Nonlinear system - Knowledge (XXG) (2024)

1867:

1859:

1451:, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.620:

behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.

1836:

can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.

1349:

The general root-finding algorithms apply to polynomial roots, but, generally they do not find all the roots, and when they fail to find a root, this does not imply that there is no roots. Specific methods for polynomials allow finding all roots or the

2679:

This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact)

2114:

615:

are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble

1966:

2426:

935:

2257:

1404:. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related2674:

823:

2473:

1428:. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the2298:

766:

3270:

1813:

is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly linear). Sometimes, the equation may be transformed into one or more

1439:

One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of

3616:

1657:

1521:

2608:

2499:

1565:

2535:

will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.

2324:

2570:

2354:

1365:, that is finding the common zeros of a set of several polynomials in several variables is a difficult problem for which elaborated algorithms have been designed, such as1347:

2023:

592:

that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is

4011:

2533:

2191:

4135:

4016:

3254:

1804:

2158:

1989:

1284:

1117:

976:

3879:

1707:

1594:

1203:

1166:

1075:

1012:

688:

4006:

1046:

1727:

1680:

1408:

and analysis procedures. These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.

1227:

1137:

3609:

3666:

2792:

492:

2163:

Another way to approach the problem is to linearize any nonlinearity (the sine function term in this case) at the various points of interest through

3779:

3221:

2035:

1798:

580:

which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a

3823:

3602:

3047:

Mosconi, Francesco; Julou, Thomas; Desprat, Nicolas; Sinha, Deepak Kumar; Allemand, Jean-François; Vincent Croquette; Bensimon, David (2008).

3717:

3543:

3521:

3495:

3476:

3267:

4128:

4001:

1991:

is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use

2782:

1788:

Existence of solutions of Finite-Duration, which can happen under specific conditions for some non-linear ordinary differential equations.

596:

if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.

3926:

3707:

4291:

3231:

2996:

Korenberg, Michael J.; Hunter, Ian W. (March 1996). "The identification of nonlinear biological systems: Volterra kernel approaches".

239:

122:

3712:

3676:

3457:

2981:

1896:

2816:

2362:

553:, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler2940:

3774:

3686:

3385:

Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013

863:

857:

is additive but not hom*ogeneous. The conditions of additivity and hom*ogeneity are often combined in the superposition principle

4425:

4121:

3742:

2861:

2696:– any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system1405:

1362:

4057:

3813:

2787:

2199:

4483:

3656:

1814:

1460:

406:

3828:

1241:

A nonlinear system of equations consists of a set of equations in several variables such that at least one of them is not a

273:

4387:

3671:

3661:

1810:

599:

As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (

526:

485:

277:

181:

36:

2616:

1433:

773:

4617:

3681:

3651:

1825:

1448:

1380:

and its variants. Generally they may provide a solution, but do not provide any information on the number of solutions.

43:

3570:

2802:

2434:

1429:

4233:

3916:

3911:

3786:

3747:

1736:

Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield

126:

2772:

4612:

4607:

4478:

4203:

2740:

2327:

1425:

603:). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as102:

3956:

2265:

701:

632:

Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals.

4261:

4092:

4062:

4037:

1841:

214:

32:

31:

This article is about "nonlinearity" in mathematics, physics and other sciences. For video and film editing, see

2123:. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the4397:

4339:

4266:

4198:

3951:

612:

478:

465:

355:

219:

189:

135:

3961:

1606:

2330:

corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at

1473:

4402:

4357:

4296:

4228:

4188:

3931:

3921:

3818:

2832:

2575:

1833:

1818:

1464:

1444:

1373:

695:

243:

194:

157:

94:

4488:

4329:

2842:

2124:

1875:

1853:

1829:

1741:

1737:

1417:

1287:

589:

585:

577:

456:

2478:

1824:

Another common (though less mathematical) tactic, often exploited in fluid and heat mechanics, is to use

4301:

4218:

4077:

4072:

3966:

3890:

3869:

3737:

3732:

3727:

3722:

3646:

3625:

1529:

1421:

1230:

569:

384:

379:

247:

4324:

451:

2303:

4460:

4420:

4362:

3971:

3895:

3864:

3445:

3303:

3176:

3121:

3060:

2837:

1883:

1821:, which is always useful whether or not the resulting ordinary differential equation(s) is solvable.1752:

1440:

1078:

2541:

4440:

4334:

4276:

4173:

3981:

3874:

3768:

2797:

2505:, and note that unlike the small angle approximation, this approximation is unstable, meaning that2502:

1782:

1401:

1389:

1355:

1300:

836:

460:

435:

418:

327:

227:

161:

84:

2333:

1747:

Common methods for the qualitative analysis of nonlinear ordinary differential equations include:

1307:

4586:

4581:

4536:

4493:

4455:

4412:

4271:

4102:

3941:

3936:

3859:

3691:

3412:

3335:

3248:

3145:

3111:

3084:

3029:

2767:

2762:

2757:

2026:

1994:

1777:

1756:

581:

430:

351:

347:

289:

281:

152:

2702:– values of a system cannot be predicted indefinitely far into the future, and fluctuations are1377:

2508:

2170:

1396:

as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the

4498:

4470:

4193:

4163:

3539:

3531:

3517:

3491:

3472:

3453:

3327:

3319:

3227:

3202:

3137:

3076:

3021:

3013:

2977:

2876:

2826:

2777:

2120:

1762:

1730:

1169:

637:

263:

223:

114:

98:

67:

2130:

1974:

1251:

4508:

4286:

4213:

4087:

4067:

3404:

3366:

3311:

3192:

3184:

3129:

3068:

3005:

2969:

2866:

2821:

2811:

2751:

2726:

2164:

1769:

1087:

946:

550:

371:

339:

297:

235:

110:

1685:

1139:

can be any sensible mathematical object (number, vector, function, etc.), and the function

4556:

4503:

4392:

4349:

4281:

4238:

4223:

4183:

4047:

3976:

3274:

2896:

2847:

2693:

1570:

1242:

1179:

1173:

1142:

1051:

988:

829:

664:

367:

363:

293:

231:

185:

147:

130:

118:

89:

4052:

3450:

Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness

3165:"Topological properties of a self-assembled electrical network via ab initio calculation"3102:

Gintautas, V. (2008). "Resonant forcing of nonlinear systems of differential equations".

