水文系統之碎形分析與渾沌預測(Ⅱ)

  



   (II)

Fractal Analysis and Chaotic Prediction in Hydrologic Systems (II)

 NSC 88-2218-E-006-022

 87  8  1  88  7  31 

 !"

#$%&'()*+,-./

 


                 Lyapunov !"#$%&' ()* +3,4-./0123 4+56789:;<=> ?@ABCDEFGHIGJ

Lorenz"#.KCDEFBL MANO

PQ8R1STU.V23WXYZ ,[\]^H_=>`abc/ defghijkhl8mnopq rc.  stGG"#G =>u

Abstract

This study used the chaotic dynamics to analyze the temporal variation of hydrologic variables. The correlation dimension and maximum Lyapunov exponent are calculated. The results show that the chaotic attractor exists and suggests that it may be described by 3 or 4 variables. Four types of data are selected and the complexity is white noise, daily rainfall, daily streamflow and Lorenz attractor in descending order. Only the white noise is apparently in disorder, the others have some kinds of pattern. Once the data are transformed by the integration or moving av-erage, the complexity is decreased. It implies that the model can be established with a sim-pler way and have a longer lead-time pre-dictability.

Keywords: Phase Space, Correlation Dimen-sion, Chaotic Attractor, Complexity     vBwxMAy8Az{|} ~€‚ƒ„…†‡ˆ‰Š‹B MŒq.Ž&‘’ “”|•– —˜™&’ <“–ušZ›‘.œžŸ  ¡uv¢£y¤„$¥¦§S ˆ¢£¨U&ty©Bª«¬­„ "®#—˜™¯°§S±šZ –.²³´œBwx§SX+µ ¶‡˜™m·¸¬"®#¹º»¼w x.&½­¾wx¿À˜™Á ZUÃS$tµ¶Äu ¹¾ÅvÆÇWXžÈ.  

1. tJO¸i[1,2] É { , , , }x x_{1 2} L x_N BƍR­¢£Ê $AËOyN+23n,23© .|Ì`­AËÍάmÏÐt  Rm<sub>y^­-ÇÑ҇ƍAˬ</sub> `¨U“)ÇÑÒy$ÓÔ+ X_k =( ,x x_{k k+1}, ,L x_{k m+ −1}) ( )1 hy k=1 2, , ,L N_mÊ N_m = − +1.N m Õ(Takens[2])Ö$ÍÎÁ­d" #cÍά2d+1ty×/ØÙ U¢£µ¶. 2. ÍÎ ÚÛmty­¢ËÇÑÒÜ ÉOy­Ý"ÇV"BX J_i X Oy_j i ≠ j l i J j =1 L,2, ,N_m.Þ߄­àápâ r¬Bãä8{åVÇ(X_i, X_j)$ pâB?¬ rØÙpâ?¬ r ÇV&) 8ÇVy)æç5èYZé

„êC_m(r)[3][4]²“s

(

)

∑ ∑

_{= =} − − ≠ − = Nm m i N j i j m m m r X X i j N N r C 1 1 ) 1 ( 1 ) ( θ ( 2) hyθ+Heavisideé„ê+s6z>0 ‡θ( =z) 1ë6z≤0‡θ( =z) 0. ì`„ê+d ²“s <sub>m</sub> r r C d r m ln ) ( ln lim 0 → = ( 3) V¬­íî§S d<sub>m</sub>èÌíïmèðñ Ê¡uðñòóôõ­ö÷èëV ¬­„…§S d<sub>m</sub>èÌ&R­µ„mèø Êù¬ú„Ê`ú„è,ö÷è„ê+ ûU¢£$"#d. 3. Lyapunov Lyapunov+tyØÙR­ü Úýþ$
ýþOZ¦…­1 ”|&¢£y$V Õu.V¬­Ýƍ23 ÇWolf [5]JWolf[6]¥ØÙ Lyapunov„gŒ„겓s λ1 2 1 1 ₁ = ′ =

