20:
831:
1671:
1680:, named after V. N. Nikolaevsky who introudced the equation in 1989, whereas the first two equations has been introduced recently in the context of transitions near tricritical points, i.e., change in the sign of the fourth derivative term with the plus sign approaching a Kuramoto–Sivashinsky type and the minus sign approaching a
63:
flame front. It was later and independently derived by G. M. Homsy and A. A. Nepomnyashchii in 1974, in connection with the stability of liquid film on an inclined plane and by R. E. LaQuey et. al. in 1975 in connection with trapped-ion instability. The
Kuramoto–Sivashinsky equation is known for its
1310:
2432:
Nikolaevskii, V. N. (1989). Dynamics of viscoelastic media with internal oscillators. In Recent
Advances in Engineering Science: A Symposium dedicated to A. Cemal Eringen June 20–22, 1988, Berkeley, California (pp. 210-221). Berlin, Heidelberg: Springer Berlin
1290:
432:
1666:{\displaystyle {\begin{aligned}u_{t}+qu_{xx}+u_{xxxx}-u_{xxxxxx}+uu_{x}&=0,\quad q>0,\\u_{t}+u_{xx}-u_{xxxxxx}+uu_{x}&=0,\\u_{t}+qu_{xx}-u_{xxxx}-u_{xxxxxx}+uu_{x}&=0,\quad q>-1/4\\\end{aligned}}}
1106:
176:
985:
A third-order derivative term represneting dispersion of wavenumbers are often encountered in many applications. The disperseively modified
Kuramoto–Sivashinsky equation, which is often called as the
2379:
Akrivis, G., Papageorgiou, D. T., & Smyrlis, Y. S. (2012). Computational study of the dispersively modified
Kuramoto–Sivashinsky equation. SIAM Journal on Scientific Computing, 34(2), A792-A813.
1315:
860:. After some time the system returns to its initial state, only translated slightly (~4 units) to the left. This particular solution has three unstable directions and three marginal directions.
269:
588:
894:
1136:
680:
486:
1714:
820:
455:
325:
779:
630:
541:
2451:
Matthews, P. C., & Cox, S. M. (2000). One-dimensional pattern formation with
Galilean invariance near a stationary bifurcation. Physical Review E, 62(2), R1473.
858:
966:
942:
918:
740:
720:
700:
292:
1144:
340:
2345:
Chang, H. C.; Demekhin, E. A.; Kopelevich, D. I. (1993). "Laminarizing effects of dispersion in an active-dissipative nonlinear medium".
2169:; Davidchack, Ruslan L.; Siminos, Evangelos (2010). "On the State Space Geometry of the Kuramoto–Sivashinsky Flow in a Periodic Domain".
1991:"Backpropagation algorithms and Reservoir Computing in Recurrent Neural Networks for the forecasting of complex spatiotemporal dynamics"
1709:
995:
82:
56:
334:
The
Kuramoto–Sivashinsky equation can also be generalized to higher dimensions. In spatially periodic domains, one possibility is
1138:
is real parameter. A fifth-order derivative term is also often included, which is the modified
Kawahara equation and is given by
19:
2553:
1789:
Sivashinsky, G.I. (1977). "Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations".
722:
is a velocity, this change of variable describes a transformation into a frame that is moving with constant relative velocity
2442:
Tribelsky, M. I., & Tsuboi, K. (1996). New scenario for transition to turbulence?. Physical review letters, 76(10), 1631.
1693:
44:
1681:
864:
Solutions of the
Kuramoto–Sivashinsky equation possess rich dynamical characteristics. Considered on a periodic domain
969:
187:
1863:. Lectures in Applied Mathematics. Vol. 15. Providence: American Mathematical Society. pp. 191–194.
1692:
Applications of the
Kuramoto–Sivashinsky equation extend beyond its original context of flame propagation and
546:
2558:
834:
A converged relative periodic orbit for the KS equation with periodic boundary conditions for a domain size
1942:"Model-Free Prediction of Large Spatiotemporally Chaotic Systems from Data: A Reservoir Computing Approach"
1989:
Vlachas, P.R.; Pathak, J.; Hunt, B.R.; Sapsis, T.P.; Girvan, M.; Ott, E.; Koumoutsakos, P. (2020-03-21).
2568:
1300:
Three forms of the sixth-order
Kuramoto–Sivashinsky equations are encountered in applications involving
2166:
1882:
Nepomnyashchii, A. A. (1975). "Stability of wavy conditions in a film flowing down an inclined plane".
