Important Notes on Lyapunov Exponents

Transcription

1 Imporan Noes on Lyapunov Exponens Keying Guan Sciense College, Beijing Jiaoong Universiy, Beijing, China, Absrac: I is shown ha he famous Lyapunov exponens canno be used as he numerical characerisic for disinguishing differen kinds of aracors, such as he equilibrium poin he limi closed curve he sable orus and he srange aracor 1 Inroducion The concepion Lyapunov Exponen has been used widely in he sudy of dynamical sysem Usually, he Lyapunov exponen or Lyapunov characerisic exponen of a dynamical sysem is a quaniy ha characerizes he rae of separaion of infiniesimally close rajecories Z () and Z ( ) in phase space Le Z( ) Z( ) - Z ( ), Z Z ) - () ( Z, if Z( ) e Z (1) hen is reaed as he Lyapunov exponen If he rajecory Z () is given by a n-dimensional linear ordinary differenial equaion sysem wih consan coefficiens Z A Z f() () and if he consan coefficien marix has n differen eigenvalues,,,, 1, n hen he real pars of he n differen eigenvalues are naurally he Lyapunov

2 exponens However, if he dynamical sysem is no given by (), for insance, if he dynamical sysem is a nonlinear polynomial auonomous sysem, he concepion on Lyapunov exponens becomes complicaed Consider he commonly acceped definiion of Lyapunov Exponens, which are quoed from he reference [1], and from he Wikipedia on Lyapunov exponen (hp://enwikipediaorg/wiki/lyapunov_exponen) A The maximal Lyapunov exponen can be defined as follows: max 1 Z( ) lim lim ln (3) Z Z The limi Z ensures he validiy of he linear approximaion a any ime I is required ha he wo limis canno be exchanged, oherwise, in bounded aracors, he resul would be rivially B For a dynamical sysem wih evoluion equaion f in an n dimensional phase space, he specrum of Lyapunov exponens {,,, } n 1 in general, depends on he saring poin x The Lyapunov exponens describe he behavior of vecors in he angen space of he phase space and are defined from he Jacobian marix df ( x) J ( x ) (4) dx x The J marix describes how a small change a he poin x propagaes o he final poin f x ) The limi ( lim ( 1 J Transpose ( J )) (5)

3 defines a marix V x ) (he condiions for he exisence of he limi are given by ( he Oseledec heorem) If (Λ i x ) are he eigenvalues of V ( x ) Lyapunov exponens i are defined by, hen he x ) Ln Λ ( ) (6) i( i x Based on he experience of he linear sysem () and some plausible hinking, for a dissipaive sysem, as crierions, i is proposed in he reference [1] ha, if he aracor reduces o (a) sable fixed poin, all he exponens are negaive; (b) limi cycle, an exponen is zero and he remaining ones are all negaive; (c) k-dimensional sable orus, he firs k LEs vanish and he remaining ones are negaive; (d) for srange aracor generaed by a chaoic dynamics a leas one exponen is posiive The above-menioned definiion on Lyapunov exponens and proposed crierions abou he relaions beween he characerisic of LE and he properies of he aracors are widely used Some noes on he definiion of Lyapunov Exponens I should be poin ou firs ha he expression (3) is no sricly, since he value of righ hand side of i is usually no uniquely deermined for a given rajecory In fac he righ hand side of (3) depends on how Z To see his fac, wihou los of generaliy, assume ha he given dynamical sysem is a hree-dimensional auonomous sysem dx d dy d dz d Or for simple, (7) can be wrien as dr F(r) d where T r ( x, y, z), F ( P, Q, R) P( x, y, z) Q( x, y, z) R( x, y, z) T (7) (7 ) Le

