56:
372:
908:
229:
1061: ≠0, the eigenvalues have non-zero real part. Thus, this equilibrium point is a hyperbolic equilibrium point. The linearized system will behave similar to the non-linear system near (0, 0). When
630:
559:
1055:
691:
148:
786:
483:
430:
781:
206:
997:
1073:
In the case of an infinite dimensional system—for example systems involving a time delay—the notion of the "hyperbolic part of the spectrum" refers to the above property.
367:{\displaystyle {\begin{bmatrix}x_{n+1}\\y_{n+1}\end{bmatrix}}={\begin{bmatrix}1&1\\1&2\end{bmatrix}}{\begin{bmatrix}x_{n}\\y_{n}\end{bmatrix}}}
1092:
565:
494:
1005:
1178:
1153:
1124:
646:
103:
903:{\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=y,\\{\frac {dy}{dt}}&=-x-x^{3}-\alpha y,~\alpha \neq 0\end{aligned}}}
436:
383:
209:
752:
756:
1226:
159:
29:
176:
913:(0, 0) is the only equilibrium point. The Jacobian matrix of the linearization at the equilibrium point is
52:'—but it has become standard." Several properties hold about a neighborhood of a hyperbolic point, notably
760:
41:
88:
1221:
919:
220:
37:
1174:
1149:
1120:
216:
78:
17:
717:
171:
71:
65:
45:
33:
1142:
1087:
1215:
1113:
764:
55:
1203:
697:
59:
Orbits near a two-dimensional saddle point, an example of a hyperbolic equilibrium.
49:
1198:
1082:
1065: = 0, the system has a nonhyperbolic equilibrium at (0, 0).
949:
48:
notes that "hyperbolic is an unfortunate name—it sounds like it should mean '
54:
625:{\displaystyle \lambda _{2}={\frac {\ln(3-{\sqrt {5}})}{2}}<1}
554:{\displaystyle \lambda _{1}={\frac {\ln(3+{\sqrt {5}})}{2}}>1}
1050:{\displaystyle {\frac {-\alpha \pm {\sqrt {\alpha ^{2}-4}}}{2}}}
44:
system resemble hyperbolas. This fails to hold in general.
686:{\displaystyle F\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}
143:{\displaystyle T\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}
755:
states that the orbit structure of a dynamical system in a
77:
The dynamics on the invariant set can be represented via
478:{\displaystyle \lambda _{2}={\frac {3-{\sqrt {5}}}{2}}}
425:{\displaystyle \lambda _{1}={\frac {3+{\sqrt {5}}}{2}}}
329:
293:
238:
1008:
922:
784:
649:
568:
497:
439:
386:
232:
179:
106:
1141:
1112:
1049:
991:
902:
685:
624:
553:
477:
424:
366:
200:
142:
740:. Hyperbolic fixed points may also be called
732:has no eigenvalues with zero real parts then
8:
1169:Abraham, Ralph; Marsden, Jerrold E. (1978).
1027:
1021:
1009:
1007:
948:
921:
866:
826:
789:
785:
783:
677:
673:
672:
662:
658:
657:
648:
600:
582:
573:
567:
529:
511:
502:
496:
488:We know that the Lyapunov exponents are:
462:
453:
444:
438:
409:
400:
391:
385:
350:
336:
324:
288:
265:
245:
233:
231:
178:
134:
130:
129:
119:
115:
114:
105:
219:whose only fixed point is hyperbolic is
1103:
40:point the orbits of a two-dimensional,
1093:Normally hyperbolic invariant manifold
201:{\displaystyle \operatorname {D} T(p)}
759:of a hyperbolic equilibrium point is
7:
1173:. Reading Mass.: Benjamin/Cummings.
1002:The eigenvalues of this matrix are
377:Since the eigenvalues are given by
180:
14:
84:A natural measure can be defined,
635:Therefore it is a saddle point.
68:and an unstable manifold exist,
1148:. Cambridge University Press.
938:
926:
775:Consider the nonlinear system
763:to the orbit structure of the
668:
607:
591:
536:
520:
195:
189:
125:
1:
992:{\displaystyle J(0,0)=\left.}
1197:Eugene M. Izhikevich (ed.).
1115:Nonlinear Dynamics and Chaos
212:on the complex unit circle.
22:hyperbolic equilibrium point
1243:
1144:Chaos in Dynamical Systems
746:elementary critical points
742:hyperbolic critical points
1111:Strogatz, Steven (2001).
1171:Foundations of Mechanics
761:topologically equivalent
753:Hartman–Grobman theorem
32:that does not have any
1051:
993:
904:
700:with a critical point
687:
626:
555:
479:
426:
368:
202:
168:hyperbolic fixed point
144:
60:
26:hyperbolic fixed point
1052:
994:
905:
688:
627:
556:
480:
427:
369:
203:
145:
58:
1140:Ott, Edward (1994).
