36:
2417:
Dynamics of strongly driven quantum systems are often examined using
Floquet theory. In superconducting circuits, Floquet framework has been leveraged to shed light on the quantum electrodynamics of drive-induced multiqubit
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252:
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2619:
Huang, Ziwen; Mundada, Pranav S.; Gyenis, András; Schuster, David I.; Houck, Andrew A.; Koch, Jens (2021-03-22). "Engineering
Dynamical Sweet Spots to Protect Qubits from 1 / f Noise".
414:
359:
2580:
Deng, Chunqing; Shen, Feiruo; Ashhab, Sahel; Lupascu, Adrian (2016-09-27). "Dynamics of a two-level system under strong driving: Quantum-gate optimization based on
Floquet theory".
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Nguyen, L.B.; Kim, Y.; Hashim, A.; Goss, N.; Marinelli, B.; Bhandari, B.; Das, D.; Naik, R.K.; Kreikebaum, J.M.; Jordan, A.; Santiago, D.I.; Siddiqi, I. (16 January 2024).
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65:
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2020:
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1413:
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M.S.P. Eastham, "The
Spectral Theory of Periodic Differential Equations", Texts in Mathematics, Scottish Academic Press, Edinburgh, 1973.
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87:
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1108:
104:
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205:
2720:
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108:
48:
2715:
58:
52:
44:
2737:
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1282:
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2454:. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) . Vol. 19. Berlin: Springer-Verlag. pp. x+247.
364:
314:
69:
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2514:
424:
2028:
1774:
1201:
2506:
309:
1077:
117:
806:
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2710:
2164:
2401:
2397:
2389:
2265:
is a characteristic multiplier of the system. Notice that
Floquet exponents are not unique, since
1880:
1543:
1375:
566:
is the identity. A principal fundamental matrix can be constructed from a fundamental matrix using
1832:
533:
170:
2671:
2628:
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2414:
in intense laser fields can be described in terms of solutions obtained from the
Floquet theorem.
2366:
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977:
2090:
1047:
768:
646:
437:
428:
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1740:
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473:
258:
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2599:
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2378:
2127:
1253:
2469:
1944:
is continuous and periodic it must be bounded. Thus the stability of the zero solution for
1877:), under which our original system becomes a linear system with real constant coefficients
506:
2478:
2465:
2218:
1976:
1947:
1918:
902:
853:
434:
Note that the solutions of the linear differential equation form a vector space. A matrix
2159:
2685:
1520:
2411:
2407:
2344:
2005:
1720:
1700:
1680:
1660:
1496:
1355:
1259:
1027:
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882:
416:
that transforms the periodic system to a traditional linear system with constant, real
297:
289:
264:
17:
2731:
2642:
2552:
2548:
2365:. The zero solution is asymptotically stable if all Lyapunov exponents are negative,
305:
2447:
417:
2694:
2659:
2603:
423:
When applied to physical systems with periodic potentials, such as crystals in
2121:
1074:
be a fundamental matrix solution of this differential equation. Then, for all
643:. The solution of the linear differential equation with the initial condition
2650:
2611:
2511:
The
Operator of Translation along the Trajectories of Differential Equations
255:
2479:"Sur les équations différentielles linéaires à coefficients périodiques"
2498:
1647:{\displaystyle \phi (t)=Q(t)e^{tR}{\text{ for all }}t\in \mathbb {R} .}
1479:{\displaystyle \phi (t)=P(t)e^{tB}{\text{ for all }}t\in \mathbb {R} .}
2676:
2633:
2594:
2361:
is an integer. The real parts of the
Floquet exponents are called
503:
if all columns are linearly independent solutions and there exists
2369:
if the
Lyapunov exponents are nonpositive and unstable otherwise.