4 T r ( ) ( x( ), y( ), z( )) and r ( ) ( x( ), y( ), z( )) T are wo rajecories of (7) Formally, hey saisfy respecively r ( ) r ( ) F( r ( s)) ds (8) r( ) r () F( r( s)) ds (9) Le r ( ) r( ) r ( ) ( x( ), y( ), z( )), and (1) r r r( ) () ( T T x (), y(), z()) l(cos,cos,cos ) where l ( x()) ( y()) ( z ()), and (cos,cos,cos ) T is he uni vecor of r represened wih is azimuh When l is small enough, ha is, hese wo rajecories are close enough, hen, for fixed, r() can be reaed a small quaniy wih he same order as r So r( ) exp( J( r ( s)) ds) r o( r ) (11) where o( r ) is a higher order small quaniy of r (ref[]) Therefore, when along fixed direcion (cos,cos,cos ), r hen r( ) lim r r T J ( ( s )) ds) (cos,cos,cos ) (1) exp( r This means he limi (1) depends obviously on he direcion of r Le J1( ), J ( ),, J n( ) be he n eigenvalues of he marix J (r ( s)) ds, and assume J * () is he max real par of hese eigenvalus Then max[ lim r r( ) ] exp J r From (13), if he following limi exiss * ( ) * 1 r( ) J ( ) lim ln max[ lim ] lim r r (13) (14)

5 hen his limi can be reasonably reaed as he maximal Lyapunov exponen max And he n limis Re J i ( ) i lim (15) may called he Lyapunov exponens of he rajecory r ( ) This definiion has obvious meaning and has a close relaion wih he expeced relaion (1) However, here are some issue should be discussed on he commonly used Lyapunov exponens defined by (5) and (6) Clearly, J is equivalen o exp( J ( r ( s)) ds), and 1 ( s ds s ds ( )) Transpose( ( ( )) ) J r J r ( J Transpose( J )) exp( ) So by (5) and (6), he Lyapunov exponens are jus he eigenvalues of he following limi marix J( r ( s)) ds Transpose( J( r ( s)) ds) lim These exponens may be jus he same as hose given by (15), if J (r ( s)) ds is wih some symmeries For insance, if J ( r a ( s)) ds T J r ( s)) ds ( J( r ( s)) ds b c ( ), or if c b (16) Bu, in mos cases, he wo kinds of Lyapunov exponens are differen For example, if J ( r ( s)) ds 3 he exponens given by (5) and (6) are 1,- and 3, and he exponens given 5 5 by (15) are 1, and Someimes, he difference beween he wo kinds of exponens is subsanive For insance, if

6 J ( r ( s)) ds hen he exponens given by (5) and (6) are, 3 and 11 3, and he exponens given by (15) are 1, and 3 In he firs group of he exponens, 11 3 he maximum one is, which is posiive, and in he second group, he maximum one is 1 The wo maximum exponens indicae wo opposie sabiliies Because of he difference menioned above, he exponens given by (15) will be denoed as LE J, and he exponens given by (5) and (6) will be denoed as LE O Clearly, in he applicaion of he Lyapunov exponens, he exisence of hese numbers is very imporan Oseledec proved ha he limi marix V x ) of (5) exiss wih he excepion of a subse of iniial condiions of zero measure (ref []) This fac migh be he advanage of he definiion of he exponens given by (5) and (6) ( However, he auhor believes ha LE O has los he basic pracical significance when i is differen o LE J In he case ha he rajecory r ( ) is a closed orbi, ha is, r ( ) is periodic, he limi T J(r (s))ds J(r (s))ds lim (17) T does exis So, boh of LE J and LE O exis In he following secion, we will sudy wha hese exponens can be for he spaial limi closed orbis (including he spaial limi cycle) C The Lyapunov exponens for some spaial limi close orbi In order o ge some exac resuls, his paper will sudy firs some limi cycles, which can be represened exacly wih simple elemenary funcions A spaial limi cycle is said mea-sable in his paper, if here are some rajecories approaching o he cycle, and in is any small neighborhood, here are

7 also some rajecories going away from he neighborhood as Consider hree-dimensional auonomous sysem dx d dy d dz d y x(1 x x y(1 x y y )(1 x z z )(1 x 3 y ) y ) (18) where ( ) and are real parameers The sysem (18) has 6 differen limi cycles: (i) (ii) x 1 sin, y 1 cos, z (iii) (iv) (v) x 1 sin, y 1 cos, z (vi) x 1 sin, y 1 cos, z Since he sysem (18) is srongly symmerical, LE J and LE O are jus he same For he limi cycle (i), he eigenvalues of (17) are and, and LE J = LE O :,-, - If, he limi cycle is mea-sable when, and i is unsable when If, he limi cycle is asympoically sable when, and i is measable when For he limi cycle (ii), he eigenvalues of (17) are, i, i, and LE J = LE O :,,

