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b(doing)h(the)g(cen)o(tre)e(manifold)h(reduction,)g(i.e.) f(\014nding)j Fn(\015)1574 2054 y Fh(1)1594 2047 y Fn(;)8 b(:)g(:)g(:)15 b(;)8 b(\027)1735 2054 y Fh(10)1773 2047 y Fp(,)17 b(w)o(e)936 2854 y(13)p eop %%Page: 14 14 14 13 bop 60 53 a Fp(see)16 b(that)h(at)f(the)g(double)g(Hopf)h (bifurcation)e(\(i.e.)g(when)h Fn(\025)f Fp(=)e Fg(\000)p Fp(2)p Fn(\017)1314 60 y Fl(r)1333 53 y Fp([1)e(+)g(cos)e(\()p Fn(\031)r(=)m(N)c Fp(\)]\))79 194 y Fn(\022)102 201 y Fh(9)163 194 y Fp(=)253 160 y Fn(i\013)301 167 y Fh(2)p 247 182 80 2 v 247 228 a Fp(\012)p Fn(N)343 194 y Fp(+)11 b Fn(\017)412 201 y Fl(r)439 133 y Ff(\024)478 160 y Fp(2)p Fn(\013)533 167 y Fh(2)p 466 182 100 2 v 466 228 a Fp(\012)501 213 y Fh(2)521 228 y Fn(N)570 194 y Fp(\(1)g(+)g(cos)e (\()p Fn(\031)r(=)m(N)c Fp(\)\))899 133 y Ff(\025)932 194 y Fp(+)11 b Fn(O)q Fp(\()p Fg(j)p Fn(\017)1072 201 y Fl(r)1091 194 y Fn(;)d(\017)1133 201 y Fl(i)1147 194 y Fg(j)1161 173 y Fh(2)1181 194 y Fp(\))76 314 y Fn(\015)101 321 y Fh(8)163 314 y 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Ff(\024)514 521 y Fp(2)p Fn(\013)569 528 y Fh(3)p 490 543 124 2 v 490 589 a Fp(9\012)549 574 y Fh(2)570 589 y Fn(N)619 554 y Fp(\(cos)d(\()p Fn(\031)r(=)m(N)d Fp(\))12 b(+)f(cos)d(\(2)p Fn(\031)r(=)m(N)d Fp(\))q(\))1155 494 y Ff(\025)163 675 y Fp(+)41 b Fn(\017)262 682 y Fl(i)285 614 y Ff(\024)327 641 y Fp(2)p Fn(i\013)399 648 y Fh(3)p 311 663 V 311 709 a Fp(9)p Fn(N)5 b Fp(\012)414 695 y Fh(2)440 675 y Fp(\(cos)k(\(2)p Fn(\031)r(=)m(N)c Fp(\))11 b Fg(\000)g Fp(2)d(cos)h(\()p Fn(\031)r(=)m(N)c Fp(\))12 b Fg(\000)f Fp(3\))1095 614 y Ff(\025)1128 675 y Fp(+)g Fn(O)q Fp(\()p Fg(j)p Fn(\017)1268 682 y Fl(r)1287 675 y Fn(;)d(\017)1329 682 y Fl(i)1343 675 y Fg(j)1357 654 y Fh(2)1376 675 y Fp(\))79 795 y Fn(\022)102 802 y Fh(2)163 795 y Fp(=)253 761 y Fn(i\013)301 768 y Fh(2)p 247 783 80 2 v 247 829 a Fp(\012)p Fn(N)343 795 y Fp(+)j Fn(\017)412 802 y Fl(r)439 735 y Ff(\024)478 761 y Fp(2)p Fn(\013)533 768 y Fh(2)p 466 783 100 2 v 466 829 a Fp(\012)501 815 y Fh(2)521 829 y Fn(N)570 795 y Fp(\(1)g(+)g(cos)e(\()p Fn(\031)r(=)m(N)c Fp(\)\))899 735 y Ff(\025)932 795 y Fp(+)11 b Fn(O)q Fp(\()p Fg(j)p Fn(\017)1072 802 y Fl(r)1091 795 y Fn(;)d(\017)1133 802 y Fl(i)1147 795 y Fg(j)1161 775 y Fh(2)1181 795 y Fp(\))76 915 y Fn(\015)101 922 y Fh(6)163 915 y Fp(=)253 882 y Fn(i\013)301 889 y Fh(2)p 247 904 80 2 v 247 950 a Fp(\012)p Fn(N)343 915 y Fp(+)j Fn(\017)412 922 y Fl(r)439 855 y Ff(\024)478 882 y Fp(2)p Fn(\013)533 889 y Fh(2)p 466 904 100 2 v 466 950 a Fp(\012)501 935 y Fh(2)521 950 y Fn(N)570 915 y Fp(\(cos)e(\()p Fn(\031)r(=)m(N)c Fp(\))11 b(+)g(cos)e(\(2)p Fn(\031)r(=)m(N)c Fp(\)\))1106 855 y Ff(\025)163 1036 y Fp(+)41 b Fn(\017)262 1043 y Fl(i)285 975 y Ff(\024)315 1002 y Fp(2)p Fn(i\013)387 1009 y Fh(2)p 311 1024 V 311 1070 a Fp(\012)346 1056 y Fh(2)366 1070 y Fn(N)416 1036 y Fp(\(cos)8 b(\(2)p Fn(\031)r(=)m(N)d Fp(\))12 b Fg(\000)f Fp(1\))770 975 y Ff(\025)803 1036 y Fp(+)g Fn(O)q Fp(\()p Fg(j)p Fn(\017)943 1043 y Fl(r)962 1036 y Fn(;)d(\017)1004 1043 y Fl(i)1018 1036 y Fg(j)1032 1015 y Fh(2)1051 1036 y Fp(\))78 1156 y Fn(\027)102 1163 y Fh(4)163 1156 y Fp(=)253 1122 y Fn(i\013)301 1129 y Fh(2)p 247 1144 80 2 v 247 1190 a Fp(\012)p Fn(N)343 1156 y Fp(+)j Fn(\017)412 1163 y Fl(r)439 1095 y Ff(\024)478 1122 y Fp(2)p Fn(\013)533 1129 y Fh(2)p 466 1144 100 2 v 466 1190 a Fp(\012)501 1176 y Fh(2)521 1190 y Fn(N)570 1156 y Fp(\(cos)e(\()p Fn(\031)r(=)m(N)c Fp(\))11 b(+)g(cos)e(\(2)p Fn(\031)r(=)m(N)c Fp(\)\))1106 1095 y Ff(\025)163 1276 y Fp(+)41 b Fn(\017)262 1283 y Fl(i)285 1216 y Ff(\024)315 1243 y Fp(2)p Fn(i\013)387 1250 y Fh(2)p 311 1265 V 311 1311 a Fn(N)5 b Fp(\012)390 1296 y Fh(2)416 1276 y Fp(\(cos)j(\(2)p Fn(\031)r(=)m(N)d Fp(\))12 b Fg(\000)f Fp(1\))770 1216 y Ff(\025)803 1276 y Fp(+)g Fn(O)q Fp(\()p Fg(j)p Fn(\017)943 1283 y Fl(r)962 1276 y Fn(;)d(\017)1004 1283 y Fl(i)1018 1276 y Fg(j)1032 1256 y Fh(2)1051 1276 y Fp(\))79 1397 y Fn(\022)102 1404 y Fh(3)163 1397 y Fp(=)41 b Fg(\000)286 1363 y Fp(2)p Fn(i\013)358 1370 y Fh(1)p 286 1385 92 2 v 292 1431 a Fp(\012)p Fn(N)394 1397 y Fp(+)11 b Fn(\017)463 1404 y Fl(r)490 1336 y Ff(\024)529 1363 y Fp(4)p Fn(\013)584 1370 y