2260:
Papageorgiou, D.T.; Smyrlis, Y.S. (1991), "The route to chaos for the Kuramoto-Sivashinsky equation",
1864:
1724:
1719:
2563:
1729:
867:
743:
594:
489:
52:
2529:
2504:
2470:
2295:
2204:
2178:
2038:
2002:
1114:
897:
635:
505:
464:
2526:
2496:
2488:
2461:
Cuerno, Rodolfo; Barabási, Albert-László (1995). "Dynamic Scaling of Ion-Sputtered Surfaces".
2415:
2362:
2287:
2239:
2196:
2143:
2098:
2080:
2030:
2022:
1971:
1963:
1841:
1806:
1771:
1301:
784:
440:
297:
2480:
2405:
2354:
2325:
2277:
2269:
2231:
2188:
2133:
2125:
2088:
2072:
2012:
1953:
1922:
1891:
1833:
1798:
1761:
1697:
749:
600:
511:
458:
48:
1910:
945:
837:
1868:
1859:
Homsy, G. M. (1974). "Model equations for wavy viscous film flow". In Newell, A. (ed.).
830:
2093:
2060:
1285:{\displaystyle u_{t}+u_{xx}+\delta _{3}u_{xxx}+u_{xxxx}+\delta _{5}u_{xxxxx}+uu_{x}=0.}
951:
927:
903:
725:
705:
685:
501:
328:
277:
2547:
2358:
2299:
2235:
2042:
1824:
Sivashinsky, G. I. (1980). "On Flame Propagation Under Conditions of Stoichiometry".
1802:
2508:
2208:
2222:
Michelson, Daniel (1986). "Steady solutions of the Kuramoto-Sivashinsky equation".
2065:
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
1958:
1941:
921:
65:
60:
2061:"An in-depth numerical study of the two-dimensional Kuramoto–Sivashinsky equation"
2410:
2393:
2116:
Tadmor, Eitan (1986). "The Well-Posedness of the Kuramoto–Sivashinsky Equation".
2017:
1990:
2484:
1940:
Pathak, Jaideep; Hunt, Brian; Girvan, Michelle; Lu, Zhixin; Ott, Edward (2018).
28:
1926:
2492:
2419:
2366:
2291:
2243:
2200:
2147:
2084:
2026:
1967:
1845:
1810:
1775:
2534:
2394:"Tricritical point as a crossover between type-Is and type-IIs bifurcations"
2282:
968:
is increased. In particular, the transition to chaos occurs by a cascade of
2500:
2102:
2076:
2034:
1975:
427:{\displaystyle u_{t}+\Delta u+\Delta ^{2}u+{\frac {1}{2}}|\nabla u|^{2}=0,}
1696:. These additional applications include flows in pipes and at interfaces,
23:
A spatiotemporal plot of a simulation of the Kuramoto–Sivashinsky equation
2475:
2330:
2313:
1766:
1749:
2273:
1895:
2192:
1909:
Laquey, R. E.; Mahajan, S. M.; Rutherford, P. H.; Tang, W. M. (1975).
2138:
2129:
1837:
1700:, chemical reaction dynamics, and models of ion-sputtered surfaces.
2314:"Approximate equations for long nonlinear waves on a viscous fluid"
2007:
2183:
829:
1101:{\displaystyle u_{t}+u_{xx}+\delta _{3}u_{xxx}+u_{xxxx}+uu_{x}=0}
171:{\displaystyle u_{t}+u_{xx}+u_{xxxx}+{\frac {1}{2}}u_{x}^{2}=0}
944:, solutions may include equilibria, relative equilibria, and
2161:
2159:
2157:
2059:
Kalogirou, A.; Keaveny, E. E.; Papageorgiou, D. T. (2015).
55:, who derived the equation in the late 1970s to model the
508:
in the sense of Hadamard—that is, for given initial data
948:—all of which typically become dynamically unstable as
76:
The 1d version of the Kuramoto–Sivashinsky equation is
2054:
2052:
1313:
1147:
1117:
998:
954:
930:
906:
870:
840:
787:
752:
728:
708:
688:
638:
603:
549:
514:
467:
443:
343:
300:
280:
190:
85:
2255:
2253:
1665:
1284:
1130:
1100:
960:
936:
912:
888:
852:
814:
773:
734:
714:
694:
674:
624:
582:
535:
480:
449:
426:
319:
286:
263:
170:
1676:in which the last equation is referred to as the
1715:List of nonlinear partial differential equations
742:. On a periodic domain, the equation also has a
593:The 1d Kuramoto–Sivashinsky equation possesses
590:that depends continuously on the initial data.