8 If, he limi cycle is unsable when, and i is mea-sable when If, he limi cycle is mea-sable when, and i is asympoically sable when For he limi cycle (iii) and (iv), he eigenvalues of (17) are and, and LE J = LE O : -,, The limi cycle is asympoically sable when, and i is mea-sable when For he limi cycle (v) and (vi), he eigenvalues of (17) are and, and LE J = LE O : -,, The limi cycle is mea-sable when, and i is asympoically sable when From he above resuls, i is easy o see he signs of Lyapunov exponens for a asympoically sable limi cycle may have wo kinds of disribuion (a ) All of he hree Lyapunov exponens are negaive, ie, I happens in he case (iii) and (iv) when and, and also in he case (v) and (vi) when and, (b )One of he Lyapunov exponens is zero, and he oher wo are negaive, ie, I happens in he case (i) when and, in he case (ii) when and, and in he case (v) and (vi) when and Besides, here are sill some oher possible sign disribuions of he Lyapunov exponens, such ha he corresponding limi cycle is asympoically sable

9 Consider he following sysem dx d dy d dz d y x(1 x x y(1 x y ) 3 z z y ) 3 3 (19) I is easy o see ha his sysem has hree limi cycles if : (i ) (ii ) (iii ) In he case (ii ) and (iii ) when, he limi cycles are boh asympoically sable, and LE J = LE O :,,- In he case (I ) when, he limi cycle is also asympoically sable, and LE J = LE O :,, From his example we see ha here are oher wo possible sign disribuions of he Lyapunov exponens, such ha he corresponding limi cycle is asympoically sable: (c ) Two zeros and one negaive, ie, (d ) Three zeros, ie, Noe: From he above examples, i is easy o see ha, some exponens equals zero is usually corresponding o he criical siuaion ha he sabiliy of he limi cycle changes, or ha he number of asympoically limi cycles changes I seems a very reasonable crierion for he deerminaion of he sabiliy of he limi cycle Bu we will seem ha his crierion may no be rue in general In all of he above examples, LE J and LE O are he same In he following example, more unexpeced facs on he Lyapunov exponens for limi closed orbis

10 will appear Consider he paricular Silnikov equaion sysem dx d dy d dz d x 3 a y z () x y b z where he parameers and are posiive In [3] and [4], his sysem is proved o be an ideal sysem for he sudy of hree-dimensional differenial dynamical sysems since i has differen kinds of aracors, including spaial limi closed orbis wih differen roaion numbers I has been shown ha when he parameer is slighly smaller han, he sysem () will have a limi cycle around he origin, ie, he Hopf bifurcaion (Beiye Feng 1) and hen Hongwei Liu ) have given respecively he sric proof for he exisence of he Hopf bifurcaion) This sysem is no inegrable wih quadraure, jus as expeced by mos researchers In fac, i has been proven by Yanxia Hu 3) ha his sysem does no admi any global analyical Lie group, excep he rivial one: Therefore, here is no hope for us o represen he limi cycle of () wih he quadraure So, he calculaion of he Lyapunov exponens for is limi cycles, or more generally for he limi closed orbis wih differen roaion numbers can only be realized numerically (): The following is a series of numerical resuls of differen limi closed orbis of (n 1 ) and The () has a limi cycle of period (see Fig 1) For he limi cycle, 1) Beiye Feng, 关于在 b=1 时发生 Hopf 分支的证明, hp://blogsciencenecn/blog hml ) Hongwei Liu, e al, 一类 Silnikov 方程的 Hopf 分岔及其稳定性, o appear 3) Yanxia Hu, The Non-inegrabiliy of a Silnikov Equaion, o appear

11 Fig1 and (n ) and The sysem () has a limi cycle of period (see Fig ) For his limi cycle, Fig and (n 3 ) and The sysem () has a limi cycle of period