2393:
29:
2660:"Programmable Heisenberg interactions between Floquet qubits"
2158:
of the system. They are also the eigenvalues of the (linear)
2215:(sometimes called a characteristic exponent), is a complex
2334:{\displaystyle e^{(\mu +{\frac {2\pi ik}{T}})T}=e^{\mu T}}
470:
if the columns form a basis of the solution set. A matrix
1185:{\displaystyle \phi (t+T)=\phi (t)\phi ^{-1}(0)\phi (T).}
636:{\displaystyle \Phi (t)=\phi \,(t){\phi \,}^{-1}(t_{0})}
1829:
gives rise to a time-dependent change of coordinates (
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2347:
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2221:
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2130:
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1921:
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1523:
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1204:
1111:
1080:
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1030:
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960:
934:
905:
885:
856:
850:
be a linear first order differential equation, where
809:
771:
758:{\displaystyle x(t)=\phi \,(t){\phi \,}^{-1}(0)x_{0}}
691:
649:
572:
536:
509:
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368:
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and defines the state of the stability of solutions.
267:
247:{\displaystyle \displaystyle A(t)\in {R^{n\times n}}}
209:
208:
173:
120:
2522:, Translation of Mathematical Monographs, 19, 294p.
2486:
Annales Scientifiques de l'École Normale Supérieure
2430:Ordinary Differential Equations with Applications.
2373:Floquet theory is very important for the study of
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2203:
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2014:
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1505:
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1241:
1184:
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1066:
1036:
1016:
966:
946:
920:
891:
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842:
787:
757:
677:
635:
558:
522:
491:
456:
408:
353:
273:
246:
194:
156:
57:but its sources remain unclear because it lacks
107:relating to the class of solutions to periodic
1342:{\displaystyle e^{TB}=\phi ^{-1}(0)\phi (T),}
8:
2542:Theory and Application of Mathieu Functions
2452:Convexity methods in Hamiltonian mechanics
409:{\displaystyle \displaystyle Q(t+2T)=Q(t)}
354:{\displaystyle \displaystyle y=Q^{-1}(t)x}
27:Branch of ordinary differential equations
2693:
2675:
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2240:
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1415:
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1110:
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979:
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366:
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230:
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185:
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172:
122:
121:
119:
88:Learn how and when to remove this message
293:
2076:{\displaystyle \phi \,(t)=P(t)e^{tB}}
1822:{\displaystyle \phi \,(t)=Q(t)e^{tR}}
1242:{\displaystyle \phi ^{-1}(0)\phi (T)}
501:principal fundamental matrix solution
7:
2002:is determined by the eigenvalues of
795:is any fundamental matrix solution.
284:The main theorem of Floquet theory,
2392:) approximating the motion of the
2384:Floquet theory shows stability in
573:
537:
477:
25:
1095:{\displaystyle t\in \mathbb {R} }
157:{\displaystyle {\dot {x}}=A(t)x,}
2643:10.1103/PhysRevApplied.15.034065
843:{\displaystyle {\dot {x}}=A(t)x}
34:
2432:Springer-Verlag, New York 1999.
1256:. In addition, for each matrix
105:ordinary differential equations
2307:
2277:
2204:{\displaystyle x(t)\to x(t+T)}
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1:
2563:American Mathematical Society
2519:American Mathematical Society
1908:{\displaystyle {\dot {y}}=Ry}
1767:Consequences and applications
1568:{\displaystyle t\mapsto Q(t)}
1400:{\displaystyle t\mapsto P(t)}
1352:there is a periodic (period
1276:(possibly complex) such that
879:is a column vector of length
109:linear differential equations
103:is a branch of the theory of
1870:{\displaystyle y=Q^{-1}(t)x}
954:periodic matrix with period
559:{\displaystyle \Phi (t_{0})}
195:{\displaystyle x\in {R^{n}}}
2716:Encyclopedia of Mathematics
2087:for the fundamental matrix
1017:{\displaystyle A(t+T)=A(t)}
467:fundamental matrix solution
2759:
2695:10.1038/s41567-023-02326-7
2604:10.1103/PhysRevA.94.032323
2529:, Dover-Phoenix Editions,
2386:Hill differential equation
2156:characteristic multipliers
2110:{\displaystyle \phi \,(t)}
1067:{\displaystyle \phi \,(t)}
788:{\displaystyle \phi \,(t)}
678:{\displaystyle x(0)=x_{0}}
457:{\displaystyle \phi \,(t)}
2258:{\displaystyle e^{\mu T}}
1756:{\displaystyle n\times n}
947:{\displaystyle n\times n}
427:, the result is known as
2544:, New York: Dover, 1964.