1911:"Nonlinear Saturation of the Trapped-Ion Mode"
264:{\displaystyle v_{t}+v_{xx}+v_{xxxx}+vv_{x}=0}
1750:"Diffusion-Induced Chaos in Reaction Systems"
8:
702:is an arbitrary constant. Physically, since
504:for the 1d Kuramoto–Sivashinsky equation is
274:obtained by differentiating with respect to
1754:Progress of Theoretical Physics Supplement
920:is increased, culminating in the onset of
2474:
2409:
2329:
2281:
2182:
2171:SIAM Journal on Applied Dynamical Systems
2137:
2092:
2016:
2006:
1957:
1765:
1651:
1619:
1588:
1566:
1550:
1534:
1507:
1476:
1460:
1447:
1407:
1376:
1354:
1338:
1322:
1314:
1312:
1270:
1242:
1232:
1210:
1191:
1181:
1165:
1152:
1146:
1122:
1116:
1086:
1061:
1042:
1032:
1016:
1003:
997:
981:Dispersive Kuramoto–Sivashinsky equations
953:
929:
905:
869:
839:
786:
751:
727:
707:
687:
637:
602:
548:
513:
472:
466:
442:
409:
404:
392:
382:
370:
348:
342:
311:
299:
279:
249:
224:
208:
195:
189:
156:
151:
137:
119:
103:
90:
84:
2318:Journal of the Physical Society of Japan
18:
1740:
583:{\displaystyle u(x,0\leq t<\infty )}
976:Modified Kuramoto–Sivashinsky equation
2387:
2385:
2118:SIAM Journal on Mathematical Analysis
896:, the dynamics undergoes a series of
7:
924:behavior. Depending on the value of
2392:Rajamanickam, P.; Daou, J. (2023).
1826:SIAM Journal on Applied Mathematics
574:
469:
444:
397:
367:
357:
14:
2312:Topper, J.; Kawahara, T. (1978).
543:, there exists a unique solution
2530:"Kuramoto-Sivashinsky Equation"
2262:Theoret. Comput. Fluid Dynamics
1638:
1426:
57:diffusive–thermal instabilities
2347:Physica D: Nonlinear Phenomena
2224:Physica D: Nonlinear Phenomena
1959:10.1103/PhysRevLett.120.024102
1710:Michelson–Sivashinsky equation
809:
794:
768:
756:
663:
642:
619:
607:
577:
553:
530:
518:
405:
393:
43:) is a fourth-order nonlinear
1:
889:{\displaystyle 0\leq x\leq L}
45:partial differential equation
33:Kuramoto–Sivashinsky equation
16:Partial differential equation
2411:10.13023/psmij.2023.04-01-02
2359:10.1016/0167-2789(93)90113-F
2236:10.1016/0167-2789(86)90055-2
2018:10.1016/j.neunet.2020.02.016
1803:10.1016/0094-5765(77)90096-0
970:period-doubling bifurcations
2485:10.1103/PhysRevLett.74.4746
1131:{\displaystyle \delta _{3}}
675:{\displaystyle u(x-ct,t)-c}
481:{\displaystyle \Delta ^{2}}
327:. This is the form used in
2585:
2398:Progress in Scale Modeling
1927:10.1103/PhysRevLett.34.391
1748:Kuramoto, Yoshiki (1978).
1694:reaction–diffusion systems
632:is a solution, then so is
815:{\displaystyle -u(-x,t)}
2463:Physical Review Letters
1946:Physical Review Letters
1915:Physical Review Letters
450:{\displaystyle \Delta }
320:{\displaystyle v=u_{x}}
2554:Differential equations
2077:10.1098/rspa.2014.0932
1667:
1286:
1132:
1102:
962:
938:
914:
890:
861:
854:
816:
775:
774:{\displaystyle u(x,t)}
736:
716:
696:
676:
626:
625:{\displaystyle u(x,t)}
584:
537:
536:{\displaystyle u(x,0)}
482:
451:
428:
321:
288:
265:
172:
24:
1861:Nonlinear Wave Motion
1668:
1304:, which are given by
1296:Sixth-order equations
1287:
1133:
1103:
963:
939:
915:
891:
855:
833:
817:
776:
737:
717:
697:
677:
627:
585:
538:
483:
452:
429:
322:
289:
266:
181:An alternate form is
173:
22:
2331:10.1143/JPSJ.44.2003
1797:(11–12): 1177–1206.