12 (see Fig 3) For his limi cycle, Fig3 and (n 4 ) and The sysem () has sill one limi cycle of period (see Fig 4) For he limi cycle, Fig4 and

13 (n 5 ) and The sysem () has sill one limi cycle of period (see Fig 5) For his limi cycle, Fig5 and (n 6 ) and The sysem () has sill one limi cycle of period (see Fig 6) For his limi cycle,

14 Fig 6 and (n 7 ) and Numerically, he sysem () has sill a limi cycle of period (see Fig 7) For his limi cycle, Fig 7 and (n 8 ) and From he numerical resul, i can be seen ha he number of asympoically sable limi cycles of sysem () has become wo (see

15 Fig 8) They are symmerical abou he origin and are very close o each oher The period of each limi cycle is They have he same LE J and he same LE O, Fig 8 and wo slighly separaed limi cycles (n 9 ) and The sysem () has wo asympoically sable clearly separaed and symmerical limi cycles (see Fig 9) The period of each limi cycle is For hem

16 Fig 9 and, wo clearly separae limi ccycles (n 1 ) and The sysem () has wo asympoically sable clearly separaed and symmerical limi closed orbis Their roaion number are boh wo (see Fig 1) The period of each limi cycle is They have he same LE J and he same LE O, Fig 1 and, wo limi closed orbis wih roaion number

17 (n 11 ) and The sysem () has only one asympoically sable limi closed orbis Is roaion number is 13 (see Fig 11) I is symmeric o iself abou he origin The period of he limi closed orbi is Fig 11 and One limi closed orbi wih roaion number 13 From he above numerical resuls, we see ha, for he sysem (), LE J and LE O are quie differen, in he cases (n 1 ), (n ) and (n 3 ), he hree exponens of LE J are all negaive, while for LE O, one posiive and wo negaive In oher 8 cases, all of he exponens for he asympoically limi closed orbis, boh LE J and LE O have one posiive and wo negaive values, ie, (,-,- ) So, we have seen ha he sign disribuion of Lyaponov exponens of a hree-dimensional limi closed orbis has he following five possibiliies: (a )(,, ) (b )(,, ) (c )(,, ) (d )(,, ) (e )(,, )

18 This resuls is quie differen o he proposed crierion for disinguishing aracor given by ref [1]: (a) sable fixed poin, all he exponens are negaive; (b) limi cycle, an exponen is zero and he remaining ones are all negaive; (c) k-dimensional sable orus, he firs k LEs vanish and he remaining ones are negaive; (d) for srange aracor generaed by a chaoic dynamics a leas one exponen is posiive In addiion, for he sysem (), he fac ha one of LE J is zero and he ohers are negaive for he asympoically sable limi cycle happens around he parameer a 1and b 54 (see (n 3 )), and he fac ha he limi cycle separaed ino wo happens around he parameer a 1and b 4893 (see (n 6 ) (n 7 ) and (n 8 )) These facs show ha some exponens equals zero may no be he criical siuaion ha he sabiliy of he limi cycle, or ha he number of asympoically limi cycles changes 4 Conclusions This paper has shown ha here are wo kinds of he Lyapunov exponens LE J and LE O, When hey have differen values, he second one may lose he basic meaning in dynamical sysem heory The concree examples have shown ha neiher of hese wo exponens could be applied as he numerical characerisic for disinguishing differen aracors In anoher paper, he auhor will give an explanaion for why limi closed orbis can sill be asympoically sable when one of he hree exponens is posiive References [1] Cencini M e al, M Chaos From Simple models o complex sysems World Scienific, (1) ISBN [] Oseledec, V I (1968) A muliplicaive ergodic heorem Lyapunov characerisic numbers for dynamical sysems, Trans Mosc Mah Soc 19, p 197 [3] Keying Guan, Non-rivial Local Aracors of a Three-dimensional Dynamical Sysem arxivorg, hp://arxivorg/abs/13116(13) [4] Keying Guan, Beiye Feng, Period-doubling Cascades of a Silnikov Equaion arxivorg, hp://arxivorg/abs/13143