dg) y W ) = span {,µ, ),,µ,)} W, Dg) y W ) = span {,, )} W and Dg) y W 3 ) ={} W 3. Noice ha he ses Dg) y W j can be proper subses of W j. In his example, his comes abou since he dynamics of g are no of full dimension, which is immediaely apparen from Eq. 9).) If v V \ V

4 6 W.D. Decher, R. Gençay / Physics Leers A 76 ) hen [ ] [ ] v = α + β, α, and DJ 3 ) v = α + β. µ Here, α implies ha DJ 3 )v W \ W.Ifv V \ V 3 hen [ ] v = β, β, and DJ 3 ) v = β. Also β implies ha DJ 3 )v W \ W 3.Ifw W \ W hen w = α µ + β + γ, α, and Dg) y w = αµ µ + βµ. Hence lim ln Dg ) y w =ln µ. If w W \ W 3 hen w = β + γ, β, and Dg ) y w = βµ. Hence lim ln Dg ) y w =ln. If w W 3 \ W 4 hen w = γ, γ and Dg ) y w =. Therefore lim ln Dg) y w =. This example shows Theorem. a work. The wo larges Lyapunov exponens of g are he Lyapunov exponens of f, and in his example he spurious hird exponen of g is. 4. Spurious Lyapunov exponens In [9,] he numerical sudies demonsraed ha he n Lyapunov exponens of f urned ou o be he larges n Lyapunovexponensof g. These resuls were obained by using an observaion funcion of he form hx,x,...,x n ) = x ) which has been widely used in simulaion sudies of nonlinear dynamical sysems. Consider he following variaion o he example in he previous secion. The dynamics are he same linear dynamics of Eq. 5) and he observaion funcion is he same as Eq. 6). From his we obain he same embedding equaions as 7) and 8). Now however, consider he following funcion g: foranya R,le a gy) = a µ + µ ) aµ µ µ µ + y ) for y R 3. Noice ha his is no in he form of Eq. 4), however i does saisfy µ g J 3 x) = x = J 3 fx) µ 3 µ 3 and herefore he condiion for conjugacy is saisfied. 6 Also, a a µ + µ ) aµ µ Dg) y =. µ µ + If > a, le W = R 3, W = span{,, ),,, )}, W 3 = span{,, )} and W 4 ={}. Then if a =, Dg) y W ) = span {,µ, ),,µ,)} W, Dg) y W ) = span {,,)} W, and Dg) y W 3 ) ={} W 3. 6 This shows ha here can be many funcions which can generae he same dynamics. In our case we are ineresed in he impac ha he observer funcion has on his mulipliciy of represenaions, g.