1720:List of chaotic maps
1682:Ginzburg–Landau type
1678:Nikolaevsky equation
1311:
1145:
1115:
996:
952:
928:
904:
868:
853:{\displaystyle L=35}
838:
822:is also a solution.
785:
781:is a solution, then
750:
726:
706:
686:
636:
601:
547:
512:
465:
441:
341:
298:
278:
188:
83:
47:. It is named after
2167:Cvitanović, Predrag
1869:1974LApM...15.....N
1767:10.1143/PTPS.64.346
1730:Laminar flame speed
900:as the domain size
744:reflection symmetry
595:Galilean invariance
490:biharmonic operator
161:
53:Gregory Sivashinsky
2527:Weisstein, Eric W.
2274:10.1007/BF00271514
2071:(2179): 20140932.
1896:10.1007/BF01025515
1663:
1661:
1302:tricritical points
1282:
1128:
1098:
958:
934:
910:
886:
862:
850:
812:
771:
732:
712:
692:
672:
622:
580:
533:
478:
447:
424:
317:
284:
261:
168:
147:
25:
2469:(23): 4746–4749.
2193:10.1137/070705623
1791:Acta Astronautica
1725:Clarke's equation
987:Kawahara equation
961:{\displaystyle L}
937:{\displaystyle L}
913:{\displaystyle L}
735:{\displaystyle c}
715:{\displaystyle u}
695:{\displaystyle c}
390:
294:and substituting
287:{\displaystyle x}
145:
35:(also called the
2576:
2540:
2539:
2513:
2512:
2478:
2476:cond-mat/9411083
2458:
2452:
2449:
2443:
2440:
2434:
2430:
2424:
2423:
2413:
2389:
2380:
2377:
2371:
2370:
2353:(3–4): 299–320.
2342:
2336:
2335:
2333:
2309:
2303:
2302:
2285:
2283:2060/19910004329
2257:
2248:
2247:
2219:
2213:
2212:
2186:
2163:
2152:
2151:
2141:
2113:
2107:
2106:
2096:
2056:
2047:
2046:
2020:
2010:
1986:
1980:
1979:
1961:
1937:
1931:
1930:
1906:
1900:
1899:
1879:
1873:
1872:
1856:
1850:
1849:
1821:
1815:
1814:
1786:
1780:
1779:
1769:
1745:
1672:
1670:
1669:
1664:
1662:
1655:
1624:
1623:
1608:
1607:
1580:
1579:
1558:
1557:
1539:
1538:
1512:
1511:
1496:
1495:
1468:
1467:
1452:
1451:
1412:
1411:
1396:
1395:
1368:
1367:
1346:
1345:
1327:
1326:
1291:
1289:
1288:
1283:
1275:
1274:
1259:
1258:
1237:
1236:
1224:
1223:
1202:
1201:
1186:
1185:
1173:
1172:
1157:
1156:
1137:
1135:
1134:
1129:
1127:
1126:
1107:
1105:
1104:
1099:
1091:
1090:
1075:
1074:
1053:
1052:
1037:
1036:
1024:
1023:
1008:
1007:
967:
965:
964:
959:
943:
941:
940:
935:
919:
917:
916:
911:
895:
893:
892:
887:
859:
857:
856:
851:
821:
819:
818:
813:
780:
778:
777:
772:
741:
739:
738:
733:
721:
719:
718:
713:
701:
699:
698:
693:
681:
679:
678:
673:
631:
629:
628:
623:
589:
587:
586:
581:
542:
540:
539:
534:
487:
485:
484:
479:
477:
476:
459:Laplace operator
456:
454:
453:
448:
433:
431:
430:
425:
414:
413:
408:
396:
391:
383:
375:
374:
353:
352:
326:
324:
323:
318:
316:
315:
293:
291:
290:
285:
270:
268:
267:
262:
254:
253:
238:
237:
216:
215:
200:
199:
177:
175:
174:
169:
160:
155:
146:
138:
133:
132:
111:
110:
95:
94:
49:Yoshiki Kuramoto
2584:
2583:
2579:
2578:
2577:
2575:
2574:
2573:
2544:
2543:
2525:
2524:
2521:
2516:
2460:
2459:
2455:
2450:
2446:
2441:
2437:
2431:
2427:
2391:
2390:
2383:
2378:
2374:
2344:
2343:
2339:
2311:
2310:
2306:
2259:
2258:
2251:
2221:
2220:
2216:
2165:
2164:
2155:
2130:10.1137/0517063
2115:
2114:
2110:
2058:
2057:
2050:
1995:Neural Networks
1988:
1987:
1983:
1939:
1938:
1934:
1908:
1907:
1903:
1881:
1880:
1876:
1858:
1857:
1853:
1838:10.1137/0139007
1823:
1822:
1818:
1788:
1787:
1783:
1747:
1746:
1742:
1738:
1706:
1690:
1660:
1659:
1625:
1615:
1584:
1562:
1546:
1530:
1527:
1526:
1513:
1503:
1472:
1456:
1443:
1440:
1439:
1413:
1403:
1372:
1350:
1334:
1318:
1309:
1308:
1298:
1266:
1238:
1228:
1206:
1187:
1177:
1161:
1148:
1143:
1142:
1118:
1113:
1112:
1082:
1057:
1038:
1028:
1012:
999:
994:
993:
983:
978:
950:
949:
946:traveling waves
926:
925:
902:
901:
866:
865:
836:
835:
828:
783:
782:
748:
747:
724:
723:
704:
703:
684:
683:
634:
633:
599:
598:
545:
544:
510:
509:
498:
468:
463:
462:
439:
438:
403:
366:
344:
339:
338:
307:
296:
295:
276:
275:
245:
220:
204:
191:
186:
185:
115:
99:
86:
81:
80:
74:
17:
12:
11:
5:
2582:
2580:
2572:
2571:
2566:
2561:
2559:Fluid dynamics
2556:
2546:
2545:
2542:
2541:
2520:
2519:External links
2517:
2515:
2514:
2453:
2444:
2435:
2425:
2381:
2372:
2337:
2324:(2): 663–666.
2304:
2249:
2214:
2153:
2124:(4): 884–893.
2108:
2048:
1981:
1932:
1921:(7): 391–394.
1901:
1890:(3): 354–359.
1884:Fluid Dynamics
1874:
1851:
1816:
1781:
1739:
1737:
1734:
1733:
1732:
1727:
1722:
1717:
1712:
1705:
1702:
1689:
1686:
1674:
1673:
1658:
1654:
1650:
1647:
1644:
1641:
1637:
1634:
1631:
1628:
1626:
1622:
1618:
1614:
1611:
1606:
1603:
1600:
1597:
1594:
1591:
1587:
1583:
1578:
1575:
1572:
1569:
1565:
1561:
1556:
1553:
1549:
1545:
1542:
1537:
1533:
1529:
1528:
1525:
1522:
1519:
1516:
1514:
1510:
1506:
1502:
1499:
1494:
1491:
1488:
1485:
1482:
1479:
1475:
1471:
1466:
1463:
1459:
1455:
1450:
1446:
1442:
1441:
1438:
1435:
1432:
1429:
1425:
1422:
1419:
1416:
1414:
1410:
1406:
1402:
1399:
1394:
1391:
1388:
1385:
1382:
1379:
1375:
1371:
1366:
1363:
1360:
1357:
1353:
1349:
1344:
1341:
1337:
1333:
1330:
1325:
1321:
1317:
1316:
1297:
1294:
1293:
1292:
1281:
1278:
1273:
1269:
1265:
1262:
1257:
1254:
1251:
1248:
1245:
1241:
1235:
1231:
1227:
1222:
1219:
1216:
1213:
1209:
1205:
1200:
1197:
1194:
1190:
1184:
1180:
1176:
1171:
1168:
1164:
1160:
1155:
1151:
1125:
1121:
1109:
1108:
1097:
1094:
1089:
1085:
1081:
1078:
1073:
1070:
1067:
1064:
1060:
1056:
1051:
1048:
1045:
1041:
1035:
1031:
1027:
1022:
1019:
1015:
1011:
1006:
1002:
989:, is given by
982:
979:
977:
974:
957:
933:
909:
885:
882:
879:
876:
873:
849:
846:
843:
827:
824:
811:
808:
805:
802:
799:
796:
793:
790:
770:
767:
764:
761:
758:
755:
731:
711:
691:
671:
668:
665:
662:
659:
656:
653:
650:
647:
644:
641:
621:
618:
615:
612:
609:
606:
579:
576:
573:
570:
567:
564:
561:
558:
555:
552:
532:
529:
526:
523:
520:
517:
502:Cauchy problem
497:
494:
475:
471:
446:
435:
434:
423:
420:
417:
412:
407:
402:
399:
395:
389:
386:
381:
378:
373:
369:
365:
362:
359:
356:
351:
347:
331:applications.
