1、1l E9 Kv Lyapunov ) do V (Z +v0 , q 430033) K 1 ) l E9 Kv Lyapunov M15bl E 3 aH ( | ( 1 z Z b ( E f / I nK s V 1 9 T b4 4 B L u E K vLyapunov i L u/K) F. 29 TYb 1oM Lyapunov HW s Discussion on the Method for Calculating Largest Lyapunov Exponents from Small Data Sets Lu Zhen-bo Cai Zhi-ming Jiang Ke
2、-yu (College of Electronic Engineering, Navy Engineering University, WuHan 430033, China) Abstract: The problems on the method for calculating largest Lyapunov exponents from small data sets are discussed. The algorithm is robust the changes in the following quantities: embedding dimension, reconstr
3、uction delay and mean period of the time series. We indicate that the algorithm is still accurate without considering temporal separation, if the mean period is unknown. A new criterion of selecting linear zone for fitting the largest Lyapunov exponents is presented. In the end, the effects of addit
4、ive noise is discussed. Key Words: chaos, Lyapunov exponents, time series analysis 1 + M | # + X mM bWs I K # jXjXjjXjXXdjmin)0( = (2) I K # S )0(jd jjXjX V U=S b E I K # BEL K s 1 p jXjXpjj (3) p HW ( VYV q ( q 9 b * Kv Lyapunov 1 V YVE (? q9 b I 9 # HW jXjXi )(idjijijjXXid+=)( (4) ),min(,2,1 jMjMi
5、 = LL I K # jXjX 1 ? q * )(1)(tijjeCid= )0(jjdC = (5) (5) T | )ln()(ln)(ln1tiCidjj+= (6) (6) T V A wLBS = L1“ wL| qiidj)(ln t1 b yN% p (“i j )(ln idjt )(iy=qjjidtqiy1)(ln1)( (7) q d , “ b 4 wL B L u i Kl =E T B L L| q Kv Lyapunov )(idjiiy )(1 b 23M1) #_ L (1)(7) T V AKv Lyapunov 1 p #/ 5 (1)1 HW ZE
6、V H 3 p H1M1E (Auto-Correlation, AC)3a E (Mutual Information, MI)6 p 3 G-P E5aLK #E (False Nearest Neighbor, FNN)7b ZE TB B RosensteinD 2YVv _ L L l E 3 H |i 1 z Zb yN V Hm 3 ZE 91 M1E G-P E pKv Lyapunov m1 Bb Logistic Lorenz xs _ Lb Logistic S )(1)(4)1( nxnxnx =+ (8) Lorenz Runge-Kutta Ess 0.01 =+=
7、bzxyzyrxxzyyxx& )(9) 4,92.45,16 = br b “V Logistic 500 Lorenz xs 5000 bV 1 Logistic Lorenz xs Kv Lyapunov Yb V 1 Kv Lyapunov Y System N m p Calculated1 Expected1ref.% errorLogistic 500 1 2 (G-P) 50 0.698 0.6938+0.72 1 (FNN) 0.702 +1.30 Lorenz 5000 12 (AC) 5 (G-P) 50 1.467 1.5001-2.20 11 (MI) 3 (FNN)
8、 1.471 -1.93 Logistic S 3yN H9 | 1 G-P E p 3 2LK# E p 3 1 3 pKv Lyapunov 1 sY 0.698 0.702 X 0.693V bLorenz %3.1 xs 1M1E G-P E p H 12a 3 5EL K #E p H 11a 3 3 3 pKvLyapunov 1 sY 1.467 1.471 X 1.500 V b ( %2.2 p ( |501 ( hB A (2) b (2)1 HW ( T (3) E I K # BEL K s b I K #Wv ( D 2 ( VYV q ( q 9 b1 ( 5Bi
9、1 v pbBZ jXjX3 q i1 A 6BZ t 0 LiB b O ) Z E ( YV_?C l E ( |9 1 z Zb Logistic Lorenz xs I ( 4Kv Lyapunov YTnV 2b V 2 ( Kv Lyapunov Y System N m p Calculated1 Expected1ref.error%Logistic 500 1 2 1 0.703 0.6938+1.44 50 0.698 +0.72 Lorenz 5000 12 5 1 1.454 1.5001-3.07 50 1.467 -2.20 V V A ( psY | 1a50 H
10、 l E p Logistic Lorenz xs Kv Lyapunov 1 V byN ( E f / V I nK s !%1.31=p 9 V1 z9 Tb (3)1L u - V 4 wLB L u iKl=ETBL L| q Kv Lyapunov iiy )(1 b I n )(iyf = B f )1()()()(lim0=iyiyiiiyiydidffi(10) X f / T L1 “ * iiy )( f B by N wL K r - | wLiiy )(iiyiy )1()( iMM l u XL ubm 1 m 2 Logistic Lorenz xs 9 Tb m