5 W.D. Decher, R. Gençay / Physics Leers A 76 ) If a hen Dg) y W ) = W, Dg) y W ) = W and Dg) y W 3 ) = W 3. If v V \ V hen [ ] [ ] v = α + β, α, and DJ 3 ) v = α + β. µ Here, α implies ha DJ 3 )v W \ W.Ifv V \ V 3 hen [ ] v = β, β, and DJ 3) v = β. Also β implies ha DJ 3 )v W \ W 3.Ifw W \ W hen w = α µ + β + γ, α and Dg ) y w = αµ µ + βµ + γa. Hence lim ln Dg ) y w =ln µ. If w W \ W 3 hen w = β + γ, β, and Dg ) y w = βµ + γa. Hence lim ln Dg ) y w =ln. If w W 3 \ W 4 hen w = γ, γ, and Dg ) y w = γ a. Therefore lim ln Dg) yw =ln a. Noe ha if a = hen his hird spurious Lyapunov exponen is. If µ > a > hen he subspace W 3 above needs o be changed so ha W 3 = span{,, )}. Then Dg) y W ) = W, Dg) y W ) = W and Dg) y W 3 ) = W 3. The hree Lyapunov exponens are: ln µ, ln a, ln.if a > µ hen change he subspaces so ha W = span{,µ, ),,, )}, W 3 = span{,, )} and again Dg) yw ) = W, Dg) y W ) = W and Dg) y W 3 ) = W 3 will hold. The hree Lyapunov exponens are: ln a, ln µ, ln. Noice ha in all cases he wo Lyapunov exponens of f are wo of he Lyapunov exponens of g. The hird Lyapunov exponen of g can be of any magniude. The problem comes from he fac ha he parial derivaives of g do no necessarily lie in he angen space of he image of he aracor under he Takens embedding ). I raises he quesion of how o idenify he n rue Lyapunov exponens of f from he m n spurious Lyapunov exponens ha make up he Lyapunov exponens of g. References [] W.A. Brock, Disinguishing random and deerminisic sysems: abridged version, J. Econ. Theory 4 986) [] W. Brock, C. Sayers, Is he business cycle characerized by deerminisic chaos?, J. Moneary Econ. 988) 7 9. [3] A. Wolf, B. Swif, J. Swinney, J. Vasano, Deermining Lyapunov exponens from a ime series, Physica D 6 985) [4] H.D.I. Abarbanel, R. Brown, M.B. Kennel, Variaion of Lyapunov exponens on a srange aracor, J. Nonlinear Sci. 99) [5] H.D.I. Abarbanel, R. Brown, M.B. Kennel, Lyapunov exponens in chaoic sysems: heir imporance and heir evaluaion using observed daa, In. J. Mod. Phys. B 5 99) [6] H.D.I. Abarbanel, R. Brown, M.B. Kennel, Local Lyapunov exponens compued from observed daa, J. Nonlinear Sci. 99) [7] S. Ellner, A.R. Gallan, D.F. McGaffrey, D. Nychka, Convergence raes and daa requiremens for he Jacobian-based esimaes of Lyapunov exponens from daa, Phys. Le. A 53 99) [8] D. McCaffrey, S. Ellner, A.R. Gallan, D. Nychka, Esimaing Lyapunov exponens wih nonparameric regression, J. Am. Sa. Assoc ) [9] R. Gençay, W.D. Decher, An algorihm for he n Lyapunov exponens of an n-dimensional unknown dynamical sysem, Physica D 59 99) [] W.D. Decher, R. Gençay, The opological invariance of Lyapunov exponens in embedded dynamics, Physica D 9 996) 4 55.

6 64 W.D. Decher, R. Gençay / Physics Leers A 76 ) [] R. Gençay, Nonlinear predicion of noisy ime series wih feedforward neworks, Phys. Le. A ) [] R. Mañé, On he dimension of he compac invarian ses of cerain nonlinear maps, in: D. Rand, L.S. Young Eds.), Dynamical Sysems and Turbulence, Lecure Noes in Mahemaics 898, Springer, Berlin, 98. [3] F. Takens, Deecing srange aracors in urbulence, in: D. Rand, L.S. Young Eds.), Dynamical Sysems and Turbulence, Lecure Noes in Mahemaics 898, Springer, Berlin, 98. [4] J.-P. Eckmann, D. Ruelle, Ergodic heory of chaos and srange aracors, Rev. Mod. Phys ) [5] J.-P. Eckmann, S.O. Kamphors, D. Ruelle, S. Cilibero, Lyapunov exponens from ime series, Phys. Rev. A ) [6] V.I. Oseledec, A muliplicaive ergodic heorem. Liapunov characerisic numbers for dynamical sysem, Trans. Moscow Mah. Soc ) 97. [7] J.E. Cohen, J. Kesen, C.M. Newman, Random Marices and Their Applicaion, Conemporary Mahemaics, Vol. 5, American Mahemaical Sociey, Providence, RI, 986. [8] M.S. Raghunahan, A proof of Oseledec s muliplicaive ergodic heorem, Israel J. Mah ) [9] D. Ruelle, Ergodic heory of differeniable dynamical sysems, Publ. Mah. Ins. Haues Éudes Sci ) [] J. Guckenheimer, P. Holmes, Nonlinear Oscillaions, Dynamical Sysems and Bifurcaions of Vecor Fields, Springer- Verlag, New York, 983. [] M. Sano, Y. Sawada, Measuremen of Lyapunov specrum from a chaoic ime series, Phys. Rev. Le ) [] W.D. Decher, R. Gençay, Lyapunov exponens as a nonparameric diagnosic for sabiliy analysis, J. Appl. Economerics 7 99) S4 S6.