∑

+ N L L m k M t t k k log ( )4 hyNm‘„êM+( ,L Ltk t′k+1))¼ °Ý L_tk +Ç( ,x x_{tk tk+1}, ,L x_{tk m+ −1})O 
ÇÏÐpâ ′L_{tk 1+} ‡+ t_{k +1}$ L_tk n.6λ1> ûAË0 B­§SëÊ6λ1≤ ‡”!û0 AË|†‡$§S. 4. LZ=> ÅLempelZiv[7])el $ØÙgŒe+LZ=>u.ûŒ€ KolmogorovVO=>uÆÇÅ 1ef !$ØÙîkTA ËÌ)*R1 !"#+=>u  . V ¬ ­   A Ë Ê $  Kaspar  Schuster[8]ÅLZ=>u%&'L ûçLyapunovmc&†‡u| Š'm()»³.=>u„ê° ) ( ) ( ) ( s s s B C ′ = Ψ ( 5) hyC(s)B*LempelZivٌ)$= >B′(s)++¹ù¬M,J-î…./ $=>.ì`Å0Uã#­ 12AË,U¢£$=>S. 5. ;< 301234K)·I N4/-523JCDEF23 WX67çV.8Å01  < Z4BBox-Cox ¡u;<G9ZÙGY ZÙJ[\]:­1;<]8Oü  ;<.3LZ=>uWXç dì^=4UÃS8A,MAÊ V?±>O=>[email protected] ?Z)m>&kh¼i¸ üÕ(J²:BVqrCDuEF.
  
+ c G H   ABI+§ S/JKL+wxV}©Ø825 -M$233NÇÙNOP23 l©¬1000Q15MR/Z. ØÙ:ÝAË&SÍÎty$‘ pâ)YZZ4VYZ pâTVò!U%&V¡WXYO Z…/?[\Œ]^+. U1+15Ý23$VÍÎ U*U_s6ÍÎðñ &`‰Šaù¬ö÷±^%& "#ÊO"#è!¬Uy.bJ KLAË%&X+‡) * cû¢£ d+3,4.$$ &kh` Uµu^À3,4- ef g_`  xh+:­i'_ÌB^] EF. -*t¸iJVÍÎ  _ØÙ$AË%&" #±^^c+§S¬BV d8u+`ØÙLyapunov j"fk.V15Ý23ØÙO&Í Îtl1m5Lyapunov²”1 )Ë. !A&ÍÎd?t ^cónopè&¹wx±q–»

³rÍÎ{T¬"#$@ AØÙ.ã#°³&¬O"# LyapunovOè]+/— ”!JKAËfst8¢£ ¶uj"v'°³.     +c>qçu»³K3„IJ N4N3TLorenz23JCDEF 23.w_Lorenz23B5T§S ÊCDEF23‡xyjíî§S.m¬ S1;<(z&Box-Cox;<gBü λ{311-èZ4Bs =λ -2.0 (0.5) +3.0.&9ZYZ$;<gBK|A Ë(m=0)N9ZYZST10}Z4 Bs =m 1 (1) 10.8&[\]gB Å’~L=5+‘+€K|A Ë(L=0)N[\]{T10}Z4Bs = L 5 (5) 50.»Ê$ƍ23K¹º7 NOP:­1 ;<]3101 ü . &`Xã#M9:;<$01| 23ö÷=>_sLorenz"#+ 0.0754 (0.0008) G C D E F + 1.0279 (0.0005)GI+0.9356 (0.0015)J +0.4239 (0.0029)‚ƒ„…BOd †9.lû (è^L =ZO9? uLorenz"#‡0”!ˆ‰8Auë CDEF‡1Š íîuµ¶ëI ~d‡1”!Oíîu‹ˆ‰ˆb/ †Œ9=> ?‡–Ž) 3I‹8­„STU,8 Auë‡0.4”!R1ST UdˆB8Au¢£.ã#U2$:1; <234Oö÷=>uV;<ü;< è$ @™^`‰uù‘ ±^0-|234$=>@AZ4 BsCDEF>I>>Lorenz" #Ê&‘;< “{‹€ `­@A’“}9Zóno”@A.  ! &Z $tÃS=>u •V–—€= 25 -œ˜}ƍ M$I23J–—™Ñ=L(J KGš›G“œ›)$23WXZ !HIÃS=>S6“\]è ‡ 0.9ʝÃS$=>\]è_ ¬ 0.5 !OUÃSQ8R1TU.m ¬Q“¬øQ`½­¾À˜™ ÁÂ)°³7­o.8=>V)3 $tì#(²žJ“S)òML  ù‘^ì`=>±c!+U ¢£­-yj.m¬=> dm9ò7|•Õuû ° ³½­¾À˜™ÁÂ)°³ò Ÿ‘*¬QØÙ$ÜÉ ò ‘rò'Oqo°³^7+çd. "#$%&
1. )Z$%&"#) * OX+ +3,4Ê/ Lyapunov”!V Õ u`j­"¡LOX+. 2. 01‘23TU$=>@A *Ê?ÕA+CDEFGHIG JLorenz"#KCDEFBMA¢£ NOP] !Q8R1TU,»¼B x¬8A¢£. 3. 6=>uÀ¬‘;<  “$1234çd|•OˆÊ 8=¢cm¬ûV¬8A u –Ž“¬MAu. 4. Ì|23WXYZ,[\]$ ;<£Á^8¤H_¢£=>` ¥ïkhij¦ì`c§ef¨. 5. HIÃS=>S6“\] è‡0.9ʝÃSQ\]è_¬ 0.5 !Q8R1TU.I$=>“ ¬Q`½­¾°³7­o.

