% This TeX file was generated by /public/bin/texpsinclude.new % from the original TeX file cgjk.tex % and the PostScript files Figure1.ps Figure2.ps Figure3.ps Figure4.ps Figure5.ps Figure6.ps. % TeX writes out the included PostScript to files. % % Here is part of cgjk.tex: %\documentstyle[pre,aps,amsfonts,eqsecnum]{revtex} \documentstyle[pre,aps,preprint,psfig,amsfonts,eqsecnum]{revtex} % Here are the % TeX macros for dumping included Postscript to files. % Adapted from Knuth's \answer macro in the TeXbook. % Jamie Stephens, jamies@math.utexas.edu, 28 Nov 94 \def\endofps{EndOfTheIncludedPostscriptMagicCookie} \chardef\other=12 \newwrite\psdumphandle \outer\def\psdump#1{\par\medbreak \immediate\openout\psdumphandle=#1 \copytoblankline} \def\copytoblankline{\begingroup\setupcopy\copypsline} \def\setupcopy{\def\do##1{\catcode`##1=\other}\dospecials \catcode`\\=\other \obeylines} {\obeylines \gdef\copypsline#1 {\def\next{#1}% \ifx\next\endofps\let\next=\endgroup % \else\immediate\write\psdumphandle{\next} \let\next=\copypsline\fi\next}} \outer\def\closepsdump{ \immediate\closeout\psdumphandle} % Here is the PostScript for Figure1.ps: \message{Writing file Figure1.ps} \psdump{Figure1.ps}%!PS-Adobe-2.0 %%Creator: dvipsk 5.55a Copyright 1986, 1994 Radical Eye Software %%Title: fig1.dvi %%Pages: 1 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: /usr/local/lib/texmf/bin/hpux/dvips -M -O 0.0cm,1.7cm %+ -o fig1.ps fig1 %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1998.01.27:1010 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{/nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{/sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0]N df-tail}B /E{ pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get} B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{ rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv} B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail} B /c{-4 M}B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{ 3 M}B /k{4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{ 3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet TeXDict begin 39158280 55380996 1000 300 300 (fig1.dvi) @start /Fa 1 4 df<13C0A538E0C1C0EAF0C33838C700EA0EDCEA03F0EA00C0EA03F0EA 0EDCEA38C738F0C3C0EAE0C13800C000A512157D9619>3 D E /Fb 1 49 df48 D E /Fc 2 73 df14 D<90270FFFFE01B512C0A2D9003FC73807E000023E5DA44A4A 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1072 2007 V 1075 2008 V 1078 2010 V 1080 2011 V 1083 2013 V 1086 2015 V 1089 2016 V 1092 2018 V 1095 2019 V 1098 2021 V 1101 2023 V 1104 2024 V 1107 2026 V 1110 2027 V 1113 2029 V 1116 2030 V 1119 2032 V 1122 2033 V 1125 2035 V 1128 2036 V 1130 2038 V 1133 2039 V 1136 2041 V 1140 2042 V 1143 2044 V 1146 2045 V 1149 2047 V 1152 2048 V 1155 2050 V 1158 2051 V 1161 2052 V 1164 2054 V 1167 2055 V 1170 2057 V 1173 2058 V 1176 2059 V 1179 2061 V 1182 2062 V 1185 2064 V 1188 2065 V 1192 2066 V 1195 2068 V 1198 2069 V 1201 2071 V 1204 2072 V 1207 2073 V 1210 2075 V 1214 2076 V 1217 2077 V 1220 2079 V 1223 2080 V 1226 2081 V 1229 2082 V 1233 2084 V 1236 2085 V 1239 2086 V 1242 2088 V 1245 2089 V 1249 2090 V 1252 2091 V 1255 2093 V 1258 2094 V 1262 2095 V 1265 2096 V 1268 2098 V 1271 2099 V 1275 2100 V 1278 2101 V 1281 2102 V 1285 2104 V 1288 2105 V 1291 2106 V 1294 2107 V 1298 2108 V 1301 2110 V 1304 2111 V 1308 2112 V 1311 2113 V 1314 2114 V 1318 2115 V 1321 2116 V 1324 2118 V 1328 2119 V 1331 2120 V 1335 2121 V 1338 2122 V 1341 2123 V 1345 2124 V 1348 2125 V 1352 2126 V 1355 2127 V 1358 2128 V 1362 2129 V 1365 2131 V 1369 2132 V 1372 2133 V 1376 2134 V 1379 2135 V 1383 2136 V 1386 2137 V 1390 2138 V 1393 2139 V 1397 2140 V 1400 2141 V 1404 2142 V 1407 2143 V 1411 2144 V 1414 2145 V 1418 2146 V 1421 2146 V 1425 2147 V 1428 2148 V 1432 2149 V 1435 2150 V 1439 2151 V 1443 2152 V 1446 2153 V 1450 2154 V 1453 2155 V 1457 2156 V 1461 2157 V 1464 2157 V 1468 2158 V 1471 2159 V 1475 2160 V 1479 2161 V 1482 2162 V 1486 2163 V 1490 2163 V 1493 2164 V 1497 2165 V 1501 2166 V 1504 2167 V 1508 2167 V 1512 2168 V 1515 2169 V 1519 2170 V 1523 2171 V 1526 2171 V 1530 2172 V 1534 2173 V 1538 2174 V 1541 2174 V 1545 2175 V 1549 2176 V 1553 2177 V 1556 2177 V 1560 2178 V 1564 2179 V 383 2238 V 386 2237 V 389 2236 V 393 2235 V 396 2234 V 399 2233 V 403 2232 V 406 2231 V 409 2230 V 412 2229 V 416 2228 V 419 2227 V 422 2226 V 425 2225 V 429 2224 V 432 2223 V 435 2222 V 438 2221 V 442 2220 V 445 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1933 V 955 1931 V 957 1929 V 959 1927 V 962 1925 V 964 1923 V 966 1921 V 969 1919 V 971 1917 V 973 1914 V 976 1912 V 978 1910 V 980 1908 V 982 1906 V 985 1904 V 987 1902 V 989 1899 V 992 1897 V 994 1895 V 996 1893 V 998 1891 V 1001 1888 V 1003 1886 V 1005 1884 V 1007 1882 V 1010 1880 V 1012 1877 V 1014 1875 V 1016 1873 V 1019 1871 V 1021 1868 V 1023 1866 V 1025 1864 V 1027 1862 V 1030 1859 V 1032 1857 V 1034 1855 V 1036 1852 V 1038 1850 V 1041 1848 V 1043 1846 V 1045 1843 V 1047 1841 V 1049 1839 V 1051 1836 V 1053 1834 V 1056 1832 V 1058 1829 V 1060 1827 V 1062 1824 V 1064 1822 V 1066 1820 V 1068 1817 V 1070 1815 V 1073 1813 V 1075 1810 V 1077 1808 V 1079 1805 V 1081 1803 V 1083 1801 V 1085 1798 V 1087 1796 V 1089 1793 V 1091 1791 V 1093 1788 V 1095 1786 V 1097 1783 V 1100 1781 V 1102 1779 V 1104 1776 V 1106 1774 V 1108 1771 V 1110 1769 V 1112 1766 V 1114 1764 V 1116 1761 V 1118 1759 V 1120 1756 V 1122 1754 V 1124 1751 V 1126 1749 V 1128 1746 V 1130 1743 V 1132 1741 V 1134 1738 V 1135 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figm.ps % MathWorks dictionary /MathWorks 150 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop 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1859 5630 mt 1859 5580 L 1859 611 mt 1859 661 L 1695 5799 mt () s 2586 5630 mt 2586 5580 L 2586 611 mt 2586 661 L 2422 5799 mt () s 3314 5630 mt 3314 5580 L 3314 611 mt 3314 661 L 3274 5799 mt () s 4041 5630 mt 4041 5580 L 4041 611 mt 4041 661 L 3961 5799 mt () s 4768 5630 mt 4768 5580 L 4768 611 mt 4768 661 L 4688 5799 mt () s 5495 5630 mt 5495 5580 L 5495 611 mt 5495 661 L 5415 5799 mt () s 841 5339 mt 891 5339 L 5859 5339 mt 5809 5339 L 562 5392 mt () s 841 4612 mt 891 4612 L 5859 4612 mt 5809 4612 L 562 4665 mt () s 841 3884 mt 891 3884 L 5859 3884 mt 5809 3884 L 562 3937 mt () s 841 3157 mt 891 3157 L 5859 3157 mt 5809 3157 L 726 3210 mt () s 841 2429 mt 891 2429 L 5859 2429 mt 5809 2429 L 646 2482 mt () s 841 1702 mt 891 1702 L 5859 1702 mt 5809 1702 L 646 1755 mt () s 841 975 mt 891 975 L 5859 975 mt 5809 975 L 646 1028 mt () s 841 611 mt 5859 611 L 841 5630 mt 5859 5630 L 841 5630 mt 841 611 L 5859 5630 mt 5859 611 L 841 611 mt 841 611 L 5859 611 mt 5859 611 L gs 841 611 5019 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gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 5048 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 4975 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4903 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4830 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4757 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 4684 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4612 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4539 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4466 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 4393 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4321 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4248 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4175 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 4102 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4030 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3957 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 3884 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3812 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3739 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3666 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 3593 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3521 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3448 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3375 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 3302 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3230 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3157 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3084 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 3011 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2939 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2866 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2793 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 2720 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2648 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2575 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2502 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 2429 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2357 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2284 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 2211 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2139 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2066 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1993 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 1920 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1848 4 MP gs c8 PP gr c0 DP 0 73 73 0 0 -73 1932 1775 4 MP gs c8 PP gr c0 DP 0 73 73 0 0 -73 1932 1702 4 MP gs c8 PP gr c0 DP 0 72 73 0 0 -72 1932 1629 4 MP gs c8 PP gr c0 DP 0 73 73 0 0 -73 1932 1557 4 MP gs c8 PP gr c0 DP 0 73 73 0 0 -73 1932 1484 4 MP gs c9 PP gr c0 DP 0 73 73 0 0 -73 1932 1411 4 MP gs c9 PP gr c0 DP 0 72 73 0 0 -72 1932 1338 4 MP gs c8 PP gr c0 DP 0 73 73 0 0 -73 1932 1266 4 MP gs c10 PP gr c0 DP 0 73 73 0 0 -73 1932 1193 4 MP gs c10 PP gr c0 DP 0 73 73 0 0 -73 1932 1120 4 MP gs c9 PP gr c0 DP 0 72 73 0 0 -72 1932 1047 4 MP gs c11 PP gr c0 DP 0 73 73 0 0 -73 1932 975 4 MP gs c11 PP gr c0 DP 0 73 73 0 0 -73 1932 