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Every empirical law has the disquieting quality that one does not know its limitations. We have seen that there are regularities in the events in the world around us which can be formulated in terms of mathematical concepts with an uncanny accuracy. There are, on the other hand, aspects of the world
138:. A corresponding problem exists for discrete time situations. While a closed form solution is not always possible to obtain, future values of a discrete time system can be found by iterating forward one time period per iteration, though rounding error may make this impractical over long horizons.
1165:
1616:, while each remaining on the attractor, will diverge from each other over time. Thus even on a single attractor the precise values of the initial conditions make a substantial difference for the future positions of the iterates. This feature makes accurate
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of future values difficult, and impossible over long horizons, because stating the initial conditions with exact precision is seldom possible and because rounding error is inevitable after even only a few iterations from an exact initial condition.
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such that state variables with initial conditions in that basin (and nowhere else) will evolve toward that attractor. Even nearby initial conditions could be in basins of attraction of different attractors (see for example
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681:
324: = 1 being the number of time lags in the system. The initial conditions in this linear system do not affect the qualitative nature of the future behavior of the state variable
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can exhibit a substantially richer variety of behavior than linear systems can. In particular, the initial conditions can affect whether the system diverges to infinity or whether it
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1704:"The unreasonable effectiveness of mathematics in the natural sciences. Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959"
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1198: – 1 derivatives, all at some point in time such as time zero. The initial conditions do not affect the qualitative nature of the system's behavior. The
492:. Again the initial conditions do not affect the qualitative nature of the variable's long-term evolution. The solution of this equation is found by using its
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of the system. Each attractor, a (possibly disconnected) region of values that some dynamic paths approach but never leave, has a (possibly disconnected)
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484:, so the necessary number of initial conditions to trace the system through time, either iteratively or via closed form solution, is
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1160:{\displaystyle {\frac {d^{k}x}{dt^{k}}}+a_{k-1}{\frac {d^{k-1}x}{dt^{k-1}}}+\cdots +a_{1}{\frac {dx}{dt}}+a_{0}x=0.}
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concerning which we do not believe in the existence of any accurate regularities. We call these initial conditions.
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Its behavior through time can be traced with a closed form solution conditional on an initial condition vector
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Here the number of initial conditions necessary for obtaining a closed form solution is the dimension
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976:. The initial conditions do not affect the qualitative behavior (stable or unstable) of the system.
620:{\displaystyle \lambda ^{k}-a_{1}\lambda ^{k-1}-a_{2}\lambda ^{k-2}-\cdots -a_{k-1}\lambda -a_{k}=0}
158:
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initial conditions are needed in order to trace the system's variables forward through time.
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for the state variables as a function of time and of the initial conditions is called the
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is called the vector of initial conditions or simply the initial condition, and contains
123:
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initial pieces of information will typically not be different values of the variable
108:
281:
of initial conditions on the individual variables that are stacked into the vector;
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1301:{\displaystyle \lambda ^{k}+a_{k-1}\lambda ^{k-1}+\cdots +a_{1}\lambda +a_{0}=0,}
89:
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126:
in discrete time, initial conditions affect the value of the dynamic variables (
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1460:{\displaystyle x(t)=c_{1}e^{\lambda _{1}t}+\cdots +c_{k}e^{\lambda _{k}t}.}
771:{\displaystyle x_{t}=c_{1}\lambda _{1}^{t}+\cdots +c_{k}\lambda _{k}^{t}.}
54:
964:. The number of required initial pieces of information is the dimension
73:
at some point in time designated as the initial time (typically denoted
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466:{\displaystyle x_{t}=a_{1}x_{t-1}+a_{2}x_{t-2}+\cdots +a_{k}x_{t-k}.}
130:) at any future time. In continuous time, the problem of finding a
103:
different evolving variables, which together can be denoted by an
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1612:: the iterated values of any two very nearby points on the same
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different equations based on this equation, each using one of
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18:
Parameter in differential equations and dynamical systems
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874:
A differential equation system of the first order with
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Evolution of this initial condition for an example PDE
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at different points in time, but rather the values of
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Alternatively, a dynamic process in a single variable
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1353:{\displaystyle \lambda _{1},\dots ,\lambda _{k};}
676:{\displaystyle \lambda _{1},\dots ,\lambda _{k},}
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1708:Communications on Pure and Applied Mathematics
1604:Moreover, in those nonlinear systems showing
88:, or the order of the largest derivative in
1678:(3rd ed.). London: Collier-Macmillan.
983:order linear equation in a single variable
32:The initial condition of a vibrating string
1610:sensitive dependence on initial conditions
1608:, the evolution of the variables exhibits
839:for which the specific initial condition
340:but not based on the initial conditions.