2477:Floquet, Gaston (1883),
492:{\displaystyle \Phi (t)}
425:condensed matter physics
304:solution of this common
43:This article includes a
2621:Physical Review Applied
2525:W. Magnus, S. Winkler.
1024:for all real values of
798:
72:more precise citations.
2743:Differential equations
2355:
2335:
2259:
2229:
2205:
2148:
2147:{\displaystyle e^{TB}}
2111:
2077:
2016:
1996:
1967:
1938:
1909:
1871:
1823:
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560:
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458:
410:
355:
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158:
18:Floquet's theorem
2356:
2336:
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2149:
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2017:
1997:
1968:
1939:
1910:
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1712:
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1672:
1649:
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1535:
1508:
1481:
1402:
1367:
1344:
1271:
1244:
1187:
1097:
1069:
1039:
1019:
969:
949:
923:
894:
874:
845:
790:
760:
680:
638:
561:
525:
523:{\displaystyle t_{0}}
494:
459:
411:
356:
276:
249:
197:
159:
2507:Krasnosel'skii, M.A.
2345:
2269:
2239:
2228:{\displaystyle \mu }
2219:
2165:
2128:
2091:
2029:
2006:
1995:{\displaystyle x(t)}
1977:
1966:{\displaystyle y(t)}
1948:
1937:{\displaystyle Q(t)}
1919:
1881:
1833:
1775:
1741:
1721:
1701:
1681:
1661:
1582:
1544:
1521:
1497:
1414:
1376:
1356:
1283:
1260:
1202:
1109:
1078:
1048:
1028:
978:
958:
932:
921:{\displaystyle A(t)}
903:
883:
872:{\displaystyle x(t)}
854:
807:
769:
689:
647:
570:
534:
507:
474:
438:
365:
315:
265:
256:piecewise continuous
206:
171:
118:
2686:2024NatPh..20..240N
2402:gravitational field
2398:harmonic oscillator
2390:George William Hill
2085:Floquet normal form
2025:The representation
1627: for all
1459: for all
2499:10.24033/asens.220
2363:Lyapunov exponents
2351:
2331:
2255:
2225:
2201:
2144:
2107:
2073:
2012:
1992:
1963:
1934:
1905:
1867:
1819:
1753:
1727:
1707:
1687:
1667:
1644:
1565:
1540:) matrix function
1533:{\displaystyle 2T}
1530:
1503:
1476:
1397:
1372:) matrix function
1362:
1339:
1266:
1239:
1182:
1092:
1064:
1034:
1014:
964:
944:
918:
889:
869:
840:
785:
755:
675:
633:
556:
520:
489:
454:
406:
405:
351:
350:
302:fundamental matrix
290:Gaston Floquet
271:
244:
243:
192:
154:
45:list of references
2738:Dynamical systems
2582:Physical Review A
2572:978-0-8218-8328-0
2441:978-0-7011-1936-2
2375:dynamical systems
2354:{\displaystyle k}
2305:
2015:{\displaystyle R}
1893:
1730:{\displaystyle R}
1710:{\displaystyle Q}
1690:{\displaystyle P}
1670:{\displaystyle B}
1628:
1517:periodic (period-
1506:{\displaystyle R}
1489:Also, there is a
1460:
1365:{\displaystyle T}
1269:{\displaystyle B}
1037:{\displaystyle t}
967:{\displaystyle T}
892:{\displaystyle n}
819:
799:Floquet's theorem
310:coordinate change
286:Floquet's theorem
274:{\displaystyle T}
259:periodic function
130:
98:
97:
90:
16:(Redirected from
2750:
2724:
2711:"Floquet theory"
2699:
2697:
2679:
2654:
2636:
2615:
2597:
2576:
2540:N.W. McLachlan,
2521:
2502:
2501:
2483:
2473:
2379:Mathieu equation
2360:
2358:
2357:
2352:
2340:
2338:
2337:
2332:
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2329:
2314:
2313:
2306:
2301:
2287:
2264:
2262:
2261:
2256:
2254:
2253:
2234:
2232:
2231:
2226:
2213:Floquet exponent
2210:
2208:
2207:
2202:
2153:
2151:
2150:
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2143:
2142:
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2114:
2113:
2108:
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1708:
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1668:
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1640:
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1626:
1624:
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1574:
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1571:
1566:
1539:
1537:
1536:
1531:
1512:
1510:
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1504:
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1404:
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1398:
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1346:
1345:
1340:
1314:
1313:
1298:
1297:
1275:
1273:
1272:
1267:
1254:monodromy matrix
1252:is known as the
1248:
1246:
1245:
1240:
1217:
1216:
1191:
1189:
1188:
1183:
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1101:
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1073:
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1065:
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1023:
1021:
1020:
1015:
973:
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970:
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953:
951:
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945:
927:
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919:
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895:
890:
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876:
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849:
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841:
821:
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812:
794:
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786:
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761:
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753:
735:
734:
726:
684:
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673:
642:
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634:
629:
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616:
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607:
565:
563:
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552:
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529:
527:
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240:
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189:
163:
161:
160:
155:
132:
131:
123:
93:
86:
82:
79:
73:
68:this article by
59:inline citations
38:
37:
30:
21:
2758:
2757:
2753:
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2751:
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2748:
2747:
2728:
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2709:
2706:
2657:
2618:
2579:
2573:
2547:
2527:Hill's Equation
2505:
2481:
2476:
2462:
2450:(1990). "One".
2446:
2425:
2388:(introduced by
2367:Lyapunov stable
2343:
2342:
2318:
2288:
2272:
2267:
2266:
2242:
2237:
2236:
2217:
2216:
2163:
2162:
2154:are called the
2131:
2126:
2125:
2089:
2088:
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2026:
2004:
2003:
1975:
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1945:
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504:
472:
471:
436:
435:
429:Bloch's theorem
363:
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263:
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49:related reading
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11:
5:
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2726:
2725:
2705:
2704:External links
2702:
2701:
2700:
2670:(1): 240–246.
2664:Nature Physics
2655:
2616:
2577:
2571:
2549:Teschl, Gerald
2545:
2538:
2523:
2503:
2474:
2460:
2444:
2433:
2424:
2421:
2420:
2419:
2415:
2412:bond hardening
2408:Bond softening
2405:
2400:in a periodic
2382:
2377:, such as the
2350:
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664:
661:
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655:
652:
632:
627:
623:
619:
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594:
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584:
581:
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555:
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546:
542:
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517:
513:
488:
485:
482:
479:
453:
450:
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383:
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349:
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328:
324:
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298:canonical form
270:
239:
236:
233:
229:
224:
221:
218:
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188:
184:
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176:
165:
164:
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101:Floquet theory
96:
95:
53:external links
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
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2556:
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2539:
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2535:0-486-49565-5
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2508:
2504:
2500:
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2480:
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2467:
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2461:3-540-50613-6
2457:
2453:
2449:
2448:Ekeland, Ivar
2445:
2442:
2438:
2434:
2431:
2427:
2426:
2422:
2418:interactions.
2416:
2413:
2409:
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2403:
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2168:
2161:
2160:Poincaré maps
2157:
2139:
2136:
2132:
2123:
2118:
2101:
2094:
2086:
2068:
2065:
2061:
2054:
2048:
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2009:
1986:
1980:
1957:
1951:
1928:
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1850:
1847:
1843:
1839:
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1794:
1791:
1785:
1778:
1771:This mapping
1766:
1764:
1750:
1747:
1744:
1724:
1704:
1684:
1664:
1657:In the above
1641:
1633:
1630:
1620:
1617:
1613:
1606:
1600:
1597:
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1585:
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1577:
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1121:
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1104:
1103:
1084:
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1008:
1002:
999:
993:
990:
987:
981:
961:
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935:
912:
906:
886:
863:
857:
837:
831:
825:
822:
816:
813:
796:
779:
772:
750:
746:
739:
731:
728:
722:
714:
707:
704:
698:
692:
670:
666:
662:
656:
650:
625:
621:
612:
609:
603:
595:
588:
585:
579:
548:
544:
515:
511:
502:
483:
469:
468:
448:
441:
432:
430:
426:
421:
419:
399:
393:
390:
384:
381:
378:
375:
369:
347:
341:
333:
330:
326:
322:
319:
311:
308:. It gives a
307:
306:linear system
303:
299:
295:
291:
287:
282:
268:
260:
257:
237:
234:
231:
227:
222:
216:
210:
186:
182:
177:
174:
151:
148:
142:
136:
133:
127:
124:
114:
113:
112:
110:
106:
102:
92:
89:
81:
71:
67:
61:
60:
54:
50:
46:
41:
32:
31:
19:
2714:
2667:
2663:
2624:
2620:
2585:
2581:
2553:
2541:
2526:
2510:
2489:
2485:
2451:
2429:
2428:C. Chicone.