1057:. For all values of
1006:
920:
782:
647:
566:
495:
437:
384:
230:
177:
104:
89:structurally stable
1119:. Westview Press.
1047:
989:
980:
900:
898:
767:dynamical system.
683:
622:
551:
475:
422:
364:
358:
318:
279:
198:
140:
61:
1045:
1039:
886:
844:
807:
614:
605:
543:
534:
473:
467:
420:
414:
215:One example of a
79:symbolic dynamics
18:dynamical systems
1234:
1227:Stability theory
1208:
1185:
1184:
1166:
1160:
1159:
1147:
1137:
1131:
1130:
1118:
1108:
1056:
1054:
1053:
1048:
1046:
1041:
1040:
1032:
1031:
1022:
1010:
998:
996:
995:
990:
985:
981:
909:
907:
906:
901:
899:
884:
871:
870:
845:
843:
835:
827:
808:
806:
798:
790:
728:. If the matrix
692:
690:
689:
684:
682:
681:
676:
667:
666:
661:
631:
629:
628:
623:
615:
610:
606:
601:
583:
578:
577:
560:
558:
557:
552:
544:
539:
535:
530:
512:
507:
506:
484:
482:
481:
476:
474:
469:
468:
463:
454:
449:
448:
431:
429:
428:
423:
421:
416:
415:
410:
401:
396:
395:
373:
371:
370:
365:
363:
362:
355:
354:
341:
340:
323:
322:
284:
283:
276:
275:
256:
255:
221:Arnold's cat map
207:
205:
204:
199:
166:is said to be a
149:
147:
146:
141:
139:
138:
133:
124:
123:
118:
34:center manifolds
16:In the study of
1242:
1241:
1237:
1236:
1235:
1233:
1232:
1231:
1212:
1211:
1196:
1193:
1188:
1181:
1168:
1167:
1163:
1156:
1139:
1138:
1134:
1127:
1110:
1109:
1105:
1101:
1079:
1071:
1023:
1011:
1004:
1003:
979:
978:
970:
961:
960:
955:
944:
918:
917:
897:
896:
862:
846:
836:
828:
823:
822:
809:
799:
791:
780:
779:
773:
718:Jacobian matrix
712:) = 0, and let
671:
656:
645:
644:
641:
584:
569:
564:
563:
513:
498:
493:
492:
455:
440:
435:
434:
402:
387:
382:
381:
357:
356:
346:
343:
342:
332:
325:
317:
316:
311:
305:
304:
299:
289:
278:
277:
261:
258:
257:
241:
234:
228:
227:
175:
174:
172:Jacobian matrix
128:
113:
102:
101:
98:
66:stable manifold
42:non-dissipative
12:
11:
5:
1240:
1238:
1230:
1229:
1224:
1214:
1213:
1210:
1209:
1192:
1189:
1187:
1186:
1179:
1161:
1154:
1132:
1125:
1102:
1100:
1097:
1096:
1095:
1090:
1088:Hyperbolic set
1085:
1078:
1075:
1070:
1067:
1044:
1038:
1035:
1030:
1026:
1020:
1017:
1014:
1000:
999:
988:
984:
977:
974:
971:
969:
966:
963:
962:
959:
956:
954:
951:
950:
947:
943:
940:
937:
934:
931:
928:
925:
911:
910:
895:
892:
889:
883:
880:
877:
874:
869:
865:
861:
858:
855:
852:
849:
847:
842:
839:
834:
831:
825:
824:
821:
818:
815:
812:
810:
805:
802:
797:
794:
788:
787:
772:
769:
680:
675:
670:
665:
660:
655:
652:
640:
637:
633:
632:
621:
618:
613:
609:
604:
599:
596:
593:
590:
587:
581:
576:
572:
561:
550:
547:
542:
538:
533:
528:
525:
522:
519:
516:
510:
505:
501:
486:
485:
472:
466:
461:
458:
452:
447:
443:
432:
419:
413:
408:
405:
399:
394:
390:
375:
374:
361:
353:
349:
345:
344:
339:
335:
331:
330:
328:
321:
315:
312:
310:
307:
306:
303:
300:
298:
295:
294:
292:
287:
282:
274:
271:
268:
264:
260:
259:
254:
251:
248:
244:
240:
239:
237:
197:
194:
191:
188:
185:
182:
137:
132:
127:
122:
117:
112:
109:
97:
94:
93:
92:
87:The system is
85:
82:
75:
69:
13:
10:
9:
6:
4:
3:
2:
1239:
1228:
1225:
1223:
1220:
1219:
1217:
1206:
1205:
1200:
1199:"Equilibrium"
1195:
1194:
1190:
1182:
1180:0-8053-0102-X
1176:
1172:
1165:
1162:
1157:
1155:0-521-43799-7
1151:
1146:
1145:
1136:
1133:
1128:
1126:0-7382-0453-6
1122:
1117:
1116:
1107:
1104:
1098:
1094:
1091:
1089:
1086:
1084:
1081:
1080:
1076:
1074:
1068:
1066:
1064:
1060:
1042:
1036:
1033:
1028:
1024:
1018:
1015:
1012:
986:
982:
975:
972:
967:
964:
957:
952:
945:
941:
935:
932:
929:
923:
916:
915:
914:
893:
890:
887:
881:
878:
875:
872:
867:
863:
859:
856:
853:
850:
848:
840:
837:
832:
829:
819:
816:
813:
811:
803:
800:
795:
792:
778:
777:
776:
770:
768:
766:
762:
758:
757:neighbourhood
754:
749:
747:
743:
739:
735:
731:
727:
723:
719:
715:
711:
707:
703:
699:
696:
678:
663:
653:
650:
638:
636:
619:
616:
611:
602:
597:
594:
588:
585:
579:
574:
570:
562:
548:
545:
540:
531:
526:
523:
517:
514:
508:
503:
499:
491:
490:
489:
470:
464:
459:
456:
450:
445:
441:
433:
417:
411:
406:
403:
397:
392:
388:
380:
379:
378:
359:
351:
347:
337:
333:
326:
319:
313:
308:
301:
296:
290:
285:
280:
272:
269:
266:
262:
252:
249:
246:
242:
235:
226:
225:
224:
222:
218:
213:
211:
192:
186:
183:
173:
169:
165:
161:
157:
153:
135:
120:
110:
107:
95:
90:
86:
83:
80:
76:
73:
70:
67:
63:
62:
57:
53:
51:
47:
43:
39:
35:
31:
27:
23:
19:
1204:Scholarpedia
1202:
1170:
1164:
1143:
1135:
1114:
1106:
1072:
1062:
1058:
1001:
912:
774:
750:
745:
741:
737:
733:
729:
725:
721:
713:
709:
705:
701:
698:vector field
694:
642:
634:
487:
376:
214:
167:
163:
155:
151:
99:
50:saddle point
25:
21:
15:
1083:Anosov flow
716:denote the
210:eigenvalues
160:fixed point
30:fixed point
1222:Limit sets
1216:Categories
1191:References
765:linearized
738:hyperbolic
736:is called
38:hyperbolic
1034:−
1025:α
1019:±
1016:α
1013:−
976:α
973:−
965:−
891:≠
888:α
876:α
873:−
860:−
854:−
669:→
654::
598:−
589:
571:λ
518:
500:λ
460:−
442:λ
389:λ
184:
170:when the
126:→
111::
72:Shadowing
36:. Near a
1077:See also
1069:Comments
704:, i.e.,
154:map and
46:Strogatz
771:Example
208:has no
74:occurs,
1177:
1152:
1123:
885:
1099:Notes
693:be a
639:Flows
162:then
158:is a
150:is a
28:is a
1175:ISBN
1150:ISBN
1121:ISBN
751:The
643:Let
617:<
546:>
96:Maps
20:, a
744:or
724:at
720:of
217:map
100:If
24:or
1218::
1201:.
748:.
586:ln
515:ln
223::
64:A
1207:.
1183:.
1158:.
1129:.
1063:α
1059:α
1043:2
1037:4
1029:2
987:.
983:]
968:1
958:1
953:0
946:[
942:=
939:)
936:0
933:,
930:0
927:(
924:J
894:0
882:,
879:y
868:3
864:x
857:x
851:=
841:t
838:d
833:y
830:d
820:,
817:y
814:=
804:t
801:d
796:x
793:d
734:p
730:J
726:p
722:F
714:J
710:p
708:(
706:F
702:p
695:C
679:n
674:R
664:n
659:R
651:F
620:1
612:2
608:)
603:5
595:3
592:(
580:=
575:2
549:1
541:2
537:)
532:5
527:+
524:3
521:(
509:=
504:1
471:2
465:5
457:3
451:=
446:2
418:2
412:5
407:+
404:3
398:=
393:1
360:]
352:n
348:y
338:n
334:x
327:[
320:]
314:2
309:1
302:1
297:1
291:[
286:=
281:]
273:1
270:+
267:n
263:y
253:1
250:+
247:n
243:x
236:[
196:)
193:p
190:(
187:T
181:D
164:p
156:p
152:C
136:n
131:R
121:n
116:R
108:T
91:.
81:,
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