Fh(1)p 517 1385 100 2 v 517 1431 a Fp(\012)552 1416 y Fh(2)572 1431 y Fn(N)621 1397 y Fp(\(1)g(+)g(cos)e(\()p Fn(\031)r(=)m(N)c Fp(\)\))950 1336 y Ff(\025)983 1397 y Fp(+)11 b Fn(\017)1052 1404 y Fl(i)1074 1336 y Ff(\024)1105 1363 y Fp(8)p Fn(i\013)1177 1370 y Fh(1)p 1101 1385 V 1101 1431 a Fp(\012)1136 1416 y Fh(2)1156 1431 y Fn(N)1205 1397 y Fp(\(1)h(+)f(cos)e(\()p Fn(\031)r(=)m(N)c Fp(\)\))1535 1336 y Ff(\025)1568 1397 y Fp(+)11 b Fn(O)q Fp(\()p Fg(j)p Fn(\017)1708 1404 y Fl(r)1727 1397 y Fn(;)d(\017)1769 1404 y Fl(i)1783 1397 y Fg(j)1797 1376 y Fh(2)1816 1397 y Fp(\))79 1517 y Fn(\022)102 1524 y Fh(7)163 1517 y Fp(=)253 1483 y(2)p Fn(i\013)325 1490 y Fh(3)p 247 1505 104 2 v 247 1551 a Fp(3\012)p Fn(N)367 1517 y Fp(+)j Fn(\017)436 1524 y Fl(r)463 1456 y Ff(\024)514 1483 y Fp(4)p Fn(\013)569 1490 y Fh(3)p 490 1505 124 2 v 490 1551 a Fp(9\012)549 1537 y Fh(2)570 1551 y Fn(N)619 1517 y Fp(\(1)g(+)g(cos)e(\()p Fn(\031)r(=)m(N)c Fp(\)\))948 1456 y Ff(\025)981 1517 y Fp(+)11 b Fn(\017)1050 1524 y Fl(i)1072 1456 y Ff(\024)1099 1483 y Fg(\000)p Fp(8)p Fn(i\013)1210 1490 y Fh(3)p 1099 1505 131 2 v 1102 1551 a Fp(9)p Fn(N)5 b Fp(\012)1205 1537 y Fh(2)1235 1517 y Fp(\(1)11 b(+)g(cos)e(\()p Fn(\031)r(=)m(N)c Fp(\)\))1564 1456 y Ff(\025)1597 1517 y Fp(+)11 b Fn(O)q Fp(\()p Fg(j)p Fn(\017)1737 1524 y Fl(r)1756 1517 y Fn(;)d(\017)1798 1524 y Fl(i)1812 1517 y Fg(j)1826 1496 y Fh(2)1845 1517 y Fp(\))60 1657 y(for)13 b(small)e Fg(j)p Fn(\017)289 1664 y Fl(r)307 1657 y Fg(j)p Fn(;)d Fg(j)p Fn(\017)377 1664 y Fl(i)390 1657 y Fg(j)p Fp(.)20 b(Substituting)13 b(these)f(expansions)h(in)o(to)f(the)h(expressions)f(for)h Fn(B)i Fp(and)e Fn(C)j Fp(\(13{)60 1747 y(14\))h(w)o(e)f(obtain)156 1888 y Fn(Re)p Fg(f)p Fn(B)s Fg(g)41 b Fp(=)452 1854 y(1)p 432 1876 64 2 v 432 1922 a Fn(N)476 1907 y Fh(2)509 1814 y Ff(")533 1888 y Fn(Re)p Fg(f)p Fn(\014)646 1895 y Fh(2)666 1888 y Fg(g)11 b(\000)757 1854 y Fn(I)t(m)p Fg(f)p Fn(\013)882 1861 y Fh(1)900 1854 y Fn(\013)931 1861 y Fh(2)951 1854 y Fg(g)p 757 1876 220 2 v 849 1922 a Fp(\012)981 1814 y Ff(#)1773 1888 y Fp(\(15\))347 2020 y(+)472 1986 y Fn(\017)492 1993 y Fl(r)p 432 2008 119 2 v 432 2054 a Fn(N)476 2040 y Fh(2)496 2054 y Fp(\012)531 2040 y Fh(2)564 1959 y Ff(\024)586 2020 y Fp(2)p Fn(Re)p Fg(f)p Fn(\013)726 2027 y Fh(1)746 2020 y Fn(\013)777 2027 y Fh(2)797 2020 y Fg(g)830 1959 y Ff(\022)861 2020 y Fp(2)g(+)g(3)d(cos)1052 1959 y Ff(\022)1095 1986 y Fn(\031)p 1087 2008 45 2 v 1087 2054 a(N)1136 1959 y Ff(\023)1178 2020 y Fp(+)j(cos)1301 1959 y Ff(\022)1336 1986 y Fp(2)p Fn(\031)p 1336 2008 54 2 v 1341 2054 a(N)1395 1959 y Ff(\023\023)347 2148 y Fp(+)47 b(2)p Fg(j)p Fn(\013)501 2155 y Fh(2)521 2148 y Fg(j)535 2128 y Fh(2)563 2088 y Ff(\022)593 2148 y Fp(1)12 b(+)f(cos)751 2088 y Ff(\022)794 2115 y Fn(\031)p 787 2137 45 2 v 787 2183 a(N)836 2088 y Ff(\023\023)908 2148 y Fp(+)962 2115 y(4)p 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y Fn(;)d(\017)1698 2288 y Fl(i)1712 2281 y Fg(j)1726 2260 y Fh(2)1745 2281 y Fp(\))936 2854 y(14)p eop %%Page: 15 15 15 14 bop 60 53 a Fp(and)156 164 y Fn(Re)p Fg(f)p Fn(C)t Fg(g)42 b Fp(=)451 131 y(2)p 431 153 64 2 v 431 199 a Fn(N)475 184 y Fh(2)509 91 y Ff(")533 164 y Fn(Re)p Fg(f)p Fn(\014)646 171 y Fh(2)665 164 y Fg(g)11 b(\000)756 131 y Fn(I)t(m)p Fg(f)p Fn(\013)881 138 y Fh(1)900 131 y Fn(\013)931 138 y Fh(2)950 131 y Fg(g)p 756 153 220 2 v 848 199 a Fp(\012)980 91 y Ff(#)1773 164 y Fp(\(16\))347 297 y(+)472 263 y Fn(\017)492 270 y Fl(r)p 431 285 119 2 v 431 331 a Fn(N)475 316 y Fh(2)495 331 y Fp(\012)530 316 y Fh(2)564 236 y Ff(\024)586 297 y Fp(4)p Fn(Re)p Fg(f)p Fn(\013)726 304 y Fh(1)746 297 y Fn(\013)777 304 y Fh(2)796 297 y Fg(g)829 236 y Ff(\022)860 297 y Fp(2)h(+)f(3)d(cos) 1051 236 y Ff(\022)1094 263 y Fn(\031)p 1087 285 45 2 v 1087 331 a(N)1136 236 y Ff(\023)1177 297 y Fp(+)j(cos)1300 236 y Ff(\022)1336 263 y Fp(2)p Fn(\031)p 1336 285 54 2 v 1341 331 a(N)1394 236 y 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539 y(@)683 581 y(@)724 622 y(@)766 664 y(@)807 706 y(@)849 747 y(@)890 789 y(@)932 830 y(@)973 872 y(@)1015 913 y(@)1056 955 y(@)1098 996 y(@)1139 1038 y(@)1181 1079 y(@)1222 1121 y(@)1264 1162 y(@)1305 1204 y(@)1347 1245 y(@)1388 1287 y(@)1430 1328 y(@)1471 1370 y(@)p 