2212:
2119:
2084:
2083:is called a
2024:
1770:
1656:
1514:
1490:
1488:
1351:
1251:
1194:
802:
500:
499:is called a
465:
464:is called a
433:
422:
418:coefficients
285:
283:
261:with period
166:
111:of the form
100:
99:
84:
75:
64:Please help
56:
2122:eigenvalues
296:), gives a
70:introducing
2732:Categories
2677:2211.10383
2634:2004.12458
2595:1605.08826
2559:Providence
2515:Providence
2423:References
2235:such that
1763:matrices.
1575:such that
1407:such that
530:such that
2721:EMS Press
2651:2331-7019
2612:2469-9926
2492:: 47–88,
2324:μ
2293:π
2281:μ
2248:μ
2223:μ
2181:→
2095:ϕ
2033:ϕ
1915:. Since
1891:˙
1848:−
1779:ϕ
1748:×
1634:∈
1586:ϕ
1551:↦
1466:∈
1418:ϕ
1383:↦
1325:ϕ
1308:−
1304:ϕ
1228:ϕ
1211:−
1207:ϕ
1168:ϕ
1151:−
1147:ϕ
1134:ϕ
1113:ϕ
1085:∈
1052:ϕ
974:(that is
939:×
817:˙
773:ϕ
729:−
723:ϕ
708:ϕ
610:−
604:ϕ
589:ϕ
574:Φ
538:Φ
478:Φ
442:ϕ
331:−
300:for each
288:, due to
235:×
223:∈
178:∈
128:˙
78:July 2015
2551:(2012).
2509:(1968),
2341:, where
254:being a
2723:, 2001
2682:Bibcode
2470:1051888
1493:matrix
1044:). Let
292: (
66:improve
2649:
2610:
2569:
2533:
2468:
2458:
2439:
1513:and a
765:where
2672:arXiv
2629:arXiv
2627:(3).
2590:arXiv
2588:(3).
2482:(PDF)
2396:as a
1195:Here
361:with
167:with
51:, or
2647:ISSN
2608:ISSN
2567:ISBN
2531:ISBN
2456:ISBN
2437:ISBN
2410:and
2394:moon
2211:. A
2120:The
1973:and
1737:are
1717:and
1515:real
1491:real
899:and
803:Let
294:1883
202:and
2690:doi
2639:doi
2600:doi
2494:doi
2124:of
928:an
685:is
2734::
2719:,
2713:,
2688:.
2680:.
2668:20
2666:.
2662:.
2645:.
2637:.
2625:15
2623:.
2606:.
2598:.
2586:94
2584:.
2565:.
2561::
2557:.
2517::
2513:,
2490:12
2488:,
2484:,
2466:MR
2464:.
2117:.
2022:.
1697:,
1677:,
1102:,
431:.
420:.
55:,
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2698:.
2692::
2684::
2674::
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2602::
2592::
2575:.
2537:.
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2472:.
2443:.
2404:.
2381:.
2349:k
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2010:R
1990:)
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1981:x
1961:)
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1900:R
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1840:=
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1801:t
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1795:Q
1792:=
1789:)
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1642:.
1638:R
1631:t
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1595:)
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1430:=
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1319:0
1316:(
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1234:T
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1180:.
1177:)
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1122:+
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1116:(
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1009:t
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991:+
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214:(
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143:t
140:(
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85:(
80:)
76:(
62:.
20:)
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