'()*



[1] Packard, N. H., J. P. Crutchfield, J. D. Farmer and R. S. Shaw (1980). Geometry from a Time Series, Physical Review Letters, 45(9), 712-715. [2] Takens, F. (1981). Detecting Strange Attractors in Turbulence, In: Dynamical Systems and Tur-bulence, Warwick 1980, Lecture Notes in Mathematics, No. 898, Edited by D. A. Rand and L.-S. Young, Springer-Verlag, Berlin, 366-381.

[3] Grassberger, P. and I. Procaccia (1983a). Char-acterization of strange attractors. Physical Review Letters 50(5):346-349.

[4] Grassberger, P. and I. Procaccia (1983b). Meas-uring the Strangeness of Strange Attractors.

Physical D 9:189-208.

[5] Wolf, A., J. B. Swift, H. L. Swinney and J. A. Vastano (1985). Determining Lyapunov Expo-nents from a Time Series, Physica D, 16, 285-317.

[6] Wolf, A. (1986). Quantifying Chaos with Ly-apunov Exponents, In: Chaos, Edited by A. V. Holden, Manchester University Press, Man-chester, 273-290.

[7] Lempel, A. and J. Ziv (1976). On the Com-plexity of Finite Sequences, IEEE Transcations on Information Theory, IT-22(1), 75-81.

[8] Kaspar, F. and H. G. Schuster (1987). Easily Calculable Measure for the Complexity of Spa-tiotemporal Patterns, Physical Review A, 36(2), 842-848. ” 1 15 ÝAË&‘ÍÎ “$ Lyapunov  U 1 15 ÝAË$VÍ ÎU
 Lyapunov m=1 m=2 m=3 m=4 m=5 030-01 - 0.012 0.016 0.038 0.044 0.033 030-02 - 0.002 0.016 0.045 0.038 0.037 030-03 - 0.027 0.006 0.024 0.029 0.024 030-04 - 0.015 0.017 0.053 0.046 0.036 030-05 - 0.003 0.007 0.027 0.038 0.030 030-06 0.000 0.009 0.028 0.033 0.039 030-07 0.031 0.117 0.125 0.107 0.071 030-08 - 0.014 0.100 0.084 0.075 0.071 030-09 - 0.031 0.014 0.038 0.038 0.033 030-10 - 0.243 0.206 0.163 0.119 0.085 030-11 0.020 0.046 0.042 0.038 0.049 030-14 0.682 0.145 0.185 0.157 0.119 030-20 - 0.050 0.031 0.116 0.093 0.089 030-24 - 0.025 0.231 0.209 0.143 0.127 030-25 0.058 0.140 0.131 0.115 0.085

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5Paramete r of Box-Cox Transforma tion2.0 2.5 3.0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 S aturated L Z-C o m plexity RND PREC DISC ATT 01 2 3Order of Differencing45 6 78 910 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 S aturated L Z-C o m plexity RND PREC DISC ATT 01 2 3Order of Integral45 6 78 910 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 S aturated L Z-C o m plexity RND PREC DISC ATT 05 10 15Number of Smoothing2025 30 3540 45 50 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 S aturated L Z-C o m plexity RND PREC DISC ATT

(a) Box-Cox;< (b) 9Z;< (c) YZ;< (d) [\];<

I soline of L Z Complexity Iso line of LZ Com plexity Isoline of L Z Complexity Isoline of LZ Comp lexi ty

(a) –— (b) JK (c) š› (d) “œ›