902 4 MP gs c11 PP gr c0 DP 0 73 73 0 0 -73 1932 829 4 MP gs c11 PP gr c0 DP 0 72 73 0 0 -72 1932 756 4 MP gs c11 PP gr c0 DP 0 73 73 0 0 -73 1932 684 4 MP gs c11 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MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2648 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2575 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2502 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 2429 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2357 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2284 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 2211 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2139 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2066 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1993 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 1920 4 MP gs c8 PP gr c0 DP 0 73 72 0 0 -73 2005 1848 4 MP gs c8 PP gr c0 DP 0 73 72 0 0 -73 2005 1775 4 MP gs c8 PP gr c0 DP 0 73 72 0 0 -73 2005 1702 4 MP gs c8 PP gr c0 DP 0 72 72 0 0 -72 2005 1629 4 MP gs c9 PP gr c0 DP 0 73 72 0 0 -73 2005 1557 4 MP gs c9 PP gr c0 DP 0 73 72 0 0 -73 2005 1484 4 MP gs c9 PP gr c0 DP 0 73 72 0 0 -73 2005 1411 4 MP gs c9 PP gr c0 DP 0 72 72 0 0 -72 2005 1338 4 MP gs c10 PP gr c0 DP 0 73 72 0 0 -73 2005 1266 4 MP gs c10 PP gr c0 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gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 3230 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 3157 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 3084 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 2077 3011 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2939 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2866 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2793 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 2077 2720 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2648 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2575 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2502 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 2077 2429 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2357 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2284 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 2077 2211 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2139 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2066 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 1993 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 2077 1920 4 MP gs c8 PP gr c0 DP 0 73 73 0 0 -73 2077 1848 4 MP gs c8 PP gr c0 DP 0 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{pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath fill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { 2 copy moveto lineto stroke } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0336 6804 csm 560 563 5348 5269 MR c np 18 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef /Helvetica /ISOLatin1Encoding 144 FMSR 1 j c1 0 0 6476 6468 PR 6 w DO 4 w SO 6 w c0 841 5630 mt 5859 5630 L 841 611 mt 5859 611 L 5859 5630 mt 5859 611 L 841 5630 mt 841 611 L 5859 5630 mt 5859 5630 L 841 5630 mt 841 5630 L 841 5630 mt 5859 5630 L 841 5630 mt 841 611 L 841 5630 mt 841 5630 L 1132 5630 mt 1132 5580 L 1132 611 mt 1132 661 L 968 5799 mt () s 1859 5630 mt 1859 5580 L 1859 611 mt 1859 661 L 1695 5799 mt () s 2586 5630 mt 2586 5580 L 2586 611 mt 2586 661 L 2422 5799 mt () s 3314 5630 mt 3314 5580 L 3314 611 mt 3314 661 L 3274 5799 mt () s 4041 5630 mt 4041 5580 L 4041 611 mt 4041 661 L 3961 5799 mt () s 4768 5630 mt 4768 5580 L 4768 611 mt 4768 661 L 4688 5799 mt () s 5495 5630 mt 5495 5580 L 5495 611 mt 5495 661 L 5415 5799 mt () s 841 5339 mt 891 5339 L 5859 5339 mt 5809 5339 L 562 5392 mt () s 841 4612 mt 891 4612 L 5859 4612 mt 5809 4612 L 562 4665 mt () s 841 3884 mt 891 3884 L 5859 3884 mt 5809 3884 L 562 3937 mt () s 841 3157 mt 891 3157 L 5859 3157 mt 5809 3157 L 726 3210 mt () s 841 2429 mt 891 2429 L 5859 2429 mt 5809 2429 L 646 2482 mt () s 841 1702 mt 891 1702 L 5859 1702 mt 5809 1702 L 646 1755 mt () s 841 975 mt 891 975 L 5859 975 mt 5809 975 L 646 1028 mt () s 841 611 mt 5859 611 L 841 5630 mt 5859 5630 L 841 5630 mt 841 611 L 5859 5630 mt 5859 611 L 841 611 mt 841 611 L 5859 611 mt 5859 611 L gs 841 611 5019 5020 MR c np 0 73 73 0 0 -73 841 5630 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 5557 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 5485 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 5412 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 5339 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 5266 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 5194 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 5121 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 5048 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 4975 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4903 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4830 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4757 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 4684 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4612 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4539 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4466 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 4393 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4321 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4248 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4175 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 4102 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4030 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3957 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 3884 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3812 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3739 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3666 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 3593 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3521 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3448 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3375 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 3302 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3230 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3157 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3084 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 3011 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2939 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2866 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2793 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 2720 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2648 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2575 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2502 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 2429 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2357 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2284 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 2211 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2139 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2066 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1993 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 1920 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1848 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1775 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1702 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 1629 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1557 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1484 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1411 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 1338 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1266 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1193 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1120 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 1047 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 975 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 902 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 829 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 756 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 684 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 5630 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 914 5557 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 5485 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 5412 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 5339 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 914 5266 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 5194 4 MP 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c1 PP gr c0 DP 0 73 73 0 0 -73 1132 5121 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1132 5048 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1132 4975 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1132 4903 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1132 4830 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1132 4757 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1132 4684 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1132 4612 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1132 4539 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1132 4466 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1132 4393 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1132 4321 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1132 4248 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1132 4175 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1132 4102 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1132 4030 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1132 3957 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1132 3884 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1132 3812 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1132 