1360:these are used in the solution equation
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1531:given the known initial conditions on
1625:Empirical laws and initial conditions
1539:– 1 derivatives' values at some time
1478:equations that can be solved for the
1599:Newton's method#Basins of attraction
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927:{\displaystyle {\frac {dX}{dt}}=AX.}
46:Evolution from the initial condition
1524:{\displaystyle c_{1},\dots ,c_{k},}
13:
1674:Economic Dynamics: An Introduction
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820:{\displaystyle c_{1},\dots ,c_{k}}
316:being the dimension of the vector
14:
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1474:– 1 derivatives form a system of
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827:are found by solving a system of
683:for use in the solution equation
77: = 0). For a system of
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480: = 1 and the order is
247:{\displaystyle X_{t}=A^{t}X_{0}}
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878:variables stacked in a vector
347:having multiple time lags is
197:{\displaystyle X_{t+1}=AX_{t}}
1:
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1470:This equation and its first
1202:of this dynamic equation is
84:(the number of time lags in
69:, is a value of an evolving
65:, in some contexts called a
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153:matrix difference equation
1562:Another initial condition
631:solutions, which are the
332:or unstable based on the
254:predicated on the vector
204:has closed form solution
1308:whose solutions are the
1200:characteristic equation
627:to obtain the latter's
494:characteristic equation
312:pieces of information,
122:in continuous time and
1779:Differential equations
1756:Quotations related to
1728:10.1002/cpa.3160130102
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120:differential equations
1737:on February 12, 2021.
1647:Initialization vector
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957:{\displaystyle X_{0}}
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57:and particularly in
1720:1960CPAM...13....1W
1594:basin of attraction
1182:. In this case the
781:Here the constants
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1668:Baumol, William J.
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1700:Wigner, Eugene P.
1649:, in cryptography
1614:strange attractor
1582:Nonlinear systems
1547:Nonlinear systems
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873:
836:
832:
828:
780:
628:
489:
485:
481:
477:
475:
344:
342:
337:
325:
321:
317:
313:
309:
150:
117:
112:
104:
100:
96:
81:
74:
66:
62:
52:
15:
1714:(1): 1–14.
1482:parameters
334:eigenvalues
55:mathematics
1768:Categories
1654:References
1618:simulation
866:Is known.
67:seed value
1590:attractor
1586:converges
1503:…
1441:λ
1419:⋯
1402:λ
1339:λ
1332:…
1320:λ
1274:λ
1258:⋯
1247:−
1240:λ
1231:−
1211:λ
1100:⋯
1086:−
1062:−
1043:−
979:A single
802:…
752:λ
735:⋯
718:λ
662:λ
655:…
643:λ
599:−
596:λ
588:−
577:−
574:⋯
571:−
563:−
556:λ
542:−
534:−
527:λ
513:−
504:λ
453:−
429:⋯
418:−
389:−
151:A linear
94:dimension
1702:(1960).
1670:(1970).
1636:See also
1535:and its
118:In both
71:variable
1716:Bibcode
1682:
330:stable
92:) and
1735:(PDF)
79:order
61:, an
1680:ISBN
320:and
1724:doi
1601:).
987:is
882:is
53:In
1770::
1722:.
1712:13
1710:.
1706:.
1543:.
1155:0.
486:nk
310:nk
113:nk
1726::
1718::
1688:.
1541:t
1537:k
1533:x
1519:,
1514:k
1510:c
1506:,
1500:,
1495:1
1491:c
1480:k
1476:k
1472:k
1455:.
1450:t
1445:k
1436:e
1430:k
1426:c
1422:+
1416:+
1411:t
1406:1
1397:e
1391:1
1387:c
1383:=
1380:)
1377:t
1374:(
1371:x
1348:;
1343:k
1335:,
1329:,
1324:1
1296:,
1293:0
1290:=
1285:0
1281:a
1277:+
1269:1
1265:a
1261:+
1255:+
1250:1
1244:k
1234:1
1228:k
1224:a
1220:+
1215:k
1196:k
1192:x
1188:x
1184:k
1180:k
1176:k
1172:n
1152:=
1149:x
1144:0
1140:a
1136:+
1130:t
1127:d
1122:x
1119:d
1111:1
1107:a
1103:+
1097:+
1089:1
1083:k
1079:t
1075:d
1070:x
1065:1
1059:k
1055:d
1046:1
1040:k
1036:a
1032:+
1024:k
1020:t
1016:d
1011:x
1006:k
1002:d
985:x
981:k
974:n
970:k
966:n
950:0
946:X
922:.
919:X
916:A
913:=
907:t
904:d
899:X
896:d
880:X
876:n
852:t
848:x
837:t
833:k
829:k
813:k
809:c
805:,
799:,
794:1
790:c
766:.
761:t
756:k
746:k
742:c
738:+
732:+
727:t
722:1
712:1
708:c
704:=
699:t
695:x
671:,
666:k
658:,
652:,
647:1
629:k
615:0
612:=
607:k
603:a
591:1
585:k
581:a
566:2
560:k
550:2
546:a
537:1
531:k
521:1
517:a
508:k
490:k
482:k
478:n
461:.
456:k
450:t
446:x
440:k
436:a
432:+
426:+
421:2
415:t
411:x
405:2
401:a
397:+
392:1
386:t
382:x
376:1
372:a
368:=
363:t
359:x
345:x
338:A
326:X
322:k
318:X
314:n
294:0
290:X
267:0
263:X
240:0
236:X
230:t
226:A
222:=
217:t
213:X
190:t
186:X
182:A
179:=
174:1
171:+
168:t
164:X
105:n
101:n
97:n
82:k
75:t
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