329:fluid dynamics
314:
310:
306:
303:
283:
272:
271:
260:
257:
252:
248:
244:
241:
236:
233:
230:
227:
223:
219:
214:
211:
207:
203:
198:
194:
179:
178:
167:
164:
159:
154:
150:
144:
141:
136:
131:
128:
125:
122:
118:
114:
109:
106:
102:
98:
93:
89:
73:
70:
41:flame equation
15:
13:
10:
9:
6:
4:
3:
2:
2581:
2570:
2567:
2565:
2562:
2560:
2557:
2555:
2552:
2551:
2549:
2537:
2536:
2531:
2528:
2523:
2522:
2518:
2510:
2506:
2502:
2498:
2494:
2490:
2486:
2482:
2477:
2472:
2468:
2464:
2457:
2454:
2448:
2445:
2439:
2436:
2429:
2426:
2421:
2417:
2412:
2407:
2403:
2399:
2395:
2388:
2386:
2382:
2376:
2373:
2368:
2364:
2360:
2356:
2352:
2348:
2341:
2338:
2332:
2327:
2323:
2319:
2315:
2308:
2305:
2301:
2297:
2293:
2289:
2284:
2279:
2275:
2271:
2267:
2263:
2256:
2254:
2250:
2245:
2241:
2237:
2233:
2230:(1): 89–111.
2229:
2225:
2218:
2215:
2210:
2206:
2202:
2198:
2194:
2190:
2185:
2180:
2176:
2172:
2168:
2162:
2160:
2158:
2154:
2149:
2145:
2140:
2135:
2131:
2127:
2123:
2119:
2112:
2109:
2104:
2100:
2095:
2090:
2086:
2082:
2078:
2074:
2070:
2066:
2062:
2055:
2053:
2049:
2044:
2040:
2036:
2032:
2028:
2024:
2019:
2014:
2009:
2004:
2000:
1996:
1992:
1985:
1982:
1977:
1973:
1969:
1965:
1960:
1955:
1952:(2): 024102.
1951:
1947:
1943:
1936:
1933:
1928:
1924:
1920:
1916:
1912:
1905:
1902:
1897:
1893:
1889:
1885:
1878:
1875:
1870:
1866:
1862:
1855:
1852:
1847:
1843:
1839:
1835:
1831:
1827:
1820:
1817:
1812:
1808:
1804:
1800:
1796:
1792:
1785:
1782:
1777:
1773:
1768:
1763:
1759:
1755:
1751:
1744:
1741:
1735:
1731:
1728:
1726:
1723:
1721:
1718:
1716:
1713:
1711:
1708:
1707:
1703:
1701:
1699:
1695:
1687:
1685:
1683:
1679:
1656:
1652:
1648:
1645:
1642:
1639:
1635:
1632:
1629:
1627:
1620:
1616:
1612:
1609:
1604:
1601:
1598:
1595:
1592:
1589:
1585:
1581:
1576:
1573:
1570:
1567:
1563:
1559:
1554:
1551:
1547:
1543:
1540:
1535:
1531:
1523:
1520:
1517:
1515:
1508:
1504:
1500:
1497:
1492:
1489:
1486:
1483:
1480:
1477:
1473:
1469:
1464:
1461:
1457:
1453:
1448:
1444:
1436:
1433:
1430:
1427:
1423:
1420:
1417:
1415:
1408:
1404:
1400:
1397:
1392:
1389:
1386:
1383:
1380:
1377:
1373:
1369:
1364:
1361:
1358:
1355:
1351:
1347:
1342:
1339:
1335:
1331:
1328:
1323:
1319:
1307:
1306:
1305:
1303:
1295:
1279:
1276:
1271:
1267:
1263:
1260:
1255:
1252:
1249:
1246:
1243:
1239:
1233:
1229:
1225:
1220:
1217:
1214:
1211:
1207:
1203:
1198:
1195:
1192:
1188:
1182:
1178:
1174:
1169:
1166:
1162:
1158:
1153:
1149:
1141:
1140:
1139:
1123:
1119:
1095:
1092:
1087:
1083:
1079:
1076:
1071:
1068:
1065:
1062:
1058:
1054:
1049:
1046:
1043:
1039:
1033:
1029:
1025:
1020:
1017:
1013:
1009:
1004:
1000:
992:
991:
990:
988:
980:
975:
973:
971:
955:
947:
931:
923:
907:
899:
883:
880:
877:
874:
871:
847:
844:
841:
832:
825:
823:
806:
803:
800:
797:
791:
788:
765:
762:
759:
753:
745:
729:
709:
689:
669:
666:
660:
657:
654:
651:
648:
645:
639:
616:
613:
610:
604:
597:—that is, if
596:
591:
571:
568:
565:
562:
559:
556:
550:
527:
524:
521:
515:
507:
503:
495:
493:
491:
473:
460:
421:
418:
415:
410:
400:
387:
384:
379:
376:
371:
363:
360:
354:
349:
345:
337:
336:
335:
332:
330:
312:
308:
304:
301:
281:
258:
255:
250:
246:
242:
239:
234:
231:
228:
225:
221:
217:
212:
209:
205:
201:
196:
192:
184:
183:
182:
165:
162:
157:
152:
148:
142:
139:
134:
129:
126:
123:
120:
116:
112:
107:
104:
100:
96:
91:
87:
79:
78:
77:
71:
69:
67:
62:
58:
54:
50:
46:
42:
38:
34:
30:
21:
2569:Chaotic maps
2533:
2466:
2462:
2456:
2447:
2438:
2428:
2401:
2397:
2375:
2350:
2346:
2340:
2321:
2317:
2307:
2265:
2261:
2227:
2223:
2217:
2174:
2170:
2121:
2117:
2111:
2068:
2064:
1998:
1994:
1984:
1949:
1945:
1935:
1918:
1914:
1904:
1887:
1883:
1877:
1860:
1854:
1832:(1): 67–82.
1829:
1825:
1819:
1794:
1790:
1784:
1757:
1753:
1743:
1691:
1688:Applications
1677:
1675:
1299:
1110:
986:
984:
898:bifurcations
863:
592:
499:
436:
333:
273:
180:
75:
40:
36:
32:
26:
2433:Heidelberg.
2177:(1): 1–33.
2001:: 191–217.
1760:: 346–367.
37:KS equation
29:mathematics
2564:Combustion
2548:Categories
2008:1910.05266
1736:References
506:well-posed
496:Properties
72:Definition
68:behavior.
2535:MathWorld
2493:0031-9007
2420:2693-969X
2367:1872-8022
2300:116955014
2292:1432-2250
2268:: 15–42,
2244:0167-2789
2201:1536-0040
2184:0709.2944
2148:0036-1410
2139:1903/8432
2085:1364-5021
2043:211146609
2027:0893-6080
1968:0031-9007
1846:0036-1399
1811:0094-5765
1776:0375-9687
1646:−
1582:−
1560:−
1470:−
1370:−
1230:δ
1179:δ
1120:δ
1030:δ
881:≤
875:≤
826:Solutions
798:−
789:−
667:−
649:−
575:∞
566:≤
470:Δ
445:Δ
398:∇
368:Δ
358:Δ
2509:18148655
2501:10058588
2404:(1): 2.
2209:17048798
2103:26345218
2035:32248008
1976:29376715
1704:See also
682:, where
2094:4528647
1865:Bibcode
1698:plasmas
922:chaotic
488:is the
457:is the
66:chaotic
61:laminar
2507:
2499:
2491:
2418:
2365:
2298:
2290:
2242:
2207:
2199:
2146:
2101:
2091:
2083:
2041:
2033:
2025:
1974:
1966:
1844:
1809:
1774:
1111:where
461:, and
437:where
31:, the
2505:S2CID
2471:arXiv
2296:S2CID
2205:S2CID
2179:arXiv
2039:S2CID
2003:arXiv
746:: if
59:in a
2497:PMID
2489:ISSN
2416:ISSN
2363:ISSN
2288:ISSN
2240:ISSN
2197:ISSN
2144:ISSN
2099:PMID
2081:ISSN
2031:PMID
2023:ISSN
1972:PMID
1964:ISSN
1842:ISSN
1807:ISSN
1772:ISSN
1643:>
1431:>
572:<
500:The
51:and
2481:doi
2406:doi
2355:doi
2326:doi
2278:hdl
2270:doi
2232:doi
2189:doi
2134:hdl
2126:doi
2089:PMC
2073:doi
2069:471
2013:doi
1999:126
1954:doi
1950:120
1923:doi
1892:doi
1834:doi
1799:doi
1762:doi
39:or
27:In
2550::
2532:.