475 1287 208 2 v 683 1287 2 2 v 688 1287 V 693 1287 V 698 1287 V 703 1287 V 709 1287 V 714 1287 V 719 1287 V 724 1287 V 729 1287 V 735 1287 V 740 1287 V 745 1287 V 750 1287 V 755 1287 V 760 1287 V 766 1287 V 771 1287 V 776 1287 V 781 1287 V 786 1287 V 792 1287 V 797 1287 V 802 1287 V 807 1287 V 812 1287 V 818 1287 V 823 1287 V 828 1287 V 833 1287 V 838 1287 V 844 1287 V 849 1287 V 854 1287 V 859 1287 V 864 1287 V 869 1287 V 875 1287 V 880 1287 V 885 1287 V 890 1287 V 683 1287 V 683 1286 V 683 1285 V 683 1285 V 683 1284 V 683 1283 V 683 1282 V 683 1281 V 683 1280 V 683 1279 V 683 1278 V 683 1277 V 683 1276 V 683 1275 V 683 1274 V 683 1273 V 683 1272 V 683 1271 V 683 1270 V 683 1269 V 683 1268 V 683 1267 V 683 1266 V 683 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1187 V 694 1186 V 694 1185 V 694 1185 V 694 1184 V 694 1183 V 695 1182 V 695 1182 V 695 1181 V 695 1180 V 695 1179 V 696 1179 V 696 1178 V 696 1177 V 696 1176 V 696 1176 V 697 1175 V 697 1174 V 697 1173 V 697 1173 V 698 1172 V 698 1171 V 698 1170 V 698 1170 V 698 1169 V 699 1168 V 699 1168 V 699 1167 V 699 1166 V 700 1165 V 700 1165 V 700 1164 V 700 1163 V 701 1163 V 701 1162 V 701 1161 V 701 1160 V 702 1160 V 702 1159 V 702 1158 V 702 1158 V 703 1157 V 703 1156 V 703 1156 V 703 1155 V 704 1154 V 704 1154 V 704 1153 V 704 1152 V 705 1152 V 705 1151 V 705 1150 V 706 1150 V 706 1149 V 706 1148 V 706 1148 V 707 1147 V 707 1146 V 707 1146 V 708 1145 V 708 1144 V 708 1144 V 708 1143 V 709 1142 V 709 1142 V 709 1141 V 710 1140 V 710 1140 V 710 1139 V 710 1139 V 711 1138 V 711 1137 V 711 1137 V 712 1136 V 712 1135 V 712 1135 V 713 1134 V 713 1134 V 713 1133 V 714 1132 V 714 1132 V 714 1131 V 715 1130 V 715 1130 V 715 1129 V 716 1129 V 716 1128 V 716 1127 V 717 1127 V 717 1126 V 717 1126 V 718 1125 V 718 1125 V 718 1124 V 719 1123 V 719 1123 V 719 1122 V 720 1122 V 720 1121 V 720 1120 V 721 1120 V 721 1119 V 721 1119 V 722 1118 V 722 1118 V 722 1117 V 723 1117 V 723 1116 V 724 1115 V 724 1115 V 724 1114 V 725 1114 V 725 1113 V 725 1113 V 726 1112 V 726 1112 V 727 1111 V 727 1111 V 727 1110 V 728 1109 V 728 1109 V 728 1108 V 729 1108 V 729 1107 V 730 1107 V 730 1106 V 730 1106 V 731 1105 V 731 1105 V 732 1104 V 732 1104 V 732 1103 V 733 1103 V 733 1102 V 734 1102 V 734 1101 V 735 1101 V 735 1100 V 735 1100 V 736 1099 V 736 1099 V 737 1098 V 737 1098 V 737 1097 V 738 1097 V 738 1096 V 739 1096 V 739 1095 V 740 1095 V 740 1094 V 741 1094 V 741 1093 V 741 1093 V 742 1093 V 742 1092 V 743 1092 V 743 1091 V 744 1091 V 744 1090 V 745 1090 V 745 1089 V 745 1089 V 746 1088 V 746 1088 V 747 1088 V 747 1087 V 748 1087 V 748 1086 V 749 1086 V 749 1085 V 750 1085 V 750 1084 V 751 1084 V 751 1084 V 752 1083 V 752 1083 V 752 1082 V 753 1082 V 753 1082 V 754 1081 V 754 1081 V 755 1080 V 755 1080 V 756 1079 V 756 1079 V 757 1079 V 757 1078 V 758 1078 V 758 1077 V 759 1077 V 759 1077 V 760 1076 V 760 1076 V 761 1076 V 761 1075 V 762 1075 V 762 1074 V 763 1074 V 763 1074 V 764 1073 V 765 1073 V 765 1073 V 766 1072 V 766 1072 V 767 1071 V 767 1071 V 768 1071 V 768 1070 V 769 1070 V 769 1070 V 770 1069 V 770 1069 V 771 1069 V 771 1068 V 772 1068 V 773 1068 V 773 1067 V 774 1067 V 774 1067 V 775 1066 V 775 1066 V 776 1066 V 776 1065 V 777 1065 V 777 1065 V 778 1064 V 779 1064 V 779 1064 V 780 1063 V 780 1063 V 781 1063 V 781 1062 V 782 1062 V 783 1062 V 783 1061 V 784 1061 V 784 1061 V 785 1061 V 786 1060 V 786 1060 V 787 1060 V 787 1059 V 788 1059 V 788 1059 V 789 1059 V 790 1058 V 790 1058 V 791 1058 V 791 1057 V 792 1057 V 793 1057 V 793 1057 V 794 1056 V 794 1056 V 795 1056 V 796 1056 V 796 1055 V 797 1055 V 798 1055 V 798 1054 V 799 1054 V 799 1054 V 800 1054 V 801 1053 V 801 1053 V 802 1053 V 803 1053 V 803 1053 V 804 1052 V 804 1052 V 805 1052 V 806 1052 V 806 1051 V 807 1051 V 808 1051 V 808 1051 V 809 1050 V 810 1050 V 810 1050 V 811 1050 V 812 1050 V 812 1049 V 813 1049 V 814 1049 V 814 1049 V 815 1049 V 815 1048 V 816 1048 V 817 1048 V 817 1048 V 818 1048 V 819 1047 V 820 1047 V 820 1047 V 821 1047 V 822 1047 V 822 1046 V 823 1046 V 824 1046 V 824 1046 V 825 1046 V 826 1046 V 826 1045 V 827 1045 V 828 1045 V 828 1045 V 829 1045 V 830 1045 V 830 1044 V 831 1044 V 832 1044 V 833 1044 V 833 1044 V 834 1044 V 