3739 4 MP gs c1 PP gr c0 DP 0 73 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4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 1775 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 1702 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1859 1629 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 1557 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 1484 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 1411 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1859 1338 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 1266 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 1193 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 1120 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1859 1047 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 975 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 902 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 829 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1859 756 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 684 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 5630 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 5557 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 5485 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 5412 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 5339 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 5266 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 5194 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 5121 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 5048 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 4975 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4903 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4830 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4757 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 4684 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4612 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4539 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4466 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 4393 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4321 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4248 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4175 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 4102 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4030 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3957 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 3884 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3812 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3739 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3666 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 3593 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3521 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3448 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3375 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 3302 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3230 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3157 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3084 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 3011 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2939 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2866 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2793 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 2720 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2648 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2575 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2502 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 2429 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2357 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2284 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 2211 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2139 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2066 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1993 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 1920 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1848 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1775 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1702 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 1629 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1557 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1484 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1411 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 1338 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1266 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1193 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1120 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 1047 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 975 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 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0 0 -73 2005 2939 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2866 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2793 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 2720 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2648 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2575 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2502 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 2429 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2357 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2284 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 2211 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2139 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2066 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1993 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 1920 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1848 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1775 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1702 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 1629 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1557 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1484 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1411 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 1338 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1266 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1193 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1120 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 1047 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 975 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 902 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 829 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 756 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 684 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 5630 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 2077 5557 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 5485 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 5412 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 5339 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 2077 5266 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 5194 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 5121 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 5048 4 MP gs c1 PP gr c0 DP 0 72 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4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 3448 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 3375 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 2077 3302 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 3230 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 3157 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 3084 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 2077 3011 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2939 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2866 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2793 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 2077 2720 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2648 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2575 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2502 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 2077 2429 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2357 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2284 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 2077 2211 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2139 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 2066 4 MP gs c1 PP gr c0 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/ellipse{/endangle X /startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 39158280 55380996 1000 300 300 (fig.dvi) @start /Fa 2 51 df<13801203120F12F31203B3A9EA07C0EAFFFE0F217CA018>49 DI E /Fb 1 24 df23 D E /Fc 19 115 df<127812FCA212FEA2127A1202A51204A31208A212101220A2124007 147A8512>44 DI<127812FCA4127806067A8512>I48 D51 D54 D<91383FE001903901FFF803903807F01E90391F800307013EC71287017814 4F49142F4848141F4848140F485A000F150790C8FC481503121E123E003C1501127CA300 78150012F8AB1278127C1601A2123C123E121E001F15027E6D1406000715046C6C14086C 7E6C6C141001781420013E14C090391F800380903907F00F00903801FFFC9038003FE028 337CB130>67 D69 DII73 D80 D82 D<13FE380303C0380C00E00010137080003C133C003E131C141EA21208C7FCA3EB0FFEEB FC1EEA03E0EA0F80EA1F00123E123C127C481404A3143EA21278007C135E6CEB8F08390F 0307F03903FC03E01E1F7D9E21>97 D<15F0141FA214011400AFEB0FC0EB70303801C00C 3803800238070001120E001E13005AA2127C1278A212F8A71278A2127C123CA27E000E13 016C1302380380046C6C487E3A00F030FF80EB1FC021327EB125>100 DI104 D<380780FE39FF83078090388C03C0390F9001E0EA07A06E7E13C0A25BB3A2486C487E3A FFFC1FFF80A2211F7E9E25>110 D<380783E038FF8418EB887CEA0F90EA07A01438EBC0 00A35BB3487EEAFFFEA2161F7E9E19>114 D E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 324 516 a Fc(FIG.)22 b(6) 280 977 y 22376157 22376157 4867850 5328322 34206515 34206515 startTexFig 