166:
582:
Bragin, V. O.; Vagaitsev, V.I.; Kuznetsov, N. V.; Leonov, G.A. (2011). "Algorithms for
Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits".
295:
362:
418:
228:
613:"Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits"
260:
193:
392:
318:
66:
86:
The conjecture is true for the two-dimensional case. However, counterexamples have been constructed in higher dimensions. Hence, in the two-dimensional case
670:
665:
121:
33:
655:
102:
569:
Bernat, Josep; Llibre, Jaume (1996). "Counterexample to Kalman and Markus–Yamabe
Conjectures in dimension larger than 3".
507:
660:
76:
536:
265:
106:
40:, then the fixed point is asymptotically stable. Markus-Yamabe conjecture asks if a similar result holds
624:
479:
323:
29:
599:
530:
Cima, Anna; van den Essen, Arno; Gasull, Armengol; Hubbers, Engelbert; Mañosas, Francesc (1997).
105:. Analog of the conjecture for nonlinear control system with scalar nonlinearity is known as
632:
591:
555:
545:
516:
487:
397:
198:
233:
171:
80:
45:
371:
628:
503:"A proof of the two-dimensional Markus–Yamabe stability conjecture and a generalisation"
483:
448:
303:
51:
37:
492:
467:
649:
603:
72:
431:
69:
17:
637:
612:
595:
25:
550:
531:
97:
Related mathematical results concerning global asymptotic stability, which
521:
502:
468:"A solution to the bidimensional Global Asymptotic Stability Conjecture"
560:
83:
is everywhere
Hurwitz, then the fixed point is globally stable.
161:{\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}}
532:"A Polynomial Counterexample to the Markus–Yamabe Conjecture"
101:
applicable in dimensions higher than two, include various
571:
Dynamics of
Continuous, Discrete & Impulsive Systems
584:
Journal of
Computer and Systems Sciences International
400:
374:
326:
306:
268:
236:
201:
174:
124:
54:
432:"Global Stability Criteria for Differential Systems"
412:
386:
356:
312:
289:
254:
222:
187:
160:
60:
617:International Journal of Bifurcation and Chaos
320:is a global attractor of the dynamical system
449:"A Biography of the Markus–Yamabe Conjecture"
44:. Precisely, the conjecture states that if a
8:
430:Markus, Lawrence; Yamabe, Hidehiko (1960).
36:of a dynamical system at a fixed point is
636:
559:
549:
520:
491:
399:
373:
328:
327:
325:
305:
281:
277:
276:
267:
235:
200:
179:
173:
152:
148:
147:
137:
133:
132:
123:
53:
611:Leonov, G. A.; Kuznetsov, N. V. (2013).
472:Annales de l'Institut Henri Poincaré C
290:{\displaystyle x\in \mathbb {R} ^{n}}
7:
113:Mathematical statement of conjecture
90:, it can also be referred to as the
262:which is Hurwitz stable for every
14:
357:{\displaystyle {\dot {x}}=f(x)}
103:autonomous convergence theorems
351:
345:
249:
243:
211:
205:
143:
1:
671:Theorems in dynamical systems
493:10.1016/S0294-1449(16)30147-0
508:Annales Polonici Mathematici
368:The conjecture is true for
46:continuously differentiable
687:
666:Fixed points (mathematics)
466:Gutierrez, Carlos (1995).
436:Osaka Mathematical Journal
638:10.1142/S0218127413300024
596:10.1134/S106423071104006X
394:and false in general for
22:Markus–Yamabe conjecture
537:Advances in Mathematics
501:Feßler, Robert (1995).
447:Meisters, Gary (1996).
623:(1): 1330002–1330219.