11、 1 Logistic m 2 Lorenz xs 4m 1(b) H wL62=i iiyiy )1()( +B wL | qMyN Logistic L u | bm 2(b)iiy )(62=i 501=i H wL A Hiiyiy )1()( 300200=i )1()( iyiy i9v7hl v t 0 H w L B1 l S =M y N Lorenz 20050=iiiyiy )1()( xs L u |by 0 bW K I K # i KV 0yN i9vB H wL KrwL K t 0b 6 M bWy wL L u/20050=ijXjX)(idjiiy )(ii
12、yiy )1()( iiy( )1)1( + m | (4) b (4)1F. 2) L= “ H VE Y | TF. 2 ) b IF. 2l E9 TYD 2 Logistic Lorenz xs F . 2il.1 (Signal to Noise Ratio, SNR) . 2 q| q=SNR (11) i V 39 Tb V 3 F. 2Kv Lyapunov Y2System N m SNR Calculated1 Expected1ref.error%Logistic 500 1 2 1 0.704 0.6938+1.6 10 0.779 +12.4 Lorenz 5000
13、11 3 1 0.645 1.5001-57.0 10 1.184 -21.1 . 2 Kv Lyapunov kv I K # V 1 VrbyN4 .1 Kv Lyapunov vVV 3 ALogistic H q Q711=SNR 10=SNR Hl E db Logistic HT p/ |B b 1=SNRn5 I n . 2 Kv Lyapunov 9 b ! . 2 H,21 Nnnn L 3 HM bW I K # mjNjN=+=10222)(minmin)0(mkkjkjjjjNjnnNNdj(12) i HW )(idj=+=10222)()(mkikjikjijijj
14、nnNNid(13) B 1 T )0(2jd )(2ldj1,1,0 = ml L ?C H i)0(2jd )(2ldjlm 5M Fb I n 2222,minmin)(min babacbaba+=+= (14) yN B)(idj=li 1,1,0 = ml L 1 | l O )1()()0( mdddjjjL T (7) wL M ? pb iiy )( iidj)( s yN i Hin Ni ,2,1 L= M bWN . 2 V K iv 3 3 H ( “dN 0 - s ! 3 m4=m bm 3m 4sY H 5= 1= H9 Tmb m 3 . 2 , 4,5 =
15、m m 4 . 2 , 4,1 = m m 3(a) )(iy 15,10,5,0=i H | l O )15()10()5()0( yyyy wLr - E B aL ub m 4(a) iiy )(1= E lG )3(),2(),1(),0( yyyy)3()2()1()0( yyyy bm 4(a) wL r - V| L u iiy )(41=i )3()2()1()0( yyyy M bW Q # ? qyN| L u pb 3 | H9 V MTb 41=i mV 3 Logistic H f p9L uTbm 5 Logistic HTmm 6 |“ H 3 H . 2Tmbm
16、 5 T4 L u * Kv Lyapunov 1=SNR1=SNR21=i1 0.706 D 2TM 0.002 m 6T94 L uKv Lyapunov 21=i1 0.704D 2 Logistic HT Mb1m 5m 6 ?C m“ Y wLbyN V Logistic 1=SNR1=SNR HF . 2Kv Lyapunov 9 V6 T p4 wL L u V7 D 2 Logistic iiy )( 1=SNRHpTb m 5 Logistic , 2,1 = m , 1=SNR m 6 . 2 , 2,1 = m V E Kv Lyapunov 1 p9 wL L u/ii
17、y )(1)1( + m b . 2 Kv Lyapunov kv l EE9 B. 2 Kv Lyapunov Vm 3(a)am 4(a)m 6(a) ( V A 1)1( += miH wL zBQrb iiy )(4 YV 5)/ (1)l E 3 H | 1 z Z V H 3 ZE91M1E G-P E pKv Lyapunov 1 B (2) ( E f / I nK s 9 V1 9T (3) wL Kr - wLiiy )( iiyiy )1()( iMM l u XL uL u/ 1)1( + m (4) . 2 Kv Lyapunov kv M bW # “y l EE
18、9 B. 2 Kv Lyapunov 7 O .1 . 2YAb ID 1 A.Wolf J.B.Swift, H.L.Swinney and J.A.Vastano. Determining Lyapunov exponents from a time series, Physica 16D, 71985, 285317. 2 M.T.Rosenstein, J.J.Collins and C.J.De luca. A practical method for calculating largest Lyapunov exponents from small data sets, Physi
19、ca D, 1993, 65: 117134. 3 W.Liebert, H.G.Schuster, Proper choice of the time delay for the analysis of chaotic time series. Phys.Lett.A 142 (1989) 107. 4 F.Takens. Determing strange attractors in turbulenceJ. Lecture notes in Math. 1981,898:361-381. 5 P.Grassberger, I.Procaccia. Measuring the strang
20、eness of strange attractorsJ. Physica D, 1983, 9:189-208 6 A.M.Fraser, H.L.Swinney. Independent coordinates for strange attractors form time seriesJ. Phys. Rev. A. 1986,33:1134-1140. 7 M.B.Kennel, R.Brown, H.D.I.Abarbanel. Determining embedding dimension for phase-space reconstruction using a geometrical constructionJ. Phys. Rev. A 1992,45:3403. 8 J.-P. Eckmann, D.Ruelle, Ergodic theory of chaos and strange attractorsJ, Rev. Mod. Phys. 57(1985) 617 Te do (1978-) 3 i p V 3 Z_ 2 |) a HW sa T MY ? YqZ +v0 2 i 430033 “ 13476281645 (Mobile), 027-83444036 (Office) E-mail : http:/8