835 1044 V 835 1043 V 836 1043 V 837 1043 V 838 1043 V 838 1043 V 839 1043 V 840 1043 V 840 1042 V 841 1042 V 842 1042 V 843 1042 V 843 1042 V 844 1042 V 845 1042 V 846 1042 V 846 1042 V 847 1041 V 848 1041 V 849 1041 V 849 1041 V 850 1041 V 851 1041 V 851 1041 V 852 1041 V 853 1041 V 854 1041 V 855 1040 V 855 1040 V 856 1040 V 857 1040 V 858 1040 V 858 1040 V 859 1040 V 860 1040 V 861 1040 V 861 1040 V 862 1040 V 863 1040 V 864 1040 V 864 1039 V 865 1039 V 866 1039 V 867 1039 V 868 1039 V 868 1039 V 869 1039 V 870 1039 V 871 1039 V 872 1039 V 872 1039 V 873 1039 V 874 1039 V 875 1039 V 876 1039 V 876 1039 V 877 1039 V 878 1039 V 879 1039 V 880 1039 V 880 1039 V 881 1039 V 882 1039 V 883 1038 V 884 1038 V 884 1038 V 885 1038 V 886 1038 V 887 1038 V 888 1038 V 889 1038 V 889 1038 V 890 1038 V 683 1287 V 683 1279 V 683 1271 V 684 1264 V 685 1256 V 686 1249 V 687 1241 V 689 1234 V 691 1228 V 693 1221 V 696 1215 V 698 1209 V 701 1203 V 705 1197 V 708 1192 V 712 1186 V 716 1181 V 720 1176 V 725 1172 V 729 1167 V 735 1163 V 740 1159 V 745 1155 V 751 1151 V 757 1148 V 764 1145 V 770 1142 V 777 1139 V 784 1136 V 792 1134 V 799 1132 V 807 1130 V 815 1128 V 824 1127 V 833 1125 V 842 1124 V 851 1123 V 860 1122 V 870 1122 V 880 1122 V 890 1121 V 683 1287 V 683 1283 V 683 1279 V 684 1275 V 685 1272 V 686 1268 V 687 1264 V 689 1261 V 691 1258 V 693 1254 V 696 1251 V 698 1248 V 701 1245 V 705 1242 V 708 1240 V 712 1237 V 716 1234 V 720 1232 V 725 1230 V 729 1227 V 735 1225 V 740 1223 V 745 1221 V 751 1219 V 757 1218 V 764 1216 V 770 1215 V 777 1213 V 784 1212 V 792 1211 V 799 1210 V 807 1209 V 815 1208 V 824 1207 V 833 1206 V 842 1206 V 851 1205 V 860 1205 V 870 1205 V 880 1205 V 890 1204 V 932 1050 a Fk(Z)966 1057 y Fb(2)988 1050 y Fp(\()p Fn(\024)p Fp(\))932 1133 y Fk(Z)966 1140 y Fb(2)988 1133 y Fp(\()p Fn(\024;)8 b(\031)r Fp(\))937 1200 y Ff(e)932 1216 y Fk(Z)966 1223 y Fb(N)p 1305 1038 208 2 v 1513 1038 2 2 v 1518 1038 V 1523 1038 V 1528 1038 V 1534 1038 V 1539 1038 V 1544 1038 V 1549 1038 V 1554 1038 V 1560 1038 V 1565 1038 V 1570 1038 V 1575 1038 V 1580 1038 V 1586 1038 V 1591 1038 V 1596 1038 V 1601 1038 V 1606 1038 V 1611 1038 V 1617 1038 V 1622 1038 V 1627 1038 V 1632 1038 V 1637 1038 V 1643 1038 V 1648 1038 V 1653 1038 V 1658 1038 V 1663 1038 V 1669 1038 V 1674 1038 V 1679 1038 V 1684 1038 V 1689 1038 V 1694 1038 V 1700 1038 V 1705 1038 V 1710 1038 V 1715 1038 V 1720 1038 V 1513 1038 V 1513 1034 V 1513 1030 V 1514 1026 V 1515 1023 V 1516 1019 V 1518 1015 V 1519 1012 V 1521 1009 V 1523 1005 V 1526 1002 V 1529 999 V 1532 996 V 1535 993 V 1538 990 V 1542 988 V 1546 985 V 1550 983 V 1555 981 V 1560 978 V 1565 976 V 1570 974 V 1576 972 V 1582 970 V 1588 969 V 1594 967 V 1601 966 V 1607 964 V 1615 963 V 1622 962 V 1630 961 V 1638 960 V 1646 959 V 1654 958 V 1663 957 V 1672 957 V 1681 956 V 1690 956 V 1700 956 V 1710 955 V 1720 955 V 1768 951 a Ff(e)1762 967 y Fk(Z)1796 974 y Fb(N)p 1513 1038 V 1513 1034 V 1512 1030 V 1512 1026 V 1511 1023 V 1510 1019 V 1508 1015 V 1507 1012 V 1505 1009 V 1502 1005 V 1500 1002 V 1497 999 V 1494 996 V 1491 993 V 1487 990 V 1484 988 V 1480 985 V 1475 983 V 1471 981 V 1466 978 V 1461 976 V 1456 974 V 1450 972 V 1444 970 V 1438 969 V 1432 967 V 1425 966 V 1418 964 V 1411 963 V 1404 962 V 1396 961 V 1388 960 V 1380 959 V 1372 958 V 1363 957 V 1354 957 V 1345 956 V 1335 956 V 1326 956 V 1316 955 V 1305 955 V 1513 1038 V 1513 1030 V 1512 1022 V 1512 1014 V 1511 1007 V 1510 1000 V 1508 992 V 1507 985 V 1505 979 V 1502 972 V 1500 966 V 1497 960 V 1494 954 V 1491 948 V 1487 943 V 1484 937 V 1480 932 V 1475 927 V 1471 923 V 1466 918 V 1461 914 V 1456 910 V 1450 906 V 1444 902 V 1438 899 V 1432 896 V 1425 893 V 1418 890 V 1411 887 V 1404 885 V 1396 883 V 1388 881 V 1380 879 V 1372 877 V 1363 876 V 1354 875 V 1345 874 V 1335 873 V 1326 873 V 1316 872 V 1305 872 V 1160 967 a Fk(Z)1194 974 y Fb(2)1217 967 y Fp(\()p Fn(\024)p Fp(\))1106 884 y Fk(Z)1140 891 y Fb(2)1163 884 y Fp(\()p Fn(\024;)g(\031)r Fp(\))p 973 540 208 2 v 1181 540 2 2 v 1186 540 V 1191 540 V 1196 540 V 1202 540 V 1207 540 V 1212 540 V 1217 540 V 1222 540 V 1227 540 V 1233 540 V 1238 540 V 1243 540 V 1248 540 V 1253 540 V 1259 540 V 1264 540 V 1269 540 V 1274 540 V 1279 540 V 1285 540 