2503:.
2495:.
2487:.
2479:.
2467:74
2465:.
2414:.
2400:.
2396:.
2384:^
2361:.
2351:63
2349:.
2322:44
2320:.
2316:.
2294:,
2286:,
2276:,
2264:,
2252:^
2238:.
2228:19
2226:.
2203:.
2195:.
2187:.
2173:.
2156:^
2142:.
2132:.
2122:17
2120:.
2097:.
2087:.
2079:.
2067:.
2063:.
2051:^
2037:.
2029:.
2021:.
2011:.
1997:.
1993:.
1970:.
1962:.
1948:.
1944:.
1919:34
1917:.
1913:.
1886:.
1840:.
1830:39
1828:.
1805:.
1793:.
1770:.
1758:64
1756:.
1752:.
1684:.
1280:0.
972:.
848:35
492:.
2538:.
2511:.
2483::
2473::
2422:.
2408::
2402:4
2369:.
2357::
2334:.
2328::
2280::
2272::
2266:3
2246:.
2234::
2211:.
2191::
2181::
2175:9
2150:.
2136::
2128::
2105:.
2075::
2045:.
2015::
2005::
1978:.
1956::
1929:.
1925::
1898:.
1894::
1888:9
1871:.
1867::
1848:.
1836::
1813:.
1801::
1795:4
1778:.
1764::
1657:4
1653:/
1649:1
1640:q
1636:,
1633:0
1630:=
1621:x
1617:u
1613:u
1610:+
1605:x
1602:x
1599:x
1596:x
1593:x
1590:x
1586:u
1577:x
1574:x
1571:x
1568:x
1564:u
1555:x
1552:x
1548:u
1544:q
1541:+
1536:t
1532:u
1524:,
1521:0
1518:=
1509:x
1505:u
1501:u
1498:+
1493:x
1490:x
1487:x
1484:x
1481:x
1478:x
1474:u
1465:x
1462:x
1458:u
1454:+
1449:t
1445:u
1437:,
1434:0
1428:q
1424:,
1421:0
1418:=
1409:x
1405:u
1401:u
1398:+
1393:x
1390:x
1387:x
1384:x
1381:x
1378:x
1374:u
1365:x
1362:x
1359:x
1356:x
1352:u
1348:+
1343:x
1340:x
1336:u
1332:q
1329:+
1324:t
1320:u
1277:=
1272:x
1268:u
1264:u
1261:+
1256:x
1253:x
1250:x
1247:x
1244:x
1240:u
1234:5
1226:+
1221:x
1218:x
1215:x
1212:x
1208:u
1204:+
1199:x
1196:x
1193:x
1189:u
1183:3
1175:+
1170:x
1167:x
1163:u
1159:+
1154:t
1150:u
1124:3
1096:0
1093:=
1088:x
1084:u
1080:u
1077:+
1072:x
1069:x
1066:x
1063:x
1059:u
1055:+
1050:x
1047:x
1044:x
1040:u
1034:3
1026:+
1021:x
1018:x
1014:u
1010:+
1005:t
1001:u
956:L
932:L
908:L
884:L
878:x
872:0
845:=
842:L
810:)
807:t
804:,
801:x
795:(
792:u
769:)
766:t
763:,
760:x
757:(
754:u
730:c
710:u
690:c
670:c
664:)
661:t
658:,
655:t
652:c
646:x
643:(
640:u
620:)
617:t
614:,
611:x
608:(
605:u
578:)
569:t
563:0
560:,
557:x
554:(
551:u
531:)
528:0
525:,
522:x
519:(
516:u
474:2
422:,
419:0
416:=
411:2
406:|
401:u
394:|
388:2
385:1
380:+
377:u
372:2
364:+
361:u
355:+
350:t
346:u
313:x
309:u
305:=
302:v
282:x
259:0
256:=
251:x
247:v
243:v
240:+
235:x
232:x
229:x
226:x
222:v
218:+
213:x
210:x
206:v
202:+
197:t
193:v
166:0
163:=
158:2
153:x
149:u
143:2
140:1
135:+
130:x
127:x
124:x
121:x
117:u
113:+
108:x
105:x
101:u
97:+
92:t
88:u
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