551:10.1006/aima.1997.1673
414:
413:{\displaystyle n>2}
388:
358:
314:
291:
256:
224:
223:{\displaystyle f(0)=0}
189:
162:
62:
656:Disproved conjectures
522:10.4064/ap-62-1-45-74
415:
389:
359:
315:
292:
257:
255:{\displaystyle Df(x)}
225:
190:
188:{\displaystyle C^{1}}
163:
92:Markus–Yamabe theorem
63:
398:
372:
324:
304:
266:
234:
199:
172:
122:
52:
30:asymptotic stability
629:2013IJBC...2330002L
484:1995AIHPC..12..627G
387:{\displaystyle n=2}
107:Kalman's conjecture
410:
384:
354:
310:
287:
252:
220:
185:
158:
58:
336:
313:{\displaystyle 0}
61:{\displaystyle n}
678:
661:Stability theory
642:
640:
607:
578:
565:
563:
553:
526:
524:
497:
495:
462:
460:
458:
453:
443:
419:
417:
416:
411:
393:
391:
390:
385:
363:
361:
360:
355:
338:
337:
329:
319:
317:
316:
311:
296:
294:
293:
288:
286:
285:
280:
261:
259:
258:
253:
229:
227:
226:
221:
194:
192:
191:
186:
184:
183:
167:
165:
164:
159:
157:
156:
151:
142:
141:
136:
67:
65:
64:
59:
686:
685:
681:
680:
679:
677:
676:
675:
646:
645:
610:
581:
568:
529:
500:
465:
456:
454:
451:
446:
429:
426:
396:
395:
370:
369:
322:
321:
302:
301:
275:
264:
263:
232:
231:
197:
196:
175:
170:
169:
146:
131:
120:
119:
115:
81:Jacobian matrix
50:
49:
34:Jacobian matrix
12:
11:
5:
684:
682:
674:
673:
668:
663:
658:
648:
647:
644:
643:
608:
590:(5): 511–543.
579:
566:
544:(2): 453–457.
527:
498:
478:(6): 627–671.
463:
444:
425:
422:
409:
406:
403:
383:
380:
377:
366:
365:
353:
350:
347:
344:
341:
335:
332:
309:
298:
284:
279:
274:
271:
251:
248:
245:
242:
239:
219:
216:
213:
210:
207:
204:
182:
178:
155:
150:
145:
140:
135:
130:
127:
114:
111:
57:
13:
10:
9:
6:
4:
3:
2:
683:
672:
669:
667:
664:
662:
659:
657:
654:
653:
651:
639:
634:
630:
626:
622:
618:
614:
609:
605:
601:
597:
593:
589:
585:
580:
577:(3): 337–379.
576:
572:
567:
562:
557:
552:
547:
543:
539:
538:
533:
528:
523:
518:
514:
510:
509:
504:
499:
494:
489:
485:
481:
477:
473:
469:
464:
450:
445:
442:(2): 305–317.
441:
437:
433:
428:
427:
423:
421:
407:
404:
401:
381:
378:
375:
348:
342:
339:
333:
330:
307:
299:
282:
272:
269:
246:
240:
237:
230:and Jacobian
217:
214:
208:
202:
180:
176:
153:
138:
128:
125:
117:
116:
112:
110:
108:
104:
100:
95:
93:
89:
84:
82:
78:
74:
71:
68:-dimensional
55:
47:
43:
39:
35:
31:
27:
23:
19:
620:
616:
587:
583:
574:
570:
541:
535:
512:
506:
475:
471:
455:. Retrieved
439:
435:
367:
98:
96:
91:
87:
85:
73:vector space
41:
21:
15:
561:2066/112453
457:October 20,
77:fixed point
18:mathematics
650:Categories
424:References
79:, and its
48:map on an
28:on global
26:conjecture
515:: 45–74.
334:˙
273:∈
195:map with
144:→
32:. If the
604:21657305
42:globally
625:Bibcode
480:Bibcode
38:Hurwitz
602:
75:has a
20:, the
600:S2CID
452:(PDF)
300:Then
168:be a
24:is a
459:2023
405:>
118:Let
88:only
70:real
633:doi
592:doi
556:hdl
546:doi
542:131
517:doi
488:doi
99:are
16:In
652::
631:.
621:23
619:.
615:.
598:.
588:50
586:.
573:.
554:.
540:.
534:.
513:62
511:.
505:.
486:.
476:12
474:.
470:.
440:12
438:.
434:.
420:.
109:.
94:.
641:.
635::
627::
606:.
594::
575:2
564:.
558::
548::
525:.
519::
496:.
490::
482::
461:.
408:2
402:n
382:2
379:=
376:n
364:.
352:)
349:x
346:(
343:f
340:=
331:x
308:0
297:.
283:n
278:R
270:x
250:)
247:x
244:(
241:f
238:D
218:0
215:=
212:)
209:0
206:(
203:f
181:1
177:C
154:n
149:R
139:n
134:R
129::
126:f
56:n
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.