V 1290 540 V 1295 540 V 1300 540 V 1305 540 V 1311 540 V 1316 540 V 1321 540 V 1326 540 V 1331 540 V 1336 540 V 1342 540 V 1347 540 V 1352 540 V 1357 540 V 1362 540 V 1368 540 V 1373 540 V 1378 540 V 1383 540 V 1388 540 V 1181 540 V 1181 536 V 1180 532 V 1180 528 V 1179 525 V 1178 521 V 1176 517 V 1174 514 V 1172 510 V 1170 507 V 1168 504 V 1165 501 V 1162 498 V 1159 495 V 1155 492 V 1152 490 V 1148 487 V 1143 485 V 1139 482 V 1134 480 V 1129 478 V 1124 476 V 1118 474 V 1112 472 V 1106 471 V 1100 469 V 1093 467 V 1086 466 V 1079 465 V 1072 464 V 1064 462 V 1056 461 V 1048 461 V 1040 460 V 1031 459 V 1022 459 V 1013 458 V 1003 458 V 993 457 V 983 457 V 973 457 V 1181 540 V 1181 532 V 1180 524 V 1180 516 V 1179 509 V 1178 501 V 1176 494 V 1174 487 V 1172 481 V 1170 474 V 1168 468 V 1165 462 V 1162 456 V 1159 450 V 1155 444 V 1152 439 V 1148 434 V 1143 429 V 1139 424 V 1134 420 V 1129 416 V 1124 412 V 1118 408 V 1112 404 V 1106 401 V 1100 398 V 1093 395 V 1086 392 V 1079 389 V 1072 387 V 1064 385 V 1056 383 V 1048 381 V 1040 379 V 1031 378 V 1022 377 V 1013 376 V 1003 375 V 993 375 V 983 374 V 973 374 V 1181 540 V 1181 528 V 1180 516 V 1180 504 V 1179 493 V 1178 482 V 1176 471 V 1174 461 V 1172 451 V 1170 441 V 1168 431 V 1165 422 V 1162 413 V 1159 405 V 1155 396 V 1152 389 V 1148 381 V 1143 374 V 1139 367 V 1134 360 V 1129 353 V 1124 347 V 1118 342 V 1112 336 V 1106 331 V 1100 326 V 1093 322 V 1086 318 V 1079 314 V 1072 310 V 1064 307 V 1056 304 V 1048 301 V 1040 299 V 1031 297 V 1022 295 V 1013 294 V 1003 293 V 993 292 V 983 291 V 973 291 V 766 303 a Fk(Z)800 310 y Fb(2)822 303 y Fp(\()p Fn(\024;)g(\031)r Fp(\))815 386 y Fk(Z)849 393 y Fb(2)872 386 y Fp(\()p Fn(\024)p Fp(\))854 453 y Ff(e)849 469 y Fk(Z)883 476 y Fb(N)p 102 706 208 2 v 309 706 2 2 v 314 706 V 319 706 V 325 706 V 330 706 V 335 706 V 340 706 V 345 706 V 351 706 V 356 706 V 361 706 V 366 706 V 371 706 V 377 706 V 382 706 V 387 706 V 392 706 V 397 706 V 402 706 V 408 706 V 413 706 V 418 706 V 423 706 V 428 706 V 434 706 V 439 706 V 444 706 V 449 706 V 454 706 V 460 706 V 465 706 V 470 706 V 475 706 V 480 706 V 485 706 V 491 706 V 496 706 V 501 706 V 506 706 V 511 706 V 517 706 V 309 706 V 309 702 V 310 698 V 310 694 V 311 691 V 312 687 V 314 683 V 315 680 V 317 676 V 320 673 V 322 670 V 325 667 V 328 664 V 331 661 V 334 658 V 338 656 V 342 653 V 347 651 V 351 648 V 356 646 V 361 644 V 366 642 V 372 640 V 378 638 V 384 637 V 390 635 V 397 633 V 404 632 V 411 631 V 418 630 V 426 628 V 434 628 V 442 627 V 450 626 V 459 625 V 468 625 V 477 624 V 487 624 V 496 624 V 506 623 V 517 623 V 309 706 V 309 698 V 310 690 V 310 682 V 311 675 V 312 667 V 314 660 V 315 653 V 317 647 V 320 640 V 322 634 V 325 628 V 328 622 V 331 616 V 334 610 V 338 605 V 342 600 V 347 595 V 351 591 V 356 586 V 361 582 V 366 578 V 372 574 V 378 570 V 384 567 V 390 564 V 397 561 V 404 558 V 411 555 V 418 553 V 426 551 V 434 549 V 442 547 V 450 545 V 459 544 V 468 543 V 477 542 V 487 541 V 496 541 V 506 540 V 517 540 V 309 706 V 309 702 V 309 698 V 308 694 V 307 691 V 306 687 V 304 683 V 303 680 V 301 676 V 299 673 V 296 670 V 293 667 V 290 664 V 287 661 V 284 658 V 280 656 V 276 653 V 272 651 V 267 648 V 262 646 V 257 644 V 252 642 V 246 640 V 240 638 V 234 637 V 228 635 V 221 633 V 214 632 V 207 631 V 200 630 V 192 628 V 184 628 V 176 627 V 168 626 V 159 625 V 150 625 V 141 624 V 131 624 V 122 624 V 112 623 V 102 623 V 537 635 a Fk(Z)571 642 y Fb(2)594 635 y Fp(\()p Fn(\024;)g(\031)r Fp(\))537 552 y Fk(Z)571 559 y Fb(2)594 552 y Fp(\()p Fn(\024)p Fp(\))24 619 y Ff(e)18 635 y Fk(Z)52 642 y Fb(N)p 102 1038 208 2 v 309 1038 2 2 v 316 1038 V 323 1038 V 330 1038 V 337 1038 V 344 1038 V 351 1038 V 357 1038 V 364 1038 V 370 1038 V 377 1038 V 383 1038 V 389 1038 V 395 1038 V 401 1038 V 406 1038 V 412 1038 V 418 1038 V 423 1038 V 428 1038 V 434 1038 V 439 1038 V 444 1038 V 449 1038 V 454 1038 V 458 1038 V 463 1038 V 467 1038 V 472 1038 V 476 1038 V 480 1038 V 484 1038 V 488 1038 V 492 1038 V 496 1038 V 500 1038 V 503 1038 V 507 1038 V 510 1038 V 513 1038 V 517 1038 V 309 1038 V 309 1038 V 309 1038 V 309 1037 V 309 1037 V 309 1037 V 309 1036 V 309 1036 V 309 1036 V 309 1035 V 309 1035 V 309 1035 V 309 1034 V 309 1034 V 309 1034 V 309 1034 