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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
149:. 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.
1176:
1627:, 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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335: = 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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1715:"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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1209: – 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
503:. 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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1171:{\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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987:. The initial conditions do not affect the qualitative behavior (stable or unstable) of the system.
631:{\displaystyle \lambda ^{k}-a_{1}\lambda ^{k-1}-a_{2}\lambda ^{k-2}-\cdots -a_{k-1}\lambda -a_{k}=0}
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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
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initial pieces of information will typically not be different values of the variable
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of initial conditions on the individual variables that are stacked into the vector;
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1312:{\displaystyle \lambda ^{k}+a_{k-1}\lambda ^{k-1}+\cdots +a_{1}\lambda +a_{0}=0,}
137:
in discrete time, initial conditions affect the value of the dynamic variables (
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1471:{\displaystyle x(t)=c_{1}e^{\lambda _{1}t}+\cdots +c_{k}e^{\lambda _{k}t}.}
782:{\displaystyle x_{t}=c_{1}\lambda _{1}^{t}+\cdots +c_{k}\lambda _{k}^{t}.}
17:
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975:. The number of required initial pieces of information is the dimension
84:
at some point in time designated as the initial time (typically denoted
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477:{\displaystyle x_{t}=a_{1}x_{t-1}+a_{2}x_{t-2}+\cdots +a_{k}x_{t-k}.}
141:) at any future time. In continuous time, the problem of finding a
114:
different evolving variables, which together can be denoted by an
1623:: 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
29:
Parameter in differential equations and dynamical systems
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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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687:{\displaystyle \lambda _{1},\dots ,\lambda _{k},}
1639:
1719:Communications on Pure and Applied Mathematics
1615:Moreover, in those nonlinear systems showing
8:
99:, or the order of the largest derivative in
1689:(3rd ed.). London: Collier-Macmillan.
994:order linear equation in a single variable
1621:sensitive dependence on initial conditions
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43:An initial condition of a vibrating string
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1636:Empirical laws and initial conditions
1550:– 1 derivatives' values at some time
1489:equations that can be solved for the
7:
1610:Newton's method#Basins of attraction
938:{\displaystyle {\frac {dX}{dt}}=AX.}
57:Evolution from the initial condition
1535:{\displaystyle c_{1},\dots ,c_{k},}
1685:Economic Dynamics: An Introduction
831:{\displaystyle c_{1},\dots ,c_{k}}
327:being the dimension of the vector
25:
1485:– 1 derivatives form a system of
983: = 1 of the system, or
838:are found by solving a system of
694:for use in the solution equation
88: = 0). For a system of
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491: = 1 and the order is
258:{\displaystyle X_{t}=A^{t}X_{0}}
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889:variables stacked in a vector
358:having multiple time lags is
208:{\displaystyle X_{t+1}=AX_{t}}
1:
1481:This equation and its first
1213:of this dynamic equation is
95:(the number of time lags in
80:, is a value of an evolving
76:, in some contexts called a
1806:
164:matrix difference equation
1573:Another initial condition
642:solutions, which are the
343:or unstable based on the
265:predicated on the vector
215:has closed form solution
1319:whose solutions are the
1211:characteristic equation
638:to obtain the latter's
505:characteristic equation
323:pieces of information,
133:in continuous time and
1790:Differential equations
1767:Quotations related to
1739:10.1002/cpa.3160130102
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131:differential equations
1748:on February 12, 2021.
1658:Initialization vector
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968:{\displaystyle X_{0}}
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68:and particularly in
1731:1960CPAM...13....1W
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1193:. In this case the
792:Here the constants
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1679:Baumol, William J.
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1769:Initial condition
1711:Wigner, Eugene P.
1660:, in cryptography
1625:strange attractor
1593:Nonlinear systems
1558:Nonlinear systems
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120:coordinate vector
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16:(Redirected from
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1180:
995:
991:
989:
984:
980:
976:
947:
890:
886:
884:
847:
843:
839:
791:
639:
500:
496:
492:
488:
486:
355:
353:
348:
336:
332:
328:
324:
320:
161:
128:
123:
115:
111:
107:
92:
85:
77:
73:
63:
26:
1725:(1): 1–14.
1493:parameters
345:eigenvalues
66:mathematics
1779:Categories
1665:References
1629:simulation
877:Is known.
78:seed value
18:Seed value
1601:attractor
1597:converges
1514:…
1452:λ
1430:⋯
1413:λ
1350:λ
1343:…
1331:λ
1285:λ
1269:⋯
1258:−
1251:λ
1242:−
1222:λ
1111:⋯
1097:−
1073:−
1054:−
990:A single
813:…
763:λ
746:⋯
729:λ
673:λ
666:…
654:λ
610:−
607:λ
599:−
588:−
585:⋯
582:−
574:−
567:λ
553:−
545:−
538:λ
524:−
515:λ
464:−
440:⋯
429:−
400:−
162:A linear
105:dimension
1713:(1960).
1681:(1970).
1647:See also
1546:and its
129:In both
82:variable
1727:Bibcode
1693:
341:stable
103:) and
1746:(PDF)
90:order
72:, an
1691:ISBN
331:and
1735:doi
1612:).
998:is
893:is
64:In
1781::
1733:.
1723:13
1721:.
1717:.
1554:.
1166:0.
497:nk
321:nk
124:nk
1737::
1729::
1699:.
1552:t
1548:k
1544:x
1530:,
1525:k
1521:c
1517:,
1511:,
1506:1
1502:c
1491:k
1487:k
1483:k
1466:.
1461:t
1456:k
1447:e
1441:k
1437:c
1433:+
1427:+
1422:t
1417:1
1408:e
1402:1
1398:c
1394:=
1391:)
1388:t
1385:(
1382:x
1359:;
1354:k
1346:,
1340:,
1335:1
1307:,
1304:0
1301:=
1296:0
1292:a
1288:+
1280:1
1276:a
1272:+
1266:+
1261:1
1255:k
1245:1
1239:k
1235:a
1231:+
1226:k
1207:k
1203:x
1199:x
1195:k
1191:k
1187:k
1183:n
1163:=
1160:x
1155:0
1151:a
1147:+
1141:t
1138:d
1133:x
1130:d
1122:1
1118:a
1114:+
1108:+
1100:1
1094:k
1090:t
1086:d
1081:x
1076:1
1070:k
1066:d
1057:1
1051:k
1047:a
1043:+
1035:k
1031:t
1027:d
1022:x
1017:k
1013:d
996:x
992:k
985:n
981:k
977:n
961:0
957:X
933:.
930:X
927:A
924:=
918:t
915:d
910:X
907:d
891:X
887:n
863:t
859:x
848:t
844:k
840:k
824:k
820:c
816:,
810:,
805:1
801:c
777:.
772:t
767:k
757:k
753:c
749:+
743:+
738:t
733:1
723:1
719:c
715:=
710:t
706:x
682:,
677:k
669:,
663:,
658:1
640:k
626:0
623:=
618:k
614:a
602:1
596:k
592:a
577:2
571:k
561:2
557:a
548:1
542:k
532:1
528:a
519:k
501:k
493:k
489:n
472:.
467:k
461:t
457:x
451:k
447:a
443:+
437:+
432:2
426:t
422:x
416:2
412:a
408:+
403:1
397:t
393:x
387:1
383:a
379:=
374:t
370:x
356:x
349:A
337:X
333:k
329:X
325:n
305:0
301:X
278:0
274:X
251:0
247:X
241:t
237:A
233:=
228:t
224:X
201:t
197:X
193:A
190:=
185:1
182:+
179:t
175:X
116:n
112:n
108:n
93:k
86:t
20:)
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