1366:

1025:

3307:

3180:

3125:

3064:

4546:

4523:

4450:

4208:

4144:

4042:

3849:

3565:

3510:

3505:

3197:

3164:

2891:

2806:

2708:

2681:

1866:

1712:

1665:

1212:

1122:

854:

847:

426:

359:

313:

106:

53:

4601:

4551:

4513:

4372:

3536:

Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition

3429:

3088:

3072:

2886:

2881:

2745:

1887:

600:

554:

542:

302:

169:

3416:

3149:

3033:

4445:

4377:

4306:

4097:

4032:

3946:

3339:

2699:

1874:

A classic, extensively studied nonlinear problem is the dynamics of a frictionless

1397:

608:

422:

252:

173:

2916:

4576:

4571:

4561:

4531:

4430:

4382:

4316:

3854:

2973:

2968:, The Frontiers Collection, Springer Berlin Heidelberg, 2007, pp.181–276,2720:

1351:

1172:, including integration or differentiation with associated constraints (such as840:

650:

506:

446:

410:

401:

375:

343:

322:

317:

285:

177:

3579:

3396:

3048:

1372:

For the general case of system of equations formed by equating to zero several

560:

Typically, the behavior of a nonlinear system is described in mathematics by a

4367:

4253:

4082:

3752:

3371:

3354:

2723:– asymptotic periodic orbits to which destabilized fixed points are attracted.1295:

1206:

654:

617:

573:

209:

165:

3323:

3080:

3017:

2109:{\displaystyle \int {\frac {d\theta }{\sqrt {C_{0}+2\cos(\theta )}}}=t+C_{1}}17:

4435:

4248:

4158:

3844:

3408:

2703:

1858:

546:

538:

534:

414:

205:

72:

3331:

3206:

3141:

1600:

tends to infinity). The equation is nonlinear because it may be written as

1467:, especially for autonomous equations. For example, the nonlinear equation3025:

2729:– feedback oscillations taking place in open dissipative physical systems.1844:

and using the methods outlined above for ordinary differential equations.

4541:

3884:

3594:

3220:

de Canete, Javier, Cipriano Galindo, and Inmaculada Garcia-Moral (2011).

2871:

1393:

565:

530:

335:

1886:, it may be shown that the motion of a pendulum can be described by the4566:

3009:

2714:

1879:

604:

510:

331:

268:

27:

System where changes of output are not proportional to changes of input

3574:

3278:

3223:

System Engineering and Automation: An Interactive Educational Approach

3188:

3133:

3583:

628:

for the study of nonlinear systems. This term is disputed by others:

522:

3315:

3291:

2941:"Nonlinear systems, Applied Mathematics - University of Birmingham"1662:

and the left-hand side of the equation is not a linear function of

4113:

3590:

The Center for Nonlinear Studies at Los Alamos National Laboratory

3566:

New England Complex Systems Institute: Concepts in Complex Systems

3116:

1865:

1857:

529:

to the change of the input. Nonlinear problems are of interest to

1443:

solutions can be used to construct general solutions through the

549:

since most systems are inherently nonlinear in nature. Nonlinear

1447:. A good example of this is one-dimensional heat transport with4117:

3598:

853:, hom*ogeneity does not follow from additivity. For example, an568:

in which the unknowns (or the unknown functions in the case of

1961:{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+\sin(\theta )=0}1828:

to simplify a general, natural equation in a certain specific

2421:{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+\pi -\theta =0}1740:

solutions, though implicit solutions and solutions involving

3560:

930:{\displaystyle f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)}42:"Nonlinear dynamics" redirects here. For the journal, see690:

is one which satisfies both of the following properties:

3589:

2252:{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+\theta =0}1596:

corresponding to the limit of the general solution when

2538:

One more interesting linearization is possible around

777:

705:

3355:"Thirty years of Polynomial System Solving, and now?"2619:

2578:

2544:

2511:

2481:

2437:

2365:

2336:

2306:

2268:

2202:

2173:

2133:

2038:

1997:

1977:

1899:

1809:

The most common basic approach to studying nonlinear

1715:

1688:

1668:

1609:

1573:

1567:

as a general solution (and also the special solution

1532:

1476:

1310:

1254:

1215:

1182:

1145:

1125:

1090:

1054:

1028:

991:

949:

866:

776:

704:

667:

4522:

4469:

4411:

4348:

4315:

4247:

4172:

4151:

4025:

3994:

3904:

3837:

3806:

3799:

3761:

3700:

3639:

3632:

2669:{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+1=0.}2356:, corresponding to the pendulum being straight up:1785:

methods (can be applied to algebraic equations too)

818:{\displaystyle \textstyle f(\alpha x)=\alpha f(x).}3509:

2668:

2602:

2564:

2527:

2493:

2467:

2420:

2348:

2318:

2292:

2251:

2185:

2152:

2108:

2017:

1983:

1960:

1774:Change of variables into something easier to study1721:

1701:

1674:

1651:

1588:

1559:

1515:

1341:

1278:

1221:

1197:

1160:

1131:

1111:

1069:

1040:

1006:

970:

929:

817:

760:

682:

576:of degree higher than one or in the argument of a4012:List of nonlinear ordinary differential equations3401:1985 24th IEEE Conference on Decision and Control2468:{\displaystyle \sin(\theta )\approx \pi -\theta }4017:List of nonlinear partial differential equations1805:List of nonlinear partial differential equations630:

4007:List of linear ordinary differential equations4129:

3610:

486:

8:

3471:(fourthed.). Oxford University Press.3253:: CS1 maint: multiple names: authors list (2293:{\displaystyle \sin(\theta )\approx \theta }761:{\displaystyle \textstyle f(x+y)=f(x)+f(y);}3561:Command and Control Research Program (CCRP)2711:– the presence of two or more stable states2193:, called the small angle approximation, is2119:which is an implicit solution involving an1761:Examination of dissipative quantities (see4136:

4122:

4114:

3803:

3636:

3617:

3603:

3595:

3430:David Tong: Lectures on Classical Dynamics1400:and the relations that define the various493:

479:

49:

3469:Nonlinear Ordinary Differential Equations3370:

3196:

3115:

2917:"Explained: Linear and nonlinear systems"2645:

2627:

2620:

2618:

2577:

2554:

2543:

2520:

2512:

2510:

2480:

2436:

2391:

2373:

2366:

2364:

2335:

2305:

2267:

2228:

2210:

2203:

2201:

2172:

2138:

2132:

2100:

2057:

2042:

2037:

2004:

1996:

1976:

1925:

1907:

1900:

1898:

1714:

1693:

1687:

1667:

1637:

1610:

1608:

1572:

1539:

1531:

1507:

1477:

1475:

1315:

1309:

1253:

1214:

1181:

1144:

1124:

1089:

1053:

1027:

990:

948:

865:

775:

703:

666:

525:in which the change of the output is not2501:. The solution to this problem involves1652:{\displaystyle {\frac {du}{dx}}+u^{2}=0}3290:Campbell, David K. (25 November 2004).2908:

1799:Nonlinear partial differential equation1516:{\displaystyle {\frac {du}{dx}}=-u^{2}}1424:is said to be nonlinear if it is not a1014:is a linear map (as defined above) and828:Additivity implies hom*ogeneity for any59:

52:

3246:

3049:"Some nonlinear challenges in biology"2603:{\displaystyle \sin(\theta )\approx 1}1682:and its derivatives. Note that if the1971:where gravity points "downwards" and1286:many methods have been designed; see7:

4002:List of named differential equations3397:"Finite Time Differential Equations"2688:Types of nonlinear dynamic behaviors2167:. For example, the linearization at1840:Other methods include examining the1832:. For example, the (very) nonlinear3927:Method of undetermined coefficients3708:Dependent and independent variables3292:"Nonlinear physics: Fresh breather"2494:{\displaystyle \theta \approx \pi }1765:) analogous to conserved quantities1729:, the problem would be linear (the4292:Evolutionary developmental biology1560:{\displaystyle u={\frac {1}{x+C}}}1248:For a single equation of the form1018:otherwise. The equation is called240:Evolutionary developmental biology25:

3467:Jordan, D. W.; Smith, P. (2007).3448:and Anthony J. Pritchard (2005).3163:Stephenson, C.; et., al. (2017).2717:– self-reinforcing solitary waves564:, which is a set of simultaneous3824:Carathéodory's existence theorem3586:) a Database of Physical Systems3512:Advanced Engineering Mathematics3226:. Berlin: Springer. p.46.2998:Annals of Biomedical Engineering2793:Landau–Lifsh*tz–Gilbert equation2319:{\displaystyle \theta \approx 0}1412:Nonlinear differential equations4426:Ordinary differential equations3359:Journal of Symbolic Computation2862:Aleksandr Mikhailovich Lyapunov2783:Kadomtsev–Petviashvili equation2734:Examples of nonlinear equations2029:, which would eventually yield1815:ordinary differential equations1461:ordinary differential equations1455:Ordinary differential equations1406:nonlinear system identification1363:systems of polynomial equations407:Ordinary differential equations4484:Partial differential equations2817:Nonlinear Schrödinger equation2591:

2585:

2565:{\displaystyle \theta =\pi /2}2521:

2513:

2450:

2444:

2281:

2275:

2081:

2075:

1949:

1943:

1811:partial differential equations1793:Partial differential equations1463:are often exactly solvable by1392:defines successive terms of a1384:Nonlinear recurrence relations1264:

1258:

1192:

1186:

1155:

1149:

1100:

1094:

1064:

1058:

1001:

995:

959:

953:

924:

918:

906:

900:

888:

870:

808:

802:

790:

781:

751:

745:

736:

730:

721:

709:

677:

671:

278:Partial differential equations1:

1449:Dirichlet boundary conditions562:nonlinear system of equations37:Nonlinearity (disambiguation)3652:Notation for differentiation2349:{\displaystyle \theta =\pi }1870:Linearizations of a pendulum1342:{\displaystyle x^{2}+x-1=0.}44:Nonlinear Dynamics (journal)4234:Particle swarm optimization3748:Exact differential equation3571:Nonlinear Dynamics I: Chaos3268:Nonlinear Dynamics I: Chaos2974:10.1007/978-3-540-34153-6_72018:{\displaystyle d\theta /dt}1237:Nonlinear systems equations572:) appear as variables of a127:Particle swarm optimization4636:

4479:Reaction-diffusion systems4204:Self-organized criticality3486:Khalil, Hassan K. (2001).3073:10.1088/0951-7715/21/8/T032829:for unsaturated water flow2788:Korteweg–de Vries equation2741:Algebraic Riccati equation2328:simple harmonic oscillator1862:Illustration of a pendulum1851:

1802:

1796:

1432:in fluid dynamics and the1426:system of linear equations624:Some authors use the term274:Reaction–diffusion systems103:Self-organized criticality29:

4262:Artificial neural network4058:Józef Maria Hoene-Wroński4038:Gottfried Wilhelm Leibniz3829:Cauchy–Kowalevski theorem3372:10.1016/j.jsc.2008.03.0042684:and approximate periods.2528:{\displaystyle |\theta |}2186:{\displaystyle \theta =0}215:Artificial neural network33:Non-linear editing system4340:Evolutionary game theory4267:Evolutionary computation4199:Collective consciousness3952:Finite difference method3395:Vardia T. Haimo (1985).1709:term were replaced with1434:Lotka–Volterra equations1374:differentiable functions1119:is very general in that466:Evolutionary game theory356:Second-order cybernetics220:Evolutionary computation136:Collective consciousness4358:Social network analysis4297:Artificial intelligence4229:Ant colony optimization4189:Collective intelligence3932:Variation of parameters3922:Separation of variables3819:Peano existence theorem3814:Picard–Lindelöf theorem3701:Attributes of variables3582:– (in3580:Nonlinear Model Library3409:10.1109/CDC.1985.2688322833:Self-balancing unicycle2803:Navier–Stokes equations2153:{\displaystyle C_{0}=2}1984:{\displaystyle \theta }1878:under the influence of1834:Navier-Stokes equations1819:separation of variables1742:nonelementary integrals1465:separation of variables1445:superposition principle1430:Navier–Stokes equations1279:{\displaystyle f(x)=0,}1229:, the result will be a940:An equation written as696:superposition principle244:Artificial intelligence158:Social network analysis123:Ant colony optimization95:Collective intelligence4489:Dissipative structures4330:Rational choice theory4093:Carl David Tolmé Runge3667:Differential-algebraic3626:Differential equations3403:. pp.1729–1733.2966:The Nonlinear Universe2843:Van der Pol oscillator2773:Ginzburg–Landau theory2670:

2604:

2566:

2529:

2495:

2469:

2422:

2350:

2320:

2294:

2253:

2187:

2154:

2127:(nonelementary unless2125:nonelementary integral2110:

2019:

1985:

1962:

1871:

1863:

1854:Pendulum (mathematics)1830:boundary value problem1723:

1703:

1676:

1653:

1590:

1561:

1517:

1422:differential equations1343:

1288:Root-finding algorithm1280:

1223:

1199:

1162:

1133:

1113:

1112:{\displaystyle f(x)=C}1071:

1042:

1008:

972:

971:{\displaystyle f(x)=C}931:

819:

762:

684:

642:

570:differential equations457:Rational choice theory282:Dissipative structures35:. For other uses, see4302:Evolutionary robotics4219:Agent-based modelling4078:Augustin-Louis Cauchy4073:Joseph-Louis Lagrange3967:Finite element method3957:Crank–Nicolson method3891:Numerical integration3870:Exponential stability3762:Relation to processes3647:Differential operator2964:"Nonlinear Biology",2671:

2605:

2567:

2530:

2496:

2470:

2423:

2351:

2321:

2295:

2254:

2188:

2155:

2111:

2020:

1986:

1963:

1869:

1861:

1724:

1704:

1702:{\displaystyle u^{2}}1677:

1654:

1591:

1562:

1518:

1376:, the main method is1344:

1281:

1231:differential equation1224:

1200:

1168:can literally be any1163:

1134:

1114:

1072:

1043:

1009:

973:

932:

820:

763:

685:

385:Theory of computation248:Evolutionary robotics115:Agent-based modelling4461:Coupled map lattices4421:Time series analysis4363:Small-world networks3972:Finite volume method3896:Dirac delta function3865:Asymptotic stability3807:Existence/uniqueness3672:Integro-differential3575:MIT's OpenCourseWare3446:Diederich Hinrichsen3279:MIT's OpenCourseWare2945:www.birmingham.ac.uk2838:Sine-Gordon equation2617:

2576:

2542:

2509:

2503:hyperbolic sinusoids2479:

2435:

2363:

2334:

2304:

2266:

2200:

2171:

2131:

2036:

1995:

1975:

1897:

1884:Lagrangian mechanics1753:conserved quantities1713:

1686:

1666:

1607:

1589:{\displaystyle u=0,}1571:

1530:

1474:

1441:linearly independent1402:Hofstadter sequences1308:

1290:. In the case where1252:

1213:

1198:{\displaystyle f(x)}1180:

1161:{\displaystyle f(x)}1143:

1123:

1088:

1079:hom*ogeneous function1070:{\displaystyle f(x)}1052:

1026:

1007:{\displaystyle f(x)}989:

947:

864:

837:continuous functions774:

702:

683:{\displaystyle f(x)}665:

436:Coupled map lattices402:Time series analysis162:Small-world networks4618:Concepts in physics4441:Population dynamics4335:Bounded rationality4277:Genetic programming3982:Perturbation theory3962:Runge–Kutta methods3942:Integral transforms3875:Rate of convergence3771:(discrete analogue)3452:. Springer Verlag.3353:Lazard, D. (2009).3308:2004Natur.432..455C3181:2017NatSR...741621S3126:2008Chaos..18c3118G3065:2008Nonli..21..131M1890:nonlinear equation1757:Hamiltonian systems1751:Examination of any1390:recurrence relation1356:real-root isolation1301:polynomial equation1041:{\displaystyle C=0}461:Bounded rationality419:Population dynamics328:Conversation theory228:Genetic programming153:Scale-free networks85:Collective behavior4587:Computation theory4582:Information theory4537:Operationalization4413:Nonlinear dynamics4325:Prisoner's dilemma4272:Genetic algorithms4103:Sofya Kovalevskaya3937:Integrating factor3860:Lyapunov stability3780:Stochastic partial3273:2008-02-12 at the3010:10.1007/bf026673542768:General relativity2763:Colebrook equation2758:Boltzmann equation2754:for optimal policy2666:

2600:

2562:

2525:

2491:

2465:

2418:

2346:

2316:

2290:

2249:

2183:

2150:

2106:

2027:integrating factor2015:

1981:

1958:

1872:

1864:

1778:Bifurcation theory1768:Linearization via1719:

1699:

1672:

1649:

1586:

1557:

1513:

1339:

1276:

1219:

1195:

1158:

1129:

1109:

1067:

1038:

1004:

968:

927:

815:

814:

758:

757:

680:

582:linear combination452:Prisoner's dilemma397:Nonlinear dynamics352:Operationalization348:Information theory224:Genetic algorithms4613:Dynamical systems4608:Nonlinear systems4595:

4594:

4499:Cellular automata4471:Pattern formation4403:Adaptive networks4194:Collective action4164:Self-organization4111:

4110:

3990:

3989:

3795:

3794:

3545:978-0-387-98489-63523:978-0-471-15496-93497:978-0-13-067389-33490:. Prentice Hall.3488:Nonlinear Systems3478:978-0-19-920824-13302:(7016): 455–456.3189:10.1038/srep416213134:10.1063/1.29642002877:Initial condition2827:Richards equation2778:Ishimori equation2727:Self-oscillations2652:

2398:

2235:

2165:Taylor expansions2121:elliptic integral2085:

2084:

1932:

1763:Lyapunov function1744:are encountered.1731:exponential decay1722:{\displaystyle u}1675:{\displaystyle u}1628:

1555:

1495:

1222:{\displaystyle x}1132:{\displaystyle x}626:nonlinear science551:dynamical systems545:, and many other519:non-linear system503:

502:

290:Cellular automata264:Pattern formation195:Adaptive networks99:Collective action68:Self-organization16:(Redirected from4625:

4509:Self-replication4398:Dynamic networks4287:Machine learning4214:Phase transition4138:

4131:

4124:

4115:

4088:Phyllis Nicolson4068:Rudolf Lipschitz3905:Solution methods3880:Series solutions3804:

3637:

3619:

3612:

3605:

3596:

3549:

3527:

3515:

3501:

3482:

3463:

3432:

3427:

3421:

3420:

3392:

3386:

3383:

3377:

3376:

3374:

3350:

3344:

3343:

3287:

3281:

3265:

3259:

3258:

3252:

3244:

3242:

3240:

3217:

3211:

3210:

3200:

3160:

3154:

3153:

3119:

3099:

3093:

3092:

3044:

3038:

3037:

2993:

2987:

2986:

2961:

2955:

2954:

2952:

2951:

2937:

2931:

2930:

2928:

2927:

2913:

2867:Dynamical system2822:Power-flow study2812:Nonlinear optics2798:Liénard equation2752:Bellman equation2675:

2673:

2672:

2667:

2653:

2651:

2650:

2649:

2636:

2632:

2631:

2621:

2609:

2607:

2606:

2601:

2571:

2569:

2568:

2563:

2558:

2534:

2532:

2531:

2526:

2524:

2516:

2500:

2498:

2497:

2492:

2474:

2472:

2471:

2466:

2427:

2425:

2424:

2419:

2399:

2397:

2396:

2395:

2382:

2378:

2377:

2367:

2355:

2353:

2352:

2347:

2325:

2323:

2322:

2317:

2299:

2297:

2296:

2291:

2258:

2256:

2255:

2250:

2236:

2234:

2233:

2232:

2219:

2215:

2214:

2204:

2192:

2190:

2189:

2184:

2159:

2157:

2156:

2151:

2143:

2142:

2115:

2113:

2112:

2107:

2105:

2104:

2086:

2062:

2061:

2052:

2051:

2043:

2024:

2022:

2021:

2016:

2008:

1990:

1988:

1987:

1982:

1967:

1965:

1964:

1959:

1933:

1931:

1930:

1929:

1916:

1912:

1911:

1901:

1770:Taylor expansion1755:, especially in1728:

1726:

1725:

1720:

1708:

1706:

1705:

1700:

1698:

1697:

1681:

1679:

1678:

1673:

1658:

1656:

1655:

1650:

1642:

1641:

1629:

1627:

1619:

1611:

1595:

1593:

1592:

1587:

1566:

1564:

1563:

1558:

1556:

1554:

1540:

1522:

1520:

1519:

1514:

1512:

1511:

1496:

1494:

1486:

1478:

1348:

1346:

1345:

1340:

1320:

1319:

1293:

1285:

1283:

1282:

1277:

1228:

1226:

1225:

1220:

1209:with respect to1204:

1202:

1201:

1196:

1167:

1165:

1164:

1159:

1138:

1136:

1135:

1130:

1118:

1116:

1115:

1110:

1076:

1074:

1073:

1068:

1047:

1045:

1044:

1039:

1013:

1011:

1010:

1005:

977:

975:

974:

969:

936:

934:

933:

928:

824:

822:

821:

816:

767:

765:

764:

759:

689:

687:

686:

681:

640:

515:nonlinear system495:

488:

481:

372:Systems thinking298:Self-replication236:Machine learning190:Dynamic networks111:Phase transition50:

47:

40:

21:

4635:

4634:

4628:

4627:

4626:

4624:

4623:

4622:

4598:

4597:

4596:

4591:

4557:System dynamics4518:

4504:Spatial ecology4465:

4407:

4393:Systems biology4344:

4311:

4282:Artificial life4252:

4243:

4239:Swarm behaviour4224:Synchronization4184:Social dynamics4175:

4168:

4147:

4145:Complex systems4142:

4112:

4107:

4048:Jacob Bernoulli4021:

3986:

3977:Galerkin method3900:

3838:Solution topics3833:

3791:

3757:

3696:

3628:

3623:

3557:

3552:

3546:

3532:Sontag, Eduardo3530:

3524:

3506:Kreyszig, Erwin3504:

3498:

3485:

3479:

3466:

3460:

3444:

3440:

3438:Further reading3435:

3428:

3424:

3394:

3393:

3389:

3384:

3380:

3352:

3351:

3347:

3316:10.1038/432455a3289:

3288:

3284:

3275:Wayback Machine3266:

3262:

3245:

3238:

3236:

3234:

3219:

3218:

3214:

3162:

3161:

3157:

3101:

3100:

3096:

3046:

3045:

3041:

2995:

2994:

2990:

2984:

2963:

2962:

2958:

2949:

2947:

2939:

2938:

2934:

2925:

2923:

2915:

2914:

2910:

2906:

2901:

2897:Volterra series2857:

2852:

2848:Vlasov equation2736:

2694:Amplitude death2690:

2682:phase portraits2641:

2637:

2623:

2622:

2615:

2614:

2574:

2573:

2572:, around which2540:

2539:

2507:

2506:

2477:

2476:

2433:

2432:

2387:

2383:

2369:

2368:

2361:

2360:

2332:

2331:

2302:

2301:

2264:

2263:

2224:

2220:

2206:

2205:

2198:

2197:

2169:

2168:

2134:

2129:

2128:

2096:

2053:

2044:

2034:

2033:

1993:

1992:

1973:

1972:

1921:

1917:

1903:

1902:

1895:

1894:

1856:

1850:

1842:characteristics1807:

1801:

1795:

1711:

1710:

1689:

1684:

1683:

1664:

1663:

1633:

1620:

1612:

1605:

1604:

1569:

1568:

1544:

1528:

1527:

1503:

1487:

1479:

1472:

1471:

1457:

1414:

1386:

1378:Newton's method1311:

1306:

1305:

1291:

1250:

1249:

1243:linear equation1239:

1211:

1210:

1207:differentiation1178:

1177:

1174:boundary values1141:

1140:

1121:

1120:

1086:

1085:

1084:The definition1050:

1049:

1024:

1023:

987:

986:

945:

944:

862:

861:

772:

771:

700:

699:

663:

662:

659:linear function647:

641:

636:

584:of the unknown499:

470:

469:

468:

463:

459:

454:

449:

439:

438:

433:

429:

425:

421:

417:

413:

409:

404:

399:

389:

388:

387:

382:

378:

374:

370:

368:Systems science366:

364:System dynamics362:

358:

354:

350:

346:

342:

338:

334:

330:

325:

320:

306:

305:

300:

296:

294:Spatial ecology292:

288:

284:

280:

276:

271:

266:

256:

255:

250:

246:

242:

238:

234:

232:Artificial life230:

226:

222:

217:

212:

198:

197:

192:

188:

186:Systems biology184:

180:

176:

172:

168:

164:

160:

155:

150:

140:

139:

138:

133:

131:Swarm behaviour129:

125:

121:

119:Synchronization117:

113:

109:

105:

101:

97:

92:

90:Social dynamics87:

77:

76:

75:

70:

54:Complex systems48:

41:

30:

28:

23:

22:

15:

12:

11:

5:

4633:

4632:

4629:

4621:

4620:

4615:

4610:

4600:

4599:

4593:

4592:

4590:

4589:

4584:

4579:

4574:

4569:

4564:

4559:

4554:

4549:

4547:Self-reference4544:

4539:

4534:

4528:

4526:

4524:Systems theory4520:

4519:

4517:

4516:

4511:

4506:

4501:

4496:

4491:

4486:

4481:

4475:

4473:

4467:

4466:

4464:

4463:

4458:

4453:

4451:Multistability4448:

4443:

4438:

4433:

4428:

4423:

4417:

4415:

4409:

4408:

4406:

4405:

4400:

4395:

4390:

4385:

4380:

4375:

4370:

4365:

4360:

4354:

4352:

4346:

4345:

4343:

4342:

4337:

4332:

4327:

4321:

4319:

4313:

4312:

4310:

4309:

4304:

4299:

4294:

4289:

4284:

4279:

4274:

4269:

4264:

4258:

4256:

4245:

4244:

4242:

4241:

4236:

4231:

4226:

4221:

4216:

4211:

4209:Herd mentality4206:

4201:

4196:

4191:

4186:

4180:

4178:

4170:

4169:

4167:

4166:

4161:

4155:

4153:

4149:

4148:

4143:

4141:

4140:

4133:

4126:

4118:

4109:

4108:

4106:

4105:

4100:

4095:

4090:

4085:

4080:

4075:

4070:

4065:

4063:Ernst Lindelöf4060:

4055:

4050:

4045:

4043:Leonhard Euler4040:

4035:

4029:

4027:

4026:Mathematicians4023:

4022:

4020:

4019:

4014:

4009:

4004:

3998:

3996:

3992:

3991:

3988:

3987:

3985:

3984:

3979:

3974:

3969:

3964:

3959:

3954:

3949:

3944:

3939:

3934:

3929:

3924:

3919:

3914:

3908:

3906:

3902:

3901:

3899:

3898:

3893:

3888:

3882:

3877:

3872:

3867:

3862:

3857:

3852:

3850:Phase portrait3847:

3841:

3839:

3835:

3834:

3832:

3831:

3826:

3821:

3816:

3810:

3808:

3801:

3797:

3796:

3793:

3792:

3790:

3789:

3784:

3783:

3782:

3772:

3765:

3763:

3759:

3758:

3756:

3755:

3753:On jet bundles3750:

3745:

3740:

3735:

3730:

3725:

3720:

3718:Nonhom*ogeneous3715:

3710:

3704:

3702:

3698:

3697:

3695:

3694:

3689:

3684:

3679:

3674:

3669:

3664:

3659:

3654:

3649:

3643:

3641:

3634:

3633:Classification3630:

3629:

3624:

3622:

3621:

3614:

3607:

3599:

3593:

3592:

3587:

3577:

3568:

3563:

3556:

3555:External links3553:

3551:

3550:

3544:

3528:

3522:

3502:

3496:

3483:

3477:

3464:

3458:

3441:

3439:

3436:

3434:

3433:

3422:

3387:

3378:

3365:(3): 222–231.3345:

3282:

3260:

3233:978-36422022923232:

3212:

3155:

3094:

3039:

3004:(2): 250–268.2988:

2982:

2956:

2932:

2907:

2905:

2902:

2900:

2899:

2894:

2892:Vector soliton2889:

2884:

2879:

2874:

2869:

2864:

2858:

2856:

2853:

2851:

2850:

2845:

2840:

2835:

2830:

2824:

2819:

2814:

2809:

2807:fluid dynamics2800:

2795:

2790:

2785:

2780:

2775:

2770:

2765:

2760:

2755:

2749:

2743:

2737:

2735:

2732:

2731:

2730:

2724:

2718:

2712:

2709:Multistability2706:

2697:

2689:

2686:

2677:

2676:

2665:

2662:

2659:

2656:

2648:

2644:

2640:

2635:

2630:

2626:

2599:

2596:

2593:

2590:

2587:

2584:

2581:

2561:

2557:

2553:

2550:

2547:

2523:

2519:

2515:

2490:

2487:

2484:

2464:

2461:

2458:

2455:

2452:

2449:

2446:

2443:

2440:

2429:

2428:

2417:

2414:

2411:

2408:

2405:

2402:

2394:

2390:

2386:

2381:

2376:

2372:

2345:

2342:

2339:

2315:

2312:

2309:

2289:

2286:

2283:

2280:

2277:

2274:

2271:

2260:

2259:

2248:

2245:

2242:

2239:

2231:

2227:

2223:

2218:

2213:

2209:

2182:

2179:

2176:

2149:

2146:

2141:

2137:

2117:

2116:

2103:

2099:

2095:

2092:

2089:

2083:

2080:

2077:

2074:

2071:

2068:

2065:

2060:

2056:

2050:

2047:

2041:

2014:

2011:

2007:

2003:

2000:

1980:

1969:

1968:

1957:

1954:

1951:

1948:

1945:

1942:

1939:

1936:

1928:

1924:

1920:

1915:

1910:

1906:

1852:Main article:1849:

1846:

1826:scale analysis1797:Main article:1794:

1791:

1790:

1789:

1786:

1780:

1775:

1772:

1766:

1759:

1718:

1696:

1692:

1671:

1660:

1659:

1648:

1645:

1640:

1636:

1632:

1626:

1623:

1618:

1615:

1585:

1582:

1579:

1576:

1553:

1550:

1547:

1543:

1538:

1535:

1524:

1523:

1510:

1506:

1502:

1499:

1493:

1490:

1485:

1482:

1456:

1453:

1413:

1410:

1385:

1382:

1338:

1335:

1332:

1329:

1326:

1323:

1318:

1314:

1275:

1272:

1269:

1266:

1263:

1260:

1257:

1238:

1235:

1218:

1194:

1191:

1188:

1185:

1157:

1154:

1151:

1148:

1128:

1108:

1105:

1102:

1099:

1096:

1093:

1066:

1063:

1060:

1057:

1037:

1034:

1031:

1003:

1000:

997:

994:

979:

978:

967:

964:

961:

958:

955:

952:

938:

937:

926:

923:

920:

917:

914:

911:

908:

905:

902:

899:

896:

893:

890:

887:

884:

881:

878:

875:

872:

869:

855:antilinear map826:

825:

813:

810:

807:

804:

801:

798:

795:

792:

789:

786:

783:

780:

768:

756:

753:

750:

747:

744:

741:

738:

735:

732:

729:

726:

723:

720:

717:

714:

711:

708:

694:Additivity or679:

676:

673:

670:

646:

643:

638:Stanisław Ulam634:

555:linear systems543:mathematicians501:

500:

498:

497:

490:

483:

475:

472:

471:

450:

445:

444:

441:

440:

427:Multistability400:

395:

394:

391:

390:

360:Self-reference321:

314:Systems theory312:

311:

308:

307:

267:

262:

261:

258:

257:

213:

204:

203:

200:

199:

151:

146:

145:

142:

141:

107:Herd mentality88:

83:

82:

79:

78:

71:

66:

65:

62:

61:

57:

56:

26:

24:

14:

13:

10:

9:

6:

4:

3:

2:

4631:

4630:

4619:

4616:

4614:

4611:

4609:

4606:

4605:

4603:

4588:

4585:

4583:

4580:

4578:

4575:

4573:

4570:

4568:

4565:

4563:

4560:

4558:

4555:

4553:

4552:Goal-oriented4550:

4548:

4545:

4543:

4540:

4538:

4535:

4533:

4530:

4529:

4527:

4525:

4521:

4515:

4514:Geomorphology4512:

4510:

4507:

4505:

4502:

4500:

4497:

4495:

4492:

4490:

4487:

4485:

4482:

4480:

4477:

4476:

4474:

4472:

4468:

4462:

4459:

4457:

4454:

4452:

4449:

4447:

4444:

4442:

4439:

4437:

4434:

4432:

4429:

4427:

4424:

4422:

4419:

4418:

4416:

4414:

4410:

4404:

4401:

4399:

4396:

4394:

4391:

4389:

4386:

4384:

4381:

4379:

4376:

4374:

4371:

4369:

4366:

4364:

4361:

4359:

4356:

4355:

4353:

4351:

4347:

4341:

4338:

4336:

4333:

4331:

4328:

4326:

4323:

4322:

4320:

4318:

4314:

4308:

4305:

4303:

4300:

4298:

4295:

4293:

4290:

4288:

4285:

4283:

4280:

4278:

4275:

4273:

4270:

4268:

4265:

4263:

4260:

4259:

4257:

4255:

4250:

4246:

4240:

4237:

4235:

4232:

4230:

4227:

4225:

4222:

4220:

4217:

4215:

4212:

4210:

4207:

4205:

4202:

4200:

4197:

4195:

4192:

4190:

4187:

4185:

4182:

4181:

4179:

4177:

4171:

4165:

4162:

4160:

4157:

4156:

4154:

4150:

4146:

4139:

4134:

4132:

4127:

4125:

4120:

4119:

4116:

4104:

4101:

4099:

4096:

4094:

4091:

4089:

4086:

4084:

4081:

4079:

4076:

4074:

4071:

4069:

4066:

4064:

4061:

4059:

4056:

4054:

4051:

4049:

4046:

4044:

4041:

4039:

4036:

4034:

4031:

4030:

4028:

4024:

4018:

4015:

4013:

4010:

4008:

4005:

4003:

4000:

3999:

3997:

3993:

3983:

3980:

3978:

3975:

3973:

3970:

3968:

3965:

3963:

3960:

3958:

3955:

3953:

3950:

3948:

3945:

3943:

3940:

3938:

3935:

3933:

3930:

3928:

3925:

3923:

3920:

3918:

3915:

3913:

3910:

3909:

3907:

3903:

3897:

3894:

3892:

3889:

3886:

3883:

3881:

3878:

3876:

3873:

3871:

3868:

3866:

3863:

3861:

3858:

3856:

3853:

3851:

3848:

3846:

3843:

3842:

3840:

3836:

3830:

3827:

3825:

3822:

3820:

3817:

3815:

3812:

3811:

3809:

3805:

3802:

3798:

3788:

3785:

3781:

3778:

3777:

3776:

3773:

3770:

3767:

3766:

3764:

3760:

3754:

3751:

3749:

3746:

3744:

3741:

3739:

3736:

3734:

3731:

3729:

3726:

3724:

3721:

3719:

3716:

3714:

3711:

3709:

3706:

3705:

3703:

3699:

3693:

3690:

3688:

3685:

3683:

3680:

3678:

3675:

3673:

3670:

3668:

3665:

3663:

3660:

3658:

3655:

3653:

3650:

3648:

3645:

3644:

3642:

3638:

3635:

3631:

3627:

3620:

3615:

3613:

3608:

3606:

3601:

3600:

3597:

3591:

3588:

3585:

3581:

3578:

3576:

3572:

3569:

3567:

3564:

3562:

3559:

3558:

3554:

3547:

3541:

3537:

3533:

3529:

3525:

3519:

3514:

3513:

3507:

3503:

3499:

3493:

3489:

3484:

3480:

3474:

3470:

3465:

3461:

3459:97835404412503455:

3451:

3447:

3443:

3442:

3437:

3431:

3426:

3423:

3418:

3414:

3410:

3406:

3402:

3398:

3391:

3388:

3382:

3379:

3373:

3368:

3364:

3360:

3356:

3349:

3346:

3341:

3337:

3333:

3329:

3325:

3321:

3317:

3313:

3309:

3305:

3301:

3297:

3293:

3286:

3283:

3280:

3276:

3272:

3269:

3264:

3261:

3256:

3250:

3235:

3229:

3225:

3224:

3216:

3213:

3208:

3204:

3199:

3194:

3190:

3186:

3182:

3178:

3174:

3170:

3166:

3159:

3156:

3151:

3147:

3143:

3139:

3135:

3131:

3127:

3123:

3118:

3113:

3110:(3): 033118.3109:

3105:

3098:

3095:

3090:

3086:

3082:

3078:

3074:

3070:

3066:

3062:

3058:

3054:

3050:

3043:

3040:

3035:

3031:

3027:

3023:

3019:

3015:

3011:

3007:

3003:

2999:

2992:

2989:

2985:

2983:97835403415292979:

2975:

2971:

2967:

2960:

2957:

2946:

2942:

2936:

2933:

2922:

2918:

2912:

2909:

2903:

2898:

2895:

2893:

2890:

2888:

2887:Mode coupling2885:

2883:

2882:Linear system2880:

2878:

2875:

2873:

2870:

2868:

2865:

2863:

2860:

2859:

2854:

2849:

2846:

2844:

2841:

2839:

2836:

2834:

2831:

2828:

2825:

2823:

2820:

2818:

2815:

2813:

2810:

2808:

2804:

2801:

2799:

2796:

2794:

2791:

2789:

2786:

2784:

2781:

2779:

2776:

2774:

2771:

2769:

2766:

2764:

2761:

2759:

2756:

2753:

2750:

2747:

2746:Ball and beam2744:

2742:

2739:

2738:

2733:

2728:

2725:

2722:

2719:

2716:

2713:

2710:

2707:

2705:

2701:

2698:

2695:

2692:

2691:

2687:

2685:

2683:

2663:

2660:

2657:

2654:

2646:

2642:

2638:

2633:

2628:

2624:

2613:

2612:

2611:

2597:

2594:

2588:

2582:

2579:

2559:

2555:

2551:

2548:

2545:

2536:

2517:

2504:

2488:

2485:

2482:

2462:

2459:

2456:

2453:

2447:

2441:

2438:

2415:

2412:

2409:

2406:

2403:

2400:

2392:

2388:

2384:

2379:

2374:

2370:

2359:

2358:

2357:

2343:

2340:

2337:

2329:

2313:

2310:

2307:

2287:

2284:

2278:

2272:

2269:

2246:

2243:

2240:

2237:

2229:

2225:

2221:

2216:

2211:

2207:

2196:

2195:

2194:

2180:

2177:

2174:

2166:

2161:

2147:

2144:

2139:

2135:

2126:

2122:

2101:

2097:

2093:

2090:

2087:

2078:

2072:

2069:

2066:

2063:

2058:

2054:

2048:

2045:

2039:

2032:

2031:

2030:

2028:

2012:

2009:

2005:

2001:

1998:

1978:

1955:

1952:

1946:

1940:

1937:

1934:

1926:

1922:

1918:

1913:

1908:

1904:

1893:

1892:

1891:

1889:

1888:dimensionless1885:

1881:

1877:

1868:

1860:

1855:

1847:

1845:

1843:

1838:

1835:

1831:

1827:

1822:

1820:

1817:, as seen in1816:

1812:

1806:

1800:

1792:

1787:

1784:

1781:

1779:

1776:

1773:

1771:

1767:

1764:

1760:

1758:

1754:

1750:

1749:

1748:

1745:

1743:

1739:

1734:

1732:

1716:

1694:

1690:

1669:

1646:

1643:

1638:

1634:

1630:

1624:

1621:

1616:

1613:

1603:

1602:

1601:

1599:

1583:

1580:

1577:

1574:

1551:

1548:

1545:

1541:

1536:

1533:

1508:

1504:

1500:

1497:

1491:

1488:

1483:

1480:

1470:

1469:

1468:

1466:

1462:

1454:

1452:

1450:

1446:

1442:

1437:

1435:

1431:

1427:

1423:

1419:

1411:

1409:

1407:

1403:

1399:

1395:

1391:

1383:

1381:

1379:

1375:

1370:

1368:

1364:

1359:

1357:

1353:

1336:

1333:

1330:

1327:

1324:

1321:

1316:

1312:

1303:

1302:

1297:

1289:

1273:

1270:

1267:

1261:

1255:

1246:

1244:

1236:

1234:

1232:

1216:

1208:

1189:

1183:

1175:

1171:

1152:

1146:

1126:

1106:

1103:

1097:

1091:

1082:

1080:

1061:

1055:

1035:

1032:

1029:

1021:

1017:

998:

992:

984:

965:

962:

956:

950:

943:

942:

941:

921:

915:

912:

909:

903:

897:

894:

891:

885:

882:

879:

876:

873:

867:

860:

859:

858:

856:

852:

849:

845:

842:

838:

834:

831:

811:

805:

799:

796:

793:

787:

784:

778:

770:hom*ogeneity:769:

754:

748:

742:

739:

733:

727:

724:

718:

715:

712:

706:

697:

693:

692:

691:

674:

668:

660:

656:

652:

644:

639:

633:

629:

627:

622:

619:

614:

613:singularities610:

606:

602:

601:linearization597:

595:

591:

587:

583:

579:

575:

571:

567:

563:

558:

556:

552:

548:

544:

540:

536:

532:

528:

524:

520:

516:

512:

508:

496:

491:

489:

484:

482:

477:

476:

474:

473:

467:

464:

462:

458:

453:

448:

443:

442:

437:

434:

432:

428:

424:

420:

416:

412:

408:

403:

398:

393:

392:

386:

383:

381:

377:

373:

369:

365:

361:

357:

353:

349:

345:

341:

340:Goal-oriented337:

333:

329:

324:

319:

315:

310:

309:

304:

303:Geomorphology301:

299:

295:

291:

287:

283:

279:

275:

270:

265:

260:

259:

254:

251:

249:

245:

241:

237:

233:

229:

225:

221:

216:

211:

207:

202:

201:

196:

193:

191:

187:

183:

179:

175:

171:

167:

163:

159:

154:

149:

144:

143:

137:

134:

132:

128:

124:

120:

116:

112:

108:

104:

100:

96:

91:

86:

81:

80:

74:

69:

64:

63:

58:

55:

51:

45:

38:

34:

19:

18:Non-linearity4378:Graph theory4307:Evolvability4098:Martin Kutta4053:Émile Picard4033:Isaac Newton3947:Euler method3917:Substitution3538:. Springer.3535:

3511:

3487:

3468:

3449:

3425:

3400:

3390:

3381:

3362:

3358:

3348:

3299:

3295:

3285:

3263:

3237:. Retrieved3222:

3215:

3172:

3168:

3158:

3107:

3103:

3097:

3056:

3053:Nonlinearity3052:

3042:

3001:

2997:

2991:

2965:

2959:

2948:. Retrieved2944:

2935:

2924:. Retrieved2920:

2911:

2721:Limit cycles2678:

2537:

2430:

2326:. This is a2261:

2162:

2118:

1970:

1873:

1839:

1823:

1808:

1783:Perturbation1746:

1735:

1661:

1597:

1525:

1459:First order1458:

1438:

1436:in biology.1415:

1398:logistic map1388:A nonlinear1387:

1371:

1369:algorithms.1367:Gröbner base1360:

1299:

1298:, one has a1247:

1240:

1083:

1019:

1015:

982:

980:

939:

850:

843:

832:

827:

658:

648:

631:

625:

623:

598:

593:

561:

559:

527:proportional518:

514:

504:

455:

405:

396:

326:

272:

253:Evolvability218:

174:Graph theory156:

93:

4577:Autopoiesis4572:Cybernetics4562:Sensemaking4532:Homeostasis4494:Percolation4456:Bifurcation4431:Phase space4317:Game theory4174:Collective3855:Phase space3713:hom*ogeneous3059:(8): T131.1738:closed-form1354:roots; see1020:hom*ogeneous835:, and, for651:mathematics507:mathematics447:Game theory431:Bifurcation411:Phase space376:Sensemaking344:Homeostasis323:Autopoiesis318:cybernetics286:Percolation4602:Categories4388:Robustness4368:Centrality4254:adaptation4152:Background4083:John Crank3912:Inspection3775:Stochastic3769:Difference3743:Autonomous3687:Non-linear3677:Fractional3640:Operations3239:20 January2950:2018-06-302926:2018-06-302904:References1803:See also:1733:problem).1296:polynomial981:is called839:, for any655:linear map645:Definition574:polynomial547:scientists539:physicists535:biologists415:Attractors210:adaptation182:Robustness166:Centrality4436:Attractor4249:Evolution4159:Emergence3887:solutions3845:Wronskian3800:Solutions3728:Decoupled3692:Holonomic3516:. Wiley.3324:0028-08363249:cite book3175:: 41621.3117:0803.22523089:1198082303081:0951-77153018:0090-69642704:aperiodic2634:θ2595:≈2589:θ2583:⁡2552:π2546:θ2518:θ2489:π2486:≈2483:θ2463:θ2460:−2457:π2454:≈2448:θ2442:⁡2410:θ2407:−2404:π2380:θ2344:π2338:θ2311:≈2308:θ2288:θ2285:≈2279:θ2273:⁡2241:θ2217:θ2175:θ2079:θ2073:⁡2049:θ2040:∫2002:θ1979:θ1947:θ1941:⁡1914:θ1501:−1328:−1205:contains1016:nonlinear913:β895:α883:β874:α797:α785:α590:functions586:variables566:equations531:engineers206:Evolution73:Emergence4542:Feedback4350:Networks4176:behavior3995:Examples3885:Integral3657:Ordinary3534:(1998).3508:(1998).3417:454263763332:155651393271:Archived3207:281558633169:Sci. Rep3150:183458173142:190454563034:206432062921:MIT News2872:Feedback2855:See also2715:Solitons1882:. Using1876:pendulum1394:sequence1361:Solving1304:such as846:. For a830:rational635:— 605:solitons578:function336:Feedback269:Fractals148:Networks4567:Entropy4383:Scaling3723:Coupled3662:Partial3340:44033323304:Bibcode3198:52907453177:Bibcode3122:Bibcode3061:Bibcode3026:86783571880:gravity1848:Pendula1170:mapping848:complex521:) is a511:science380:Variety332:Entropy178:Scaling4373:Motifs3738:Degree3682:Linear3584:MATLAB3542:3520:3494:3475:3456:3415:3338:3330:3322:3296:Nature3230:3205:3195:3148:3140:3087:3079:3032:3024:3016:2980:2748:system2431:since2262:since2025:as an1418:system1176:). If983:linear618:random611:, and594:linear523:system517:(or a170:Motifs60:Topics4446:Chaos3787:Delay3733:Order3413:S2CID3336:S2CID3146:S2CID3112:arXiv3104:Chaos3085:S2CID3030:S2CID2700:Chaos1294:is a1077:is a609:chaos423:Chaos4251:and3540:ISBN3518:ISBN3492:ISBN3473:ISBN3454:ISBN3328:PMID3320:ISSN3255:link3241:20183228:ISBN3203:PMID3138:PMID3077:ISSN3022:PMID3014:ISSN2978:ISBN2475:for2300:for1526:has1352:real1048:and841:real657:(or653:, a513:, a509:and316:and208:and3573:at3405:doi3367:doi3312:doi3300:4323277:at3193:PMC3185:doi3130:doi3069:doi3006:doi2970:doi2805:of2580:sin2439:sin2270:sin2160:).2070:cos1938:sin1420:of1022:if985:if649:In588:or505:In4604::3411:.3399:.3363:443361:.3357:.3334:.3326:.3318:.3310:.3298:.3294:.3251:}}3247:{{3201:.3191:.3183:.3171:.3167:.3144:.3136:.3128:.3120:.3108:183106:.3083:.3075:.3067:.3057:213055:.3051:.3028:.3020:.3012:.3002:243000:.2976:,2943:.2919:.2664:0.2610::1416:A1358:.1337:0.1245:.1233:.1081:.698::661:)607:,557:.541:,537:,533:,4137:e4130:t4123:v3618:e3611:t3604:v3548:.3526:.3500:.3481:.3462:.3419:.3407::3375:.3369::3342:.3314::3306::3257:)3243:.3209:.3187::3179::3173:73152:.3132::3124::3114::3091:.3071::3063::3036:.3008::2972::2953:.2929:.2661:=2658:12655:+2647:22643:t2639:d2629:22625:d2598:12592:)2586:(2560:22556:/2549:=2522:|2514:|2451:)2445:(2416:02413:=2401:+2393:22389:t2385:d2375:22371:d2341:=2314:02282:)2276:(2247:02244:=2238:+2230:22226:t2222:d2212:22208:d2181:02178:=2148:22145:=2140:02136:C2102:12098:C2094:+2091:t2088:=2082:)2076:(2067:22064:+2059:02055:C2046:d2013:t2010:d2006:/1999:d1956:01953:=1950:)1944:(1935:+1927:21923:t1919:d1909:21905:d1717:u1695:21691:u1670:u1647:01644:=1639:21635:u1631:+1625:x1622:d1617:u1614:d1598:C1584:,1581:01578:=1575:u1552:C1549:+1546:x1542:11537:=1534:u1509:21505:u1498:=1492:x1489:d1484:u1481:d1334:=1331:11325:x1322:+1317:21313:x1292:f1274:,1271:01268:=1265:)1262:x1259:(1256:f1217:x1193:)1190:x1187:(1184:f1156:)1153:x1150:(1147:f1127:x1107:C1104:=1101:)1098:x1095:(1092:f1065:)1062:x1059:(1056:f1036:01033:=1030:C1002:)999:x996:(993:f966:C963:=960:)957:x954:(951:f925:)922:y919:(916:f910:+907:)904:x901:(898:f892:=889:)886:y880:+877:x871:(868:f851:α844:α833:α812:.809:)806:x803:(800:f794:=791:)788:x782:(779:f755:;752:)749:y746:(743:f740:+737:)734:x731:(728:f725:=722:)719:y716:+713:x710:(707:f678:)675:x672:(669:f494:e487:t480:v46:.39:.20:)