V 309 1033 V 309 1033 V 309 1033 V 309 1032 V 309 1032 V 309 1032 V 309 1031 V 310 1031 V 310 1031 V 310 1030 V 310 1030 V 310 1030 V 310 1029 V 310 1029 V 310 1029 V 310 1028 V 310 1028 V 310 1028 V 310 1028 V 310 1027 V 310 1027 V 310 1027 V 310 1026 V 310 1026 V 310 1026 V 310 1025 V 311 1025 V 311 1025 V 311 1024 V 311 1024 V 311 1024 V 311 1024 V 311 1023 V 311 1023 V 311 1023 V 311 1022 V 311 1022 V 311 1022 V 311 1021 V 312 1021 V 312 1021 V 312 1021 V 312 1020 V 312 1020 V 312 1020 V 312 1019 V 312 1019 V 312 1019 V 312 1019 V 313 1018 V 313 1018 V 313 1018 V 313 1017 V 313 1017 V 313 1017 V 313 1017 V 313 1016 V 313 1016 V 314 1016 V 314 1015 V 314 1015 V 314 1015 V 314 1015 V 314 1014 V 314 1014 V 315 1014 V 315 1013 V 315 1013 V 315 1013 V 315 1013 V 315 1012 V 315 1012 V 315 1012 V 316 1011 V 316 1011 V 316 1011 V 316 1011 V 316 1010 V 316 1010 V 317 1010 V 317 1010 V 317 1009 V 317 1009 V 317 1009 V 317 1009 V 318 1008 V 318 1008 V 318 1008 V 318 1007 V 318 1007 V 318 1007 V 319 1007 V 319 1006 V 319 1006 V 319 1006 V 319 1006 V 319 1005 V 320 1005 V 320 1005 V 320 1005 V 320 1004 V 320 1004 V 321 1004 V 321 1004 V 321 1003 V 321 1003 V 321 1003 V 322 1003 V 322 1002 V 322 1002 V 322 1002 V 322 1002 V 323 1001 V 323 1001 V 323 1001 V 323 1001 V 324 1000 V 324 1000 V 324 1000 V 324 1000 V 324 999 V 325 999 V 325 999 V 325 999 V 325 998 V 326 998 V 326 998 V 326 998 V 326 997 V 327 997 V 327 997 V 327 997 V 327 997 V 327 996 V 328 996 V 328 996 V 328 996 V 328 995 V 329 995 V 329 995 V 329 995 V 330 994 V 330 994 V 330 994 V 330 994 V 331 994 V 331 993 V 331 993 V 331 993 V 332 993 V 332 992 V 332 992 V 332 992 V 333 992 V 333 992 V 333 991 V 334 991 V 334 991 V 334 991 V 334 990 V 335 990 V 335 990 V 335 990 V 336 990 V 336 989 V 336 989 V 337 989 V 337 989 V 337 989 V 337 988 V 338 988 V 338 988 V 338 988 V 339 988 V 339 987 V 339 987 V 340 987 V 340 987 V 340 987 V 341 986 V 341 986 V 341 986 V 342 986 V 342 985 V 342 985 V 343 985 V 343 985 V 343 985 V 344 984 V 344 984 V 344 984 V 345 984 V 345 984 V 345 984 V 346 983 V 346 983 V 346 983 V 347 983 V 347 983 V 347 982 V 348 982 V 348 982 V 349 982 V 349 982 V 349 981 V 350 981 V 350 981 V 350 981 V 351 981 V 351 981 V 351 980 V 352 980 V 352 980 V 353 980 V 353 980 V 353 979 V 354 979 V 354 979 V 355 979 V 355 979 V 355 979 V 356 978 V 356 978 V 356 978 V 357 978 V 357 978 V 358 978 V 358 977 V 358 977 V 359 977 V 359 977 V 360 977 V 360 976 V 361 976 V 361 976 V 361 976 V 362 976 V 362 976 V 363 975 V 363 975 V 363 975 V 364 975 V 364 975 V 365 975 V 365 975 V 366 974 V 366 974 V 366 974 V 367 974 V 367 974 V 368 974 V 368 973 V 369 973 V 369 973 V 370 973 V 370 973 V 370 973 V 371 973 V 371 972 V 372 972 V 372 972 V 373 972 V 373 972 V 374 972 V 374 971 V 375 971 V 375 971 V 376 971 V 376 971 V 376 971 V 377 971 V 377 970 V 378 970 V 378 970 V 379 970 V 379 970 V 380 970 V 380 970 V 381 969 V 381 969 V 382 969 V 382 969 V 383 969 V 383 969 V 384 969 V 384 969 V 385 968 V 385 968 V 386 968 V 386 968 V 387 968 V 387 968 V 388 968 V 388 968 V 389 967 V 389 967 V 390 967 V 390 967 V 391 967 V 391 967 V 392 967 V 392 967 V 393 966 V 394 966 V 394 966 V 395 966 V 395 966 V 396 966 V 396 966 V 397 966 V 397 965 V 398 965 V 398 965 V 399 965 V 399 965 V 400 965 V 401 965 V 401 965 V 402 965 V 402 964 V 403 964 V 403 964 V 404 964 V 404 964 V 405 964 V 406 964 V 406 964 V 407 964 V 407 963 V 408 963 V 408 963 V 409 963 V 410 963 V 410 963 V 411 963 V 411 963 V 412 963 V 412 963 V 413 962 V 414 962 V 414 962 V 415 962 V 415 962 V 416 962 V 417 962 V 417 962 V 418 962 V 418 962 V 419 962 V 420 961 V 420 961 V 421 961 V 421 961 V 422 961 V 423 961 V 423 961 V 424 961 V 425 961 V 425 961 V 426 961 V 426 961 V 427 960 V 428 960 V 428 960 V 429 960 V 430 960 V 430 960 V 431 960 V 431 960 V 432 960 V 433 960 V 433 960 V 434 960 V 435 959 V 435 959 V 436 959 V 437 959 V 437 959 V 438 959 V 439 959 V 439 959 V 440 959 V 441 959 V 441 959 V 442 959 V 443 959 V 443 959 V 444 959 V 445 958 V 445 958 V 446 958 V 447 958 V 447 958 V 448 958 V 449 958 V 449 958 V 450 958 V 451 958 V 451 958 V 452 958 V 453 958 V 453 958 V 454 958 V 455 958 V 455 958 V 456 957 V 457 957 V 458 957 V 458 957 V 459 957 V 460 957 V 