280 977 a %%BeginDocument: figf.ps % MathWorks dictionary /MathWorks 150 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath fill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { 2 copy moveto lineto stroke } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0336 6804 csm 560 563 5348 5269 MR c np 18 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef /Helvetica /ISOLatin1Encoding 144 FMSR 1 j c1 0 0 6476 6468 PR 6 w DO 4 w SO 6 w c0 841 5630 mt 5859 5630 L 841 611 mt 5859 611 L 5859 5630 mt 5859 611 L 841 5630 mt 841 611 L 5859 5630 mt 5859 5630 L 841 5630 mt 841 5630 L 841 5630 mt 5859 5630 L 841 5630 mt 841 611 L 841 5630 mt 841 5630 L 1132 5630 mt 1132 5580 L 1132 611 mt 1132 661 L 968 5799 mt () s 1859 5630 mt 1859 5580 L 1859 611 mt 1859 661 L 1695 5799 mt () s 2586 5630 mt 2586 5580 L 2586 611 mt 2586 661 L 2422 5799 mt () s 3314 5630 mt 3314 5580 L 3314 611 mt 3314 661 L 3274 5799 mt () s 4041 5630 mt 4041 5580 L 4041 611 mt 4041 661 L 3961 5799 mt () s 4768 5630 mt 4768 5580 L 4768 611 mt 4768 661 L 4688 5799 mt () s 5495 5630 mt 5495 5580 L 5495 611 mt 5495 661 L 5415 5799 mt () s 841 5339 mt 891 5339 L 5859 5339 mt 5809 5339 L 562 5392 mt () s 841 4612 mt 891 4612 L 5859 4612 mt 5809 4612 L 562 4665 mt () s 841 3884 mt 891 3884 L 5859 3884 mt 5809 3884 L 562 3937 mt () s 841 3157 mt 891 3157 L 5859 3157 mt 5809 3157 L 726 3210 mt () s 841 2429 mt 891 2429 L 5859 2429 mt 5809 2429 L 646 2482 mt () s 841 1702 mt 891 1702 L 5859 1702 mt 5809 1702 L 646 1755 mt () s 841 975 mt 891 975 L 5859 975 mt 5809 975 L 646 1028 mt () s 841 611 mt 5859 611 L 841 5630 mt 5859 5630 L 841 5630 mt 841 611 L 5859 5630 mt 5859 611 L 841 611 mt 841 611 L 5859 611 mt 5859 611 L gs 841 611 5019 5020 MR c np 0 73 73 0 0 -73 841 5630 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 5557 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 5485 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 5412 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 5339 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 5266 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 5194 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 5121 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 5048 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 4975 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4903 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4830 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4757 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 4684 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4612 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4539 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4466 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 4393 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4321 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4248 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4175 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 4102 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 4030 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3957 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 3884 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3812 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3739 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3666 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 3593 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3521 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3448 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3375 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 3302 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3230 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3157 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 3084 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 3011 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2939 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2866 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2793 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 2720 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2648 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2575 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2502 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 2429 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2357 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2284 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 2211 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2139 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 2066 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1993 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 1920 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1848 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1775 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1702 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 1629 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1557 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1484 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1411 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 1338 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1266 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1193 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 1120 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 1047 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 975 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 902 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 829 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 841 756 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 841 684 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 5630 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 914 5557 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 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0 72 72 0 0 -72 914 2429 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 2357 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 2284 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 914 2211 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 2139 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 2066 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 1993 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 914 1920 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 1848 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 1775 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 1702 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 914 1629 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 1557 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 1484 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 1411 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 914 1338 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 1266 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 1193 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 1120 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 914 1047 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 914 975 4 MP gs c1 PP gr 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gr c0 DP 0 72 73 0 0 -72 986 4393 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 986 4321 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 986 4248 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 986 4175 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 986 4102 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 986 4030 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 986 3957 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 986 3884 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 986 3812 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 986 3739 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 986 3666 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 986 3593 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 986 3521 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 986 3448 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 986 3375 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 986 3302 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 986 3230 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 986 3157 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 986 3084 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 986 3011 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 986 2939 4 MP 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MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 1714 975 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 1714 902 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 1714 829 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 1714 756 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 1714 684 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1786 5630 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1786 5557 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1786 5485 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1786 5412 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1786 5339 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1786 5266 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1786 5194 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1786 5121 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1786 5048 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1786 4975 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1786 4903 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1786 4830 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1786 4757 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1786 4684 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1786 4612 4 MP gs c1 PP gr c0 DP 0 73 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gr c0 DP 0 72 73 0 0 -72 1859 3593 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 3521 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 3448 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 3375 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1859 3302 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 3230 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 3157 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 3084 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1859 3011 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 2939 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 2866 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 2793 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1859 2720 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 2648 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 2575 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 2502 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1859 2429 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 2357 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1859 2284 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1859 2211 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 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PP gr c0 DP 0 73 73 0 0 -73 1932 5630 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 5557 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 5485 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 5412 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 5339 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 5266 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 5194 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 5121 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 5048 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 4975 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4903 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4830 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4757 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 4684 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4612 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4539 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4466 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 4393 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4321 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4248 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4175 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 4102 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 4030 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3957 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 3884 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3812 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3739 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3666 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 3593 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3521 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3448 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3375 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 3302 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3230 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3157 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 3084 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 3011 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2939 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2866 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2793 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 2720 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2648 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2575 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2502 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 2429 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2357 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2284 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 2211 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2139 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 2066 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1993 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 1920 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1848 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1775 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1702 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 1629 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1557 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1484 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1411 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 1338 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1266 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1193 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 1120 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 1047 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 975 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 902 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 829 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 1932 756 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 1932 684 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 5630 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 5557 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 5485 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 5412 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 5339 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 5266 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 5194 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 5121 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 5048 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 4975 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 4903 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 4830 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 4757 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 4684 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 4612 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 4539 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 4466 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 4393 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 4321 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 4248 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 4175 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 4102 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 4030 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 3957 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 3884 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 3812 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 3739 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 3666 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 3593 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 3521 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 3448 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 3375 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 3302 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 3230 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 3157 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 3084 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 3011 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2939 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2866 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2793 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 2720 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2648 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2575 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2502 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 2429 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2357 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2284 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 2211 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2139 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 2066 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1993 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 1920 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1848 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1775 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1702 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 1629 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1557 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1484 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1411 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 1338 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1266 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1193 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 1120 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 1047 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 975 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 902 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 829 4 MP gs c1 PP gr c0 DP 0 72 72 0 0 -72 2005 756 4 MP gs c1 PP gr c0 DP 0 73 72 0 0 -73 2005 684 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 5630 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 2077 5557 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 5485 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 5412 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 2077 5339 4 MP gs c1 PP 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4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 5786 1193 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 5786 1120 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 5786 1047 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 5786 975 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 5786 902 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 5786 829 4 MP gs c1 PP gr c0 DP 0 72 73 0 0 -72 5786 756 4 MP gs c1 PP gr c0 DP 0 73 73 0 0 -73 5786 684 4 MP gs c1 PP gr c0 DP gr end eplot epage end showpage %%EndDocument endTexFig 386 2394 a Fc(-30)526 b(0)c(30)256 2276 y(-30)315 1698 y(0)280 1107 y(30)1000 2465 y Fb(\027)1035 2476 y Fa(1)244 1698 y gsave currentpoint currentpoint translate 270 rotate neg exch neg exch translate 244 1698 a 30 x Fb(\027)279 1739 y Fa(2)286 1698 y currentpoint grestore moveto 286 1698 a eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF EndOfTheIncludedPostscriptMagicCookie \closepsdump % Finally, here is the rest of cgjk.tex: % \documentstyle[pre,aps,preprint,psfig,amsfonts,eqsecnum]{revtex} \begin{document} \def\u{\bbox} %\def\u{\underline} \def\d{\displaystyle} \def\mathcal#1{{\cal #1}} \def\goldenmean{\gamma} \def\phi{\varphi} \def\epsilon{\varepsilon} \def\goldenmean{\gamma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \draft \title{Universality for the breakup of invariant tori in Hamiltonian flows} \author{$^1$C. Chandre, $^1$M. Govin, $^1$H. R. Jauslin, and $^2$H. Koch} \address{$^1$Laboratoire de Physique, CNRS, Universit\'e de Bourgogne, BP 400, F-21011 Dijon, France} \address{$^2$Department of Mathematics, The University of Texas at Austin, Austin, TX 78712} \date{\today} \maketitle \begin{abstract} In this article, we describe a new renormalization-group scheme for analyzing the breakup of invariant tori for Hamiltonian systems with two degrees of freedom. The transformation, which acts on Hamiltonians that are quadratic in the action variables, combines a rescaling of phase space and a partial elimination of irrelevant (non-resonant) frequencies. It is implemented numerically for the case applying to golden invariant tori. We find a nontrivial fixed point and compute the corresponding scaling and critical indices. If one compares flows to maps in the canonical way, our results are consistent with existing data on the breakup of golden invariant circles for area-preserving maps. \end{abstract} \pacs{PACS numbers: 05.45.+b, 64.60.Ak} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} In Hamiltonian systems with two (or more) degrees of freedom, smooth invariant tori typically persist under small perturbations \cite{kolmogorov,arnold,moser,delallave}. The most stable tori appear to be the ones for which the frequency ratio is a quadratic irrational, such as the golden mean. As the strength of the perturbation passes some critical value, these tori are observed to exhibit self-similar scaling behavior, and then they break up~\cite{kadanoff,shenkerkadanoff}. To study such critical behavior, several renormalization schemes have been proposed \cite{escandedoveil,escande,mackay,mackayL,koch,govin,chandre}. \\ The idea is to use a transformation $\mathcal{R}$ that maps a Hamiltonian $H$ into a rescaled Hamiltonian $\mathcal{R}(H)$, in such a way that irrelevant degrees of freedom are contracted. The transformation $\mathcal{R}$ should have roughly the following properties: $\mathcal{R}$ has an attractive integrable fixed point $H_0$ that has a smooth invariant torus of a given frequency. Every Hamiltonian in its domain of attraction $\mathcal{D}$ has a smooth invariant torus of the desired frequency. Another non-integrable fixed point $H_*$ lies on the boundary of $\mathcal{D}$, also called critical surface, and this nontrivial fixed point attracts the Hamiltonians on the critical surface. Figure \ref{figure:flow} shows schematically the expected nature of the renormalization flow in the space of Hamiltonians. The existence of a nontrivial fixed point has strong implications concerning universal properties associated with the breakup of invariant tori. The analysis of the renormalization for area-preserving maps~\cite{mackay} gives support to the validity of this general picture.\\ The renormalization we define, following the scheme proposed in Ref.~\cite{koch}, is similar in spirit to the block spin transformation in statistical mechanics, in the sense that it uses a process of ``elimination and rescaling''. The elimination of irrelevant frequencies is done by using canonical transformations as in Kolmogorov-Arnold-Moser (KAM) theory. The frequencies we want to eliminate are the non-slow modes (non-resonant part of the perturbation), i.\ e.\ the modes which only affect the motion for a short time. The slow modes (resonant part of the perturbation), which produce the small denominators in KAM theory, are shifted towards the non-slow modes by a rescaling. They can thus be eliminated by iteration. The rescaling is the same as in Refs.\ \cite{koch,govin,chandre}. It includes a shift of the resonances, and a rescaling of the actions and of the energy. Here and in what follows, the word ``resonances'' refers to the frequency vectors ${\u \nu}_k=(p_k,q_k)$ defined by the continued fraction approximants $p_k/q_k$ for the frequency ratio of the invariant torus. The corresponding closed orbits accumulate at the invariant torus~\cite{greene}, and are often used in numerical investigations.\\ We consider the following class of Hamiltonians with two degrees of freedom, quadratic in the action variables $\u{A}=(A_1,A_2)$ and described by three scalar functions of the angles ${\u \varphi}=(\varphi_1,\varphi_2)\in {\Bbb T}^2$ (the two-dimensional torus parametrized by $[0,2\pi)^2$): \begin{equation} \label{hamiltonian} H({\u A},{\u \varphi})= \frac{1}{2}m({\u \varphi})( {\u \Omega}\cdot{\u A})^{2} +\lbrack {\u \omega}_{0} + g({\u \varphi}){\u \Omega} \rbrack \cdot{\u A} + f({\u \varphi}) \ , \end{equation} where ${\u \omega_0}$ is the frequency vector of the considered torus, and ${\u \Omega}=(1,\alpha)$ is some other constant vector, not parallel to $\u{\omega}_0$.\\ This family of Hamiltonians has been investigated in Refs.\ \cite{govin,chandre,chandrejauslin}. In particular, a KAM theorem was proven for this family in Ref.\ \cite{chandrejauslin} based on Thirring's approach~\cite{thirring}, in which the KAM transformations are constructed such that the iteration stays within the space of Hamiltonians quadratic in the actions. In order to prove the existence of a torus with frequency vector ${\u \omega}_0$ for Hamiltonian systems described by Eq.\ (\ref{hamiltonian}), it is not necessary to eliminate $m$, but only $g$ and $f$: One has to find a canonical transformation such that the equations of motion expressed in the new coordinates show trivially the existence of this torus. For Hamiltonians (\ref{hamiltonian}), if one takes $g$ and $f$ equal to zero, then the resulting equations of motion are \begin{equation} \frac{d{\u A}}{dt}=-\d\frac{1}{2}\frac{\partial m}{\partial {\u \phi}} ({\u \Omega} \cdot {\u A})^2, \, \qquad \frac{d{\u \phi}}{dt}= m({\u \phi})({\u \Omega}\cdot {\u A}){\u \Omega}+{\u \omega}_0. \end{equation} Then $\u{A}=0$ defines an invariant torus, and the motion on this torus is quasiperiodic with frequency vector ${\u \omega}_0$ (even if the resulting Hamiltonian is not integrable).\\ The elimination of $f$ and $g$ can be achieved with canonical transformations with generating functions that are linear in the action variables, and thus map the family of Hamiltonians (\ref{hamiltonian}) into itself. This is very convenient numerically, as one only works with three scalar functions $m$, $g$ and $f$. The only approximation involved in our numerical implementation of the transformation is a truncation of the Fourier series of these functions, e.\ g.\ we approximate \begin{equation} f(\u{\phi})=\sum_{\nu\in {\Bbb Z}^2} f_{\nu} e^{i\nu\cdot\phi}, \end{equation} by \begin{equation} f^{[\leq L]}(\u{\phi})=\sum_{\nu\in\mathcal{C}_L} f_{\nu} e^{i\nu\cdot\phi}, \end{equation} where $\mathcal{C}_L=\{\u{\nu}\in {\Bbb Z}^2\left| |\nu_1|\leq L, \, |\nu_2|\leq L\right.\}$.\\ In this paper, we focus on the frequency vector ${\u \omega}_0=(1/\gamma,-1)$, associated with the golden mean $\gamma=(1+\sqrt{5})/2$. We choose the set of frequencies $I^-$ describing the non-slow modes as the union of the two quadrants in the plane $(\nu_1, \nu_2)$ that contain the linear span of ${\u \omega}_0$: \begin{equation} I^-=\{ \u{\nu}\in {\Bbb Z}^2 \vert \nu_1\nu_2\leq 0\}. \end{equation} The set $I^-$ is depicted in Fig.\ \ref{figure:I-}.\\ This ``frequency cutoff'' restricts to Fourier modes (the non-slow modes) that can be eliminated in one renormalization step, without running into small denominator problems. As is common with cutoffs, there is not a single ``natural'' choice. Other possible choices of $I^-$ will be mentioned later. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{KAM-Renormalization-Group transformation} \label{sect:kamrg} The renormalization scheme described in this section is for a torus of frequency vector $\u{\omega}_0=(1/\goldenmean,-1)$ where $\goldenmean=(1+\sqrt{5})/2$. It is straightforward to adapt it to other reduced quadratic irrationals~\cite{lang}.\\ Our transformation is composed of four steps:\\ (1) A canonical change of coordinates, which acts on the Fourier mode $e^{i\nu\cdot\varphi}$ for a resonance ${\u \nu}={\u \nu}_k$ by a shift ${\u \nu}_k \mapsto {\u \nu}_{k-2}$. The resonances for $k<2$ will be eliminated in step 4, together with all the other frequency vectors in $I^-$.\\ (2) A generalized canonical change of coordinates which corresponds to a rescaling of the action variables ${\u A}$.\\ (3) A normalization corresponding to a rescaling of the energy (or time).\\ The composition of the three steps described so far defines a map $(m,g,f,\alpha)\mapsto (m',g',f',\alpha')$.\\ (4) A KAM transformation (canonical change of variables) that eliminates all non-slow modes from $g'$ and $f'$, i.\ e.