460 957 V 461 957 V 462 957 V 463 957 V 463 957 V 464 957 V 465 957 V 465 957 V 466 957 V 467 957 V 468 957 V 468 957 V 469 957 V 470 957 V 470 957 V 471 957 V 472 956 V 473 956 V 473 956 V 474 956 V 475 956 V 476 956 V 476 956 V 477 956 V 478 956 V 479 956 V 479 956 V 480 956 V 481 956 V 482 956 V 482 956 V 483 956 V 484 956 V 485 956 V 485 956 V 486 956 V 487 956 V 488 956 V 489 956 V 489 956 V 490 956 V 491 956 V 492 956 V 492 956 V 493 956 V 494 956 V 495 956 V 496 956 V 496 956 V 497 956 V 498 956 V 499 956 V 500 956 V 500 956 V 501 956 V 502 955 V 503 956 V 504 955 V 504 955 V 505 955 V 506 955 V 507 955 V 508 955 V 508 955 V 509 955 V 510 955 V 511 955 V 512 955 V 512 955 V 513 955 V 514 955 V 515 955 V 516 955 V 517 955 V 309 1038 V 309 1030 V 310 1022 V 310 1014 V 311 1007 V 312 1000 V 314 992 V 315 985 V 317 979 V 320 972 V 322 966 V 325 960 V 328 954 V 331 948 V 334 943 V 338 937 V 342 932 V 347 927 V 351 923 V 356 918 V 361 914 V 366 910 V 372 906 V 378 902 V 384 899 V 390 896 V 397 893 V 404 890 V 411 887 V 418 885 V 426 883 V 434 881 V 442 879 V 450 877 V 459 876 V 468 875 V 477 874 V 487 873 V 496 873 V 506 872 V 517 872 V 309 1038 V 309 1026 V 310 1014 V 310 1002 V 311 991 V 312 980 V 314 969 V 315 959 V 317 949 V 320 939 V 322 929 V 325 920 V 328 911 V 331 903 V 334 895 V 338 887 V 342 879 V 347 872 V 351 865 V 356 858 V 361 852 V 366 846 V 372 840 V 378 834 V 384 829 V 390 824 V 397 820 V 404 816 V 411 812 V 418 808 V 426 805 V 434 802 V 442 799 V 450 797 V 459 795 V 468 793 V 477 792 V 487 791 V 496 790 V 506 790 V 517 789 V 564 951 a Ff(e)558 967 y Fk(Z)592 974 y Fb(N)558 884 y Fk(Z)592 891 y Fb(2)615 884 y Fp(\()p Fn(\024)p Fp(\))558 801 y Fk(Z)592 808 y Fb(2)615 801 y Fp(\()p Fn(\024;)g(\031)r Fp(\))p 1388 415 2 167 v 1388 290 a Fa(6)p 1388 416 167 2 v 1513 415 a(-)1596 427 y Fn(Re)p Fg(f)p Fn(C)t 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Fp(is)g(decreased.\))936 2854 y(21)p eop %%Page: 22 22 22 21 bop 60 1558 a @beginspecial 0 @llx 0 @lly 623 @urx 303 @ury 4320 @rwi @setspecial %%BeginDocument: push.eps /$F2psDict 32 dict def $F2psDict begin $F2psDict /mtrx matrix put /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix } def newpath 0 0 0 0 0 1 DrawEllipse stroke end /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 1 setlinecap 1 setlinejoin -54 93 translate 0.000000 303.000000 translate 0.900 -0.900 scale 1.000 setlinewidth % Ellipse newpath 169 379 5 5 0 360 DrawEllipse gsave 0.000 setgray fill grestore stroke % Polyline newpath 109 274 moveto 634 274 lineto stroke % Polyline newpath 419 114 moveto 419 439 lineto stroke 2.000 setlinewidth % Polyline newpath 554 139 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scalefont setfont 629 304 moveto 1 -1 scale (Re{C}) gsave 0.000 rotate show grestore 1 -1 scale /Times-Roman findfont 18.000 scalefont setfont 439 124 moveto 1 -1 scale (Re{B}) gsave 0.000 rotate show grestore 1 -1 scale /Times-Roman findfont 18.000 scalefont setfont 564 139 moveto 1 -1 scale (Re{B}=Re{C}) gsave 0.000 rotate show grestore 1 -1 scale /Times-Roman findfont 18.000 scalefont setfont 59 319 moveto 1 -1 scale (eps=0) gsave 0.000 rotate show grestore 1 -1 scale $F2psEnd %%EndDocument @endspecial 186 x Fp(Figure)21 b(2:)32 b(Sc)o(hematic)18 b(diagram)j(sho)o(wing)h(ho)o(w)g(it)f(migh)o(t)e(b)q(e)j(p)q(ossible)f (to)h(mo)o(v)o(e)d(from)h(the)60 1834 y(line)g Fn(Re)p Fg(f)p Fn(C)t Fg(g)g Fp(=)i(2)p Fn(Re)p Fg(f)p Fn(B)s Fg(g)f Fn(<)g Fp(0,)h(whic)o(h)e(w)o(e)g(kno)o(w)h(w)o(e)f(are)h(on)g (at)g Fn(\017)1370 1841 y Fl(r)1410 1834 y Fp(=)g Fn(\017)1489 1841 y Fl(i)1525 1834 y Fp(=)g(0,)h(across)f(the)60 1924 y(line)e Fn(Re)p Fg(f)p Fn(B)s Fg(g)h Fp(=)h Fn(Re)p Fg(f)p Fn(C)t Fg(g)f Fn(<)h Fp(0)g(b)o(y)f(increasing)f Fn(\017)981 1931 y Fl(r)1000 1924 y Fp(.)34 b(Compare)19 b(with)h(Figure)g(1,)i(whic)o(h)d(sho)o(ws)60 2015 y(bifurcation)d (diagrams)g(for)g(the)g(relev)m(an)o(t)g(sectors.)936 2854 y(22)p eop %%Page: 23 23 23 22 bop 60 1853 a @beginspecial 56 @llx 200 @lly 550 @urx 602 @ury 4320 @rwi @setspecial %%BeginDocument: z3_z2k.eps % 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 /sg /setgray 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 eofill } 