\ the new functions $f''$ and $g''$ are in the range of ${\Bbb I}^-$, where ${\Bbb I}^-$ is a projection operator acting on a scalar function $f$ of the angles as \begin{equation} {\Bbb I}^-f({\u \varphi})=\sum_{\nu\in I^-}f_\nu e^{i\nu\cdot\varphi}. \end{equation} Concerning step 4 we note that, for certain purposes it can be desirable to eliminate the non-slow modes from $m'$ as well. But such a step generates terms of arbitrary order in the actions, which drastically complicates the analysis~\cite{kochwittwer}.\\ We now give a more detailed description of the four steps that define the transformation. The first step is motivated by the observation that periodic orbits for the resonant frequencies $\{ {\u\nu}_k\}$ accumulate geometrically at the (critical) ${\u \omega}_0$--torus. This suggests that the appropriate scaling (in the angle variables) is related to a shift in the sequence of resonances. In order to implement the shift ${\u \nu}_k\mapsto {\u \nu}_{k-2}$ mentioned in step 1, we use the fact that the continued fraction approximants for the golden mean $\gamma$ are $\gamma_k=F_{k+1}/F_k$, where $F_k$ is the $k$-th Fibonacci number. These numbers can be defined recursively by the equation \begin{equation} \d\left( \begin{array}{c} F_{k+1} \\ F_k \end{array} \right)= N\d\left( \begin{array}{c} F_k \\ F_{k-1} \end{array} \right), \qquad \mbox{ with } N=\d\left( \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right), \end{equation} starting with $F_0=0$ and $F_1=1$. In other words, if we define ${\u\nu}_0=(0,1)$ then ${\u\nu}_k=N{\u\nu}_{k-1}$ for $k\geq1$. Thus the desired shift ${\u \nu}_k\mapsto{\u \nu}_{k-2}$ is induced by the linear transformation $N^{-2}$, and the canonical transformation we are looking for is $(\u{A},\u{\varphi})\mapsto (N^{-2}\u{A},N^2\u{\varphi})$.\\ As the renormalization changes the scale, some of the slow modes become non-slow modes [e.\ g., $(1,1)$ is mapped into $(0,1)$ which is an element of $I^-$]. We remark that there is no intrinsic sharp difference between the slow and non-slow modes. The boundary can be set at different places. We have chosen to include the coefficients $(0,1)$ and $(1,0)$ among the non-slow modes ($I^-$, to be eliminated by the KAM transformation), and $(1,1)$ and $(2,1)$ among the slow modes. But conceptually and practically there would be no difficulty to include e.\ g.\ $(1,1)$ and $(2,1)$ (or any fixed finite number of resonances) among the non-slow modes. More generally, other choices in the splitting of $\{ e^{i\nu\cdot\varphi}\}$ into slow and non-slow modes should lead to the same results provided e.\ g.\ that the ratio $|{\u \nu}|/|{\u \omega}_0\cdot{\u \nu}|$ is bounded on $I^-$, and that $N^{-k}$ contracts vectors ${\u \nu}$ in the complement of $I^-$ for some fixed $k>0$~\cite{koch}.\\ The linear shift of the resonances multiplies ${\u \omega}_0$ by $\goldenmean^{-2}$ ($\u{\omega}_0$ is an eigenvector of $N$); therefore we rescale the energy by a factor $\goldenmean^2$ in order to keep the frequency fixed at ${\u \omega}_0$. A consequence of the shift of the resonances is that $\u{\Omega}=(1,\alpha)$ is changed into $\u{\Omega}'=(1,\alpha')$, where $\alpha'=(\alpha+1)/(\alpha+2)$.\\ Then we perform a rescaling of the action variables: we change the Hamiltonian $H$ into $$ \hat{H}({\u A},{\u \varphi})= \lambda H\left(\d\frac{{\u A}}{\lambda},{\u \varphi}\right), $$ with $\lambda=\lambda(H)$ such that the mean value of $m'$ defined as \begin{equation} \langle m' \rangle=\int_{{\Bbb T}^2} \frac{d^2{\u\varphi}}{(2\pi)^2} m'({\u\varphi}), \end{equation} is equal to $1$. Since the rescaling of energy and the shift $N^2$ transform the quadratic term of the Hamiltonian into $\goldenmean^2(2+\alpha)^2m(\u{\phi})(\u{\Omega}'\cdot \u{A})^2/2$, this condition leads to $\lambda=\goldenmean^2(2+\alpha)^2\langle m \rangle$.\\ In summary, the first three steps of our renormalization transformation rescale $m$, $g$, $f$ and $\u{\Omega}=(1,\alpha)$ into \begin{eqnarray} && m'(\u{\phi})=\d \frac{m\left(N^{-2}\u{\phi}\right)} {\langle m \rangle},\\ && g'(\u{\phi})=\goldenmean^2(2+\alpha) g\left(N^{-2}\u{\phi} \right),\\ && f'(\u{\phi})=\goldenmean^4(2+\alpha)^2 \langle m \rangle f\left(N^{-2}\u{\phi}\right),\\ && \alpha'=\frac{1+\alpha}{2+\alpha}.\label{eqn:alpha} \end{eqnarray} We remark that the map ${\u \Omega}\mapsto{\u \Omega}'$ given by Eq.\ (\ref{eqn:alpha}) has an attractive fixed point ${\u \Omega}_*=(1,\gamma^{-1})$, which is an eigenvector of $N^2$, with eigenvalue $\gamma^2>1$.\\ The fourth step is carried out via an iterative procedure, similar to a Newton algorithm. We start with $H_0=H'$. To simplify the description of the iteration step $H_n\mapsto H_{n+1}$, consider first the case where $g_n$ and $f_n$ depend on a (small) parameter $\varepsilon_n$, in such a way that ${\Bbb I}^-g_n$ and ${\Bbb I}^-f_n$ are of order $\mathcal{O}(\varepsilon_n)$. The idea is to eliminate the non-slow modes of $g_n$ and $f_n$ to first order in $\varepsilon_n$, at the expense of adding terms that are of order $\mathcal{O}(\varepsilon_n)$ in the slow modes and of order $\mathcal{O}(\varepsilon_n^2)$ in the non-slow modes.\\ We will work with Lie transformations $U_n: (\u{\varphi}_{n-1},\u{A}_{n-1})\mapsto(\u{\varphi}_{n}, \u{A}_{n})$ generated by a function $S_n$ linear in the action variables, of the form \begin{equation} \label{eqn:S} S_n(\u{A},\u{\varphi})=Y_n(\u{\varphi})\u{\Omega}'\cdot\u{A} +Z_n(\u{\varphi})+a_n\u{\Omega}'\cdot\u{\varphi}\ , \end{equation} characterized by two scalar functions $Y_n$, $Z_n$, and a constant $a_n$. The expression of the Hamiltonian in the new variables is obtained by the following equation~\cite{deprit,benettin}: \begin{eqnarray} H_{n+1}&=&H_n\circ U_n= e^{+\hat{S_n}} H_n \nonumber\\ &\equiv& H_n+\lbrace S_n,H_n \rbrace +\frac{1}{2!}\lbrace S_n,\lbrace S_n,H_n\rbrace\rbrace+\cdots . \label{eqn:exponential} \end{eqnarray} A consequence of the linearity of $S_n$ in $\u{A}$ is that the Hamiltonian $H_{n+1}$ is again quadratic in the actions, and of the form \begin{equation} \label{image} H_{n+1}({\u A},{\u \varphi})= \frac{1}{2}m_{n+1}({\u \varphi})( {\u \Omega}'\cdot{\u A})^{2} +\lbrack {\u \omega}_0 + g_{n+1}({\u \varphi}){\u \Omega}' \rbrack \cdot{\u A} + f_{n+1}({\u \varphi}). \end{equation} We notice that the vector $\u{\Omega}'$ remains unchanged from one step to the next. The functions $Y_n$, $Z_n$, and the constant $a_n$ are chosen in such a way that ${\Bbb I}^-g_{n+1}$ and ${\Bbb I}^-f_{n+1}$ vanish to order $\mathcal{O}(\varepsilon_n)$; see Appendix \ref{appendix} for details. Consequently, ${\Bbb I}^-g_{n+1}$ and ${\Bbb I}^-f_{n+1}$ are of order $\mathcal{O}(\varepsilon_0^{2^n})$, where $\varepsilon_0$ denotes the order of ${\Bbb I}^-g'$ and ${\Bbb I}^-f'$. If $\varepsilon_0$ is small, an iteration of this procedure should define a canonical transformation $U_H=U_1\circ U_2\circ \cdots \circ U_n \circ \cdots$ such that the Hamiltonian expressed in these new variables has only slow modes in $g$ and $f$. In our numerical implementation this is indeed the case, even without small parameters.\\ In summary, our renormalization-group (RG) transformation acts as follows: First, some of the slow modes (resonant part of the perturbation) are turned into non-slow modes by a frequency shift and a rescaling. Then, a KAM-type iteration eliminates these non-slow (non-resonant) modes, while producing some new slow modes. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Determination of the critical coupling; fixed point of the KAM-RG transformation} \label{sect:result} We start with the same initial Hamiltonian as in Refs.\ \cite{escande,govin,chandre} \begin{equation} \label{hamiltonieninit} H({\u A},{\u \varphi})=\frac{1}{2}({\u \Omega}\cdot{\u A})^{2} +{\u \omega}_{0}\cdot{\u A}+\varepsilon f({\u \varphi}) \ , \end{equation} where ${\u \Omega}=(1,0)$, ${\u \omega}_0 =(1/\goldenmean,-1)$, $\goldenmean=(1+\sqrt{5})/2$, and a perturbation \begin{equation} f({\u \varphi})=\cos({\u \nu}_1\cdot{\u \varphi}) + \cos({\u \nu}_2\cdot{\u \varphi}), \end{equation} where ${\u \nu}_1=(1,0)$ and ${\u \nu}_2=(1,1)$. We represent all the functions by their Fourier series truncated by retaining only the coefficients in the square $\mathcal{C}_L$ which contains $(2L+1)^2$ Fourier coefficients. For fixed $L$, we take successively larger couplings $\varepsilon$ and determine whether the KAM-RG iteration converges to a Hamiltonian with $f=0,\, g=0$ (trivial fixed point), or whether it diverges ($f,g\rightarrow\infty$). By a bisection procedure, we determine the critical coupling $\varepsilon_c(L)$. We repeat the calculation with larger numbers of Fourier coefficients, to obtain a more accurate approximation. Table I lists some values of $\varepsilon_c(L)$. For instance, for $L=10$, the result is given with 7 digits; as a comparison, the method developed in Refs.\ \cite{govin,chandre} yields 4 significant digits (in order to obtain 7 significant digits with this method one needs to calculate $\varepsilon_c(L)$ up to $L=34$). Moreover, we observe the disappearance of the oscillations of Fig.\ 1 of Ref.\ \cite{govin}, that is to say, the present method converges much faster than the one of Refs.\ \cite{govin,chandre}.\\ By iterating the transformation starting from a point on the critical surface, we observe that the process converges to a nontrivial fixed point $H_*$ (or more generally to a nontrivial fixed set related to this nontrivial fixed point by symmetries~\cite{chandre,mackay2}), which we characterize by the Fourier coefficients of the three functions $m_*,g_*,f_*$ and ${\u \Omega}_*=(1,\goldenmean^{-1})$.\\ Figures \ref{figure:m} and \ref{figure:m2} show the weight of the Fourier coefficients of $m_*$. We observe that these coefficients decrease only slowly in the direction of ${\u \omega}_0$, and in particular, along the rescaled resonances $N^{-k}{\u \nu}_1$, $k\geq 0$, which indicates that $m_*$ is not analytic. This is due to the fact that our transformation does not eliminate the non-slow modes of $m$ (for the reasons mentioned earlier).\\ Figures \ref{figure:g} and \ref{figure:f} show the weight of the Fourier coefficients for the functions $g_*$ and $f_*$, respectively. These coefficients seem to decrease exponentially, which indicates that $g_*$ and $f_*$ are real analytic. Notice also that, by construction, $g_*$ and $f_*$ have only slow modes.\\ At the fixed point $H_*$, we compute the relevant eigenvalues (critical exponents) for the linearized KAM-RG transformation. Since the breakup of invariant tori is observed in one-parameter families of Hamiltonians, indicating that the critical surface is of codimension one, one expects to find a simple eigenvalue $\delta>1$, and no other spectrum outside the open unit disk. This is precisely what we find numerically. Table I lists the values of $\delta$ as a function of $L$. We obtain $\delta\in [2.650175, 2.650234]$, which is in agreement with the value obtained by MacKay for area-preserving maps\cite{mackay} ($\delta=2.650221$).\\ If the renormalization-group picture is correct, then all the relevant informations about critical tori with frequency vector ${\u \omega}_0=(1/\gamma,-1)$ is contained in the nontrivial fixed point $H_*$. In particular, $H_*$ determines the observed critical scaling of phase space~\cite{kadanoff,shenkerkadanoff,escande}. For the scaling factor $\lambda_*=\lambda(H_*)$, we obtain numerically $\lambda_*\in [18.827910,18.828203]$; the value obtained for area-preserving maps is $\lambda_*=18.828171$ (given in Refs.\ \cite{kadanoff,shenkerkadanoff,mackay}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Conclusion} We have shown that in the renormalization-group approach to critical invariant tori, a partial elimination of non-resonant frequencies leads to substantial qualitative and quantitative improvements. Compared with previous schemes that have been studied numerically, our new KAM-RG transformation yields more accurate results and a better defined domain of attraction for the nontrivial fixed point. Conceptually it is close to the type of transformations used for the study of critical phenomena in statistical mechanics.\\ We have implemented our transformation for the case of tori with frequency vector $\u{\omega}_0=(1/\goldenmean,-1)$, where $\gamma$ is the golden mean. The extension to other reduced quadratic irrationals should be straightforward and yields similar results. It also seems possible to extend our transformation to systems with three~\cite{mackaymeissstark} and more degrees of freedom~\cite{koch}, where very little is known about the behavior of invariant tori under perturbations that are not necessarily small~\cite{maohelleman,artuso}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{acknowledgments} We thank G. Benfatto, G. Gallavotti, R. S. MacKay, and P. Wittwer for helpful discussions. H. Koch acknowledges support by the National Science Foundation under Grant No.\ DMS-9705095. Support from the EC contract No.\ ERBCHRXCT94-0460 for the project ``Stability and universality in classical mechanics'' and from the Conseil R\'egional de Bourgogne is also acknowledged. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \appendix \section{One step of the KAM iteration} \label{appendix} In this appendix, we use the notation ${\u \partial}f= \partial f/\partial {\u \varphi}$.\\ We consider a Hamiltonian of the form (\ref{hamiltonian}), with $g=(1-{\Bbb I}^-)G+\varepsilon {\Bbb I}^- G$ and $f=(1-{\Bbb I}^-)F+\varepsilon {\Bbb I}^-F$. The parameter $\varepsilon$ is introduced for bookkeeping purposes only; it will be set to one at the end. Our goal is to find a canonical transformation $U$ such that the functions $f'$ and $g'$ for the Hamiltonian $H'=H\circ U$ have only slow modes, up to terms of order $\mathcal{O}(\varepsilon^2)$. We perform a Lie transformation defined in Sec.\ \ref{sect:kamrg}. The expression of the Hamiltonian in the new variables is given by Eq.\ (\ref{eqn:exponential}): \begin{equation} \label{eqn:im} H'({\u A}',{\u \varphi}')= \frac{1}{2}m'({\u \varphi}')( {\u \Omega}\cdot{\u A}')^{2} +\lbrack {\u \omega}_0 + g'({\u \varphi}'){\u \Omega} \rbrack \cdot{\u A}' + f'({\u \varphi}'), \end{equation} where \begin{eqnarray} \label{eqn:projg} g'&=&g+\u{\omega}_0\cdot\u{\partial} Y +m(\u{\Omega}\cdot\u{\partial} Z +a\Omega^2) +\mathcal{O}(\epsilon^2),\\ \label{eqn:projf} f'&=&f+\u{\omega}_0\cdot\u{\partial} Z + a \u{\omega}_0\cdot\u{\Omega}+\mathcal{O}(\epsilon^2), \end{eqnarray} with $Y$, $Z$ and $a$ of order $\mathcal{O}(\varepsilon)$ to be determined. The generating function $S$ given by Eq.\ (\ref{eqn:S}) is determined by the projection of Eqs.\ (\ref{eqn:projg})-(\ref{eqn:projf}) on the space of non-slow modes. We recall that the condition is that ${\Bbb I}^-f'$ and ${\Bbb I}^-g'$ are of order $\mathcal{O}(\varepsilon^2)$. This leads to the following equations: \begin{eqnarray} && {\Bbb I}^-f+\u{\omega}_0\cdot\u{\partial}Z=const,\\ && {\Bbb I}^-g+\u{\omega}_0\cdot\u{\partial}Y +{\Bbb I}^-\left(m\u{\Omega}\cdot\u{\partial}Z\right) +{\Bbb I}^-m a\Omega^2 =0. \label{eqn:tcan2} \end{eqnarray} Equation (\ref{eqn:tcan2}) determines $a$: \begin{equation} a=-\frac{\langle g\rangle+\langle m\u{\Omega}\cdot\u{\partial}Z\rangle} {\Omega^2\langle m \rangle}, \end{equation} Moreover the functions $Y$ and $Z$ have only non-slow modes, and are given by the following series: \begin{eqnarray} && Z(\u{\varphi})=\sum_{\nu\in I^{-*}} \frac{i}{\u{\omega}_{0}\cdot\u{\nu}}f_{\nu} e^{i\nu\cdot\phi},\\ && Y(\u{\varphi})=\sum_{\nu\in I^{-*}} \frac{i}{\u{\omega}_{0}\cdot\u{\nu}} \left(g_{\nu}+(m\u{\Omega}\cdot\u{\partial}Z)_{\nu} +m_{\nu}a \Omega^2\right)e^{i\nu\cdot\phi}, \end{eqnarray} where $I^{-*}=I^-\setminus \{ {\u 0} \}$. The transformed Hamiltonian (\ref{eqn:im}) is constructed by defining $H^{(0)}=H$ and $H^{(i)}$ for $i=1,2,\ldots$ by the recursive relation \begin{eqnarray} H^{(i+1)}(\u{A},\u{\varphi})&=& \lbrace S(\u{A},\u{\varphi}),H^{(i)}(\u{A},\u{\varphi}) \rbrace \nonumber\\ &=& \frac{1}{2}m^{(i+1)}(\u{\varphi})(\u{\Omega}\cdot\u{A})^{2} +g^{(i+1)}(\u{\varphi})\u{\Omega}\cdot\u{A}+f^{(i+1)}(\u{\varphi}), \end{eqnarray} which leads to \begin{equation} H'=\sum_{i=0}^\infty \frac{H^{(i)}}{i!}. \end{equation} This can be expressed in terms of the image of the three scalar functions $(m,g,f)$ given by the following equations: \begin{eqnarray} && (m',g',f')=\left(\sum_{i=0}^{\infty} \frac{m^{(i)}}{i!}, \sum_{i=0}^{\infty} \frac{g^{(i)}}{i!},\sum_{i=0}^{\infty} \frac{f^{(i)}}{i!} \right),\\ && (m^{(0)},g^{(0)},f^{(0)})=(m,g,f),\\ && m^{(1)}=2m\u{\Omega}\cdot\u{\partial}Y -Y\u{\Omega}\cdot\u{\partial}m,\\ && g^{(1)}=g\u{\Omega}\cdot\u{\partial}Y -Y\u{\Omega}\cdot\u{\partial}g +m\u{\Omega}\cdot\u{\partial}Z+m a\Omega^2 +\u{\omega}_0\cdot\u{\partial}Y,\\ && f^{(1)}=-Y\u{\Omega}\cdot\u{\partial}f +g\u{\Omega}\cdot\u{\partial}Z+g a\Omega^2 +\u{\omega}_0\cdot\u{\partial} Z, \\ && m^{(i+1)}=2m^{(i)}\u{\Omega}\cdot\u{\partial}Y -Y\u{\Omega}\cdot\u{\partial}m^{(i)}, \\ && g^{(i+1)}=g^{(i)}\u{\Omega}\cdot\u{\partial}Y -Y\u{\Omega}\cdot\u{\partial}g^{(i)} +m^{(i)}\u{\Omega}\cdot\u{\partial}Z+m^{(i)}a\Omega^2,\\ && f^{(i+1)}=-Y\u{\Omega}\cdot\u{\partial}f^{(i)} +g^{(i)}\u{\Omega}\cdot\u{\partial}Z+g^{(i)}a\Omega^2, \end{eqnarray} for $i\geq 1$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{table} \label{table:par} \caption{Parameters associated with the breakup of noble tori.} \begin{tabular}{c c c c} L & Critical value $\varepsilon_c(L)$ & Area multiplier & Unstable eigenvalue\\ \hline & 0.027590 & 18.828171 & 2.650221 \\ \hline 3 & 0.027625 & 18.842654 & 2.649660 \\ 5 & 0.027579 & 18.823481 & 2.652722 \\ 8 & 0.027588 & 18.825034 & 2.649653 \\ 12 & 0.027590 & 18.829142 & 2.650484 \\ 13 & 0.027590 & 18.827464 & 2.650082 \\ 20 & 0.027590 & 18.828203 & 2.650234 \\ 21 & 0.027590 & 18.827772 & 2.650151 \\ 34 & 0.027590 & 18.827910 & 2.650175 \end{tabular} \end{table} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure} \caption{Renormalization flow in the space of Hamiltonians (\ref{hamiltonian}) associated with the universal class of ${\u \omega}_0=(1/\gamma,-1)$.} \label{figure:flow} \end{figure} \begin{figure} \caption{Non-slow modes (in the grey part) and slow modes (in the white part) associated with ${\u \omega}_0=(1/\gamma,-1)$.} \label{figure:I-} \end{figure} \begin{figure} \caption{Weight of the Fourier coefficients of $m_*$: white $<10^{-10}$, grey levels $[10^{-10},10^{-7}]$, $[10^{-7},10^{-5}]$, $[10^{-5},10^{-3}]$, $[10^{-3},5.10^{-2}]$, black $>5.10^{-2}$.} \label{figure:m} \end{figure} \begin{figure} \caption{Weight of the Fourier coefficients of $m_*$: $M(\nu_1)=\max_{\nu_2} |m_{*\nu}| $.} \label{figure:m2} \end{figure} \begin{figure} \caption{Weight of the Fourier coefficients of $g_*$: white $<10^{-10}$, grey levels $[10^{-10},10^{-7}]$, $[10^{-7},10^{-5}]$, $[10^{-5},10^{-3}]$, $[10^{-3},5.10^{-2}]$, black $>5.10^{-2}$.} \label{figure:g} \end{figure} \begin{figure} \caption{Weight of the Fourier coefficients of $f_*$: white $<10^{-10}$, grey levels $[10^{-10},10^{-7}]$, $[10^{-7},10^{-5}]$, $[10^{-5},10^{-3}]$, $[10^{-3},5.10^{-2}]$, black $>5.10^{-2}$.} \label{figure:f} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \unitlength=1cm \centerline{ \begin{picture}(18,25) \put(0,0){\psfig{figure=Figure1.ps,height=25cm,width=18cm}} \end{picture}} \newpage \unitlength=1cm \centerline{ \begin{picture}(18,25) \put(0,0){\psfig{figure=Figure2.ps,height=25cm,width=18cm}} \end{picture}} \newpage \unitlength=1cm \centerline{ \begin{picture}(18,25) \put(0,0){\psfig{figure=Figure3.ps,height=25cm,width=18cm}} \end{picture}} \newpage \unitlength=1cm \centerline{ \begin{picture}(18,25) \put(0,0){\psfig{figure=Figure4.ps,height=25cm,width=18cm}} \end{picture}} \newpage \unitlength=1cm \centerline{ \begin{picture}(18,25) \put(0,0){\psfig{figure=Figure5.ps,height=25cm,width=18cm}} \end{picture}} \newpage \unitlength=1cm \centerline{ \begin{picture}(18,25) \put(0,0){\psfig{figure=Figure6.ps,height=25cm,width=18cm}} \end{picture}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \begin{references} \bibitem{kolmogorov} A.~N.~Kolmogorov, Dokl. 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