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 { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 456 117 5934 4817 MR c np 89 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 1 j 1 sg 0 0 6917 5187 PR 6 w 0 4226 5359 0 0 -4226 899 4615 4 MP PP -5359 0 0 4226 5359 0 0 -4226 899 4615 5 MP stroke DO 4 w SO 6 w 0 sg 899 4615 mt 6258 4615 L 899 389 mt 6258 389 L 6258 4615 mt 6258 389 L 899 4615 mt 899 389 L 899 4615 mt 6258 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4561 L 899 389 mt 899 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 816 4761 mt (0.1) s 1569 4615 mt 1569 4561 L 1569 389 mt 1569 443 L 1453 4761 mt (0.15) s 2239 4615 mt 2239 4561 L 2239 389 mt 2239 443 L 2156 4761 mt (0.2) s 2909 4615 mt 2909 4561 L 2909 389 mt 2909 443 L 2793 4761 mt (0.25) s 3579 4615 mt 3579 4561 L 3579 389 mt 3579 443 L 3496 4761 mt (0.3) s 4248 4615 mt 4248 4561 L 4248 389 mt 4248 443 L 4132 4761 mt (0.35) s 4918 4615 mt 4918 4561 L 4918 389 mt 4918 443 L 4835 4761 mt (0.4) s 5588 4615 mt 5588 4561 L 5588 389 mt 5588 443 L 5472 4761 mt (0.45) s 6258 4615 mt 6258 4561 L 6258 389 mt 6258 443 L 6175 4761 mt (0.5) s 899 4615 mt 953 4615 L 6258 4615 mt 6204 4615 L 798 4659 mt (0) s 899 4192 mt 953 4192 L 6258 4192 mt 6204 4192 L 631 4236 mt (0.01) s 899 3770 mt 953 3770 L 6258 3770 mt 6204 3770 L 631 3814 mt (0.02) s 899 3347 mt 953 3347 L 6258 3347 mt 6204 3347 L 631 3391 mt (0.03) s 899 2925 mt 953 2925 L 6258 2925 mt 6204 2925 L 631 2969 mt (0.04) s 899 2502 mt 953 2502 L 6258 2502 mt 6204 2502 L 631 2546 mt (0.05) s 899 2079 mt 953 2079 L 6258 2079 mt 6204 2079 L 631 2123 mt (0.06) s 899 1657 mt 953 1657 L 6258 1657 mt 6204 1657 L 631 1701 mt (0.07) s 899 1234 mt 953 1234 L 6258 1234 mt 6204 1234 L 631 1278 mt (0.08) s 899 812 mt 953 812 L 6258 812 mt 6204 812 L 631 856 mt (0.09) s 899 389 mt 953 389 L 6258 389 mt 6204 389 L 698 433 mt (0.1) s 899 389 mt 6258 389 L 899 4615 mt 6258 4615 L 899 4615 mt 899 389 L 6258 4615 mt 6258 389 L gs 899 389 5360 4227 MR c np gr 5295 4146 mt 5345 4196 L 5345 4146 mt 5295 4196 L 4893 3766 mt 4943 3816 L 4943 3766 mt 4893 3816 L 4223 3259 mt 4273 3309 L 4273 3259 mt 4223 3309 L 3554 2752 mt 3604 2802 L 3604 2752 mt 3554 2802 L 2884 2287 mt 2934 2337 L 2934 2287 mt 2884 2337 L 2214 1906 mt 2264 1956 L 2264 1906 mt 2214 1956 L 1544 1590 mt 1594 1640 L 1594 1590 mt 1544 1640 L 874 1336 mt 924 1386 L 924 1336 mt 874 1386 L gs 899 389 5360 4227 MR c np -670 -254 -670 -316 -670 -381 -670 -465 -669 -507 -670 -507 -402 -380 5320 4171 8 MP stroke gr 1544 744 mt 1594 794 L 1594 744 mt 1544 794 L 2214 1230 mt 2264 1280 L 2264 1230 mt 2214 1280 L 2884 1780 mt 2934 1830 L 2934 1780 mt 2884 1830 L 3554 2287 mt 3604 2337 L 3604 2287 mt 3554 2337 L 4223 2878 mt 4273 2928 L 4273 2878 mt 4223 2928 L 4893 3555 mt 4943 3605 L 4943 3555 mt 4893 3605 L 5295 4024 mt 5345 4074 L 5345 4024 mt 5295 4074 L gs 899 389 5360 4227 MR c np 402 469 670 677 669 591 670 507 670 550 670 486 1569 769 7 MP stroke gr /Symbol /ISOLatin1Encoding 120 FMSR 3552 4897 mt (e) s 577 2629 mt -90 rotate (l) s 90 rotate /Helvetica /ISOLatin1Encoding 120 FMSR 577 2564 mt -90 rotate (+3) s 90 rotate /Symbol /ISOLatin1Encoding 120 FMSR 577 2428 mt -90 rotate (e) s 90 rotate /Helvetica /ISOLatin1Encoding 120 FMSR 2381 3538 mt (F) s 4790 1832 mt (G) s 1853 1398 mt (H) s end eplot epage end showpage %%EndDocument @endspecial 186 x Fp(Figure)18 b(3:)25 b(T)l(ransition)19 b(from)e(Hopf)i(bifurcation)e(to)i(a)g(stable)1258 2022 y Ff(e)1252 2039 y Fk(Z)1286 2046 y Fh(3)1324 2039 y Fp(orbit)f(to)h(Hopf)f(bifurcation)60 2129 y(to)g(a)f(stable)h Fk(Z)339 2136 y Fh(2)359 2129 y Fp(\()p Fn(\024;)8 b(\031)r Fp(\))16 b(orbit)i(as)g Fn(\017)f Fp(is)g(v)m(aried)g(in)g(equation)g (\(18\).)26 b(In)17 b(region)g(F)g(the)1641 2113 y Ff(e)1636 2129 y Fk(Z)1670 2136 y Fh(3)1707 2129 y Fp(orbit)g(is)60 2219 y(stable,)e(and)h(in)f(region)g(G,)h(the)f Fk(Z)694 2226 y Fh(2)714 2219 y Fp(\()p Fn(\024;)8 b(\031)r Fp(\))14 b(orbit)i(is)f(stable.)21 b(There)15 b(is)g(non{p)q(erio)q(dic)h(b)q (eha)o(viour)60 2309 y(in)c(the)h(w)o(edge)f(H.)g(The)h(v)o(ertical)d (co)q(ordinate)k(is)e(the)g(distance)h(in)f Fn(\025)h Fp(from)f(the)g(Hopf)h(bifurcation.)936 2854 y(23)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF