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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.
1825:
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.
1338:
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
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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)
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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
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1393:. 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 related
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1417:. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the
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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
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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
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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.
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1354:, 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 as
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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
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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.
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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
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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
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Mosconi, Francesco; Julou, Thomas; Desprat, Nicolas; Sinha, Deepak Kumar; Allemand, Jean-François; Vincent
Croquette; Bensimon, David (2008).
3256:
1980:
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
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Existence of solutions of Finite-Duration, which can happen under specific conditions for some non-linear ordinary differential equations.
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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.
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Korenberg, Michael J.; Hunter, Ian W. (March 1996). "The identification of nonlinear biological systems: Volterra kernel approaches".
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542:, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler
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Billings S.A. "Nonlinear System
Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013
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is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle
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2685:– any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system
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A nonlinear system of equations consists of a set of equations in several variables such that at least one of them is not a
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (
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and its variants. Generally they may provide a solution, but do not provide any information on the number of solutions.
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield
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592:). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as
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Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals.
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This article is about "nonlinearity" in mathematics, physics and other sciences. For video and film editing, see
2112:. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the
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corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at
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Another common (though less mathematical) tactic, often exploited in fluid and heat mechanics, is to use
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2494:, and note that unlike the small angle approximation, this approximation is unstable, meaning that
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:
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1983:
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2691:– values of a system cannot be predicted indefinitely far into the future, and fluctuations are
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as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the
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can be any sensible mathematical object (number, vector, function, etc.), and the function
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Mathematical
Systems Theory I - Modelling, State Space Analysis, Stability and Robustness
3154:"Topological properties of a self-assembled electrical network via ab initio calculation"
3091:
Gintautas, V. (2008). "Resonant forcing of nonlinear systems of differential equations".
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2015:
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Mathematical
Control Theory: Deterministic Finite Dimensional Systems. Second Edition
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A classic, extensively studied nonlinear problem is the dynamics of a frictionless
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2957:, The Frontiers Collection, Springer Berlin Heidelberg, 2007, pp. 181–276,
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1161:, including integration or differentiation with associated constraints (such as
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For the general case of system of equations formed by equating to zero several
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Typically, the behavior of a nonlinear system is described in mathematics by a
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2712:– asymptotic periodic orbits to which destabilized fixed points are attracted.
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2098:{\displaystyle \int {\frac {d\theta }{\sqrt {C_{0}+2\cos(\theta )}}}=t+C_{1}}
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tends to infinity). The equation is nonlinear because it may be written as
1456:, especially for autonomous equations. For example, the nonlinear equation
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597:
411:
241:
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2718:– feedback oscillations taking place in open dissipative physical systems.
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and using the methods outlined above for ordinary differential equations.
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3209:
de Canete, Javier, Cipriano
Galindo, and Inmaculada Garcia-Moral (2011).
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1875:, it may be shown that the motion of a pendulum can be described by the
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16:
System where changes of output are not proportional to changes of input
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3267:
3212:
System
Engineering and Automation: An Interactive Educational Approach
3177:
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4237:
4147:
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617:
for the study of nonlinear systems. This term is disputed by others:
535:
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523:
403:
194:
61:
3304:
3280:
4530:
3873:
2930:"Nonlinear systems, Applied Mathematics - University of Birmingham"
2860:
1651:
and the left-hand side of the equation is not a linear function of
1382:
554:
519:
324:
4102:
3579:
The Center for
Nonlinear Studies at Los Alamos National Laboratory
3555:
New
England Complex Systems Institute: Concepts in Complex Systems
3105:
518:
to the change of the input. Nonlinear problems are of interest to
4555:
2703:
1868:
1847:
1432:
solutions can be used to construct general solutions through the
593:
538:
since most systems are inherently nonlinear in nature. Nonlinear
499:
320:
257:
1436:. A good example of this is one-dimensional heat transport with
3572:
842:, homogeneity does not follow from additivity. For example, an
557:
in which the unknowns (or the unknown functions in the case of
511:
1950:{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+\sin(\theta )=0}
1817:
to simplify a general, natural equation in a certain specific
2410:{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+\pi -\theta =0}
1729:
solutions, though implicit solutions and solutions involving
3549:
919:{\displaystyle f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)}
31:"Nonlinear dynamics" redirects here. For the journal, see
679:
is one which satisfies both of the following properties:
3578:
3035:
2241:{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+\theta =0}
1585:
corresponding to the limit of the general solution when
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2527:
One more interesting linearization is possible around
766:
694:
3344:"Thirty years of Polynomial System Solving, and now?"
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2567:
2533:
2500:
2470:
2426:
2354:
2325:
2295:
2257:
2191:
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2122:
2027:
1986:
1966:
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The most common basic approach to studying nonlinear
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as a general solution (and also the special solution
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2658:{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+1=0.}
2345:, corresponding to the pendulum being straight up:
1774:
methods (can be applied to algebraic equations too)
1400:
807:{\displaystyle \textstyle f(\alpha x)=\alpha f(x).}
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2006:
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1949:
1763:Change of variables into something easier to study
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565:of degree higher than one or in the argument of a
4001:List of nonlinear ordinary differential equations
3390:1985 24th IEEE Conference on Decision and Control
2457:{\displaystyle \sin(\theta )\approx \pi -\theta }
1781:
1372:
1225:
4588:
4006:List of nonlinear partial differential equations
1794:List of nonlinear partial differential equations
3383:
2984:
3996:List of linear ordinary differential equations
4118:
3599:
3151:
475:
3460:(fourth ed.). Oxford University Press.
3242:: CS1 maint: multiple names: authors list (
2282:{\displaystyle \sin(\theta )\approx \theta }
750:{\displaystyle \textstyle f(x+y)=f(x)+f(y);}
3550:Command and Control Research Program (CCRP)
3455:
2700:– the presence of two or more stable states
2182:, called the small angle approximation, is
2108:which is an implicit solution involving an
1750:Examination of dissipative quantities (see
4125:
4111:
3613:
3606:
3592:
3419:David Tong: Lectures on Classical Dynamics
1389:and the relations that define the various
482:
468:
3458:Nonlinear Ordinary Differential Equations
3359:
3208:
3185:
3104:
3090:
2906:"Explained: Linear and nonlinear systems"
514:in which the change of the output is not
3493:
3278:
2490:. The solution to this problem involves
1854:
1846:
1641:{\displaystyle {\frac {du}{dx}}+u^{2}=0}
3279:Campbell, David K. (25 November 2004).
1788:Nonlinear partial differential equation
1505:{\displaystyle {\frac {du}{dx}}=-u^{2}}
1413:is said to be nonlinear if it is not a
1003:is a linear map (as defined above) and
817:Additivity implies homogeneity for any
4589:
3519:
3474:
3341:
3038:"Some nonlinear challenges in biology"
2592:{\displaystyle \sin(\theta )\approx 1}
1671:and its derivatives. Note that if the
4106:
3587:
1960:where gravity points "downwards" and
1275:many methods have been designed; see
3991:List of named differential equations
3386:"Finite Time Differential Equations"
3368:
2677:Types of nonlinear dynamic behaviors
2156:. For example, the linearization at
1829:Other methods include examining the
1821:. For example, the (very) nonlinear
4132:
3916:Method of undetermined coefficients
3697:Dependent and independent variables
3281:"Nonlinear physics: Fresh breather"
2483:{\displaystyle \theta \approx \pi }
1754:) analogous to conserved quantities
1718:, the problem would be linear (the
13:
4281:Evolutionary developmental biology
3426:
1549:{\displaystyle u={\frac {1}{x+C}}}
1237:For a single equation of the form
1007:otherwise. The equation is called
229:Evolutionary developmental biology
14:
4618:
3543:
3456:Jordan, D. W.; Smith, P. (2007).
3437:and Anthony J. Pritchard (2005).
3152:Stephenson, C.; et., al. (2017).
2706:– self-reinforcing solitary waves
553:, which is a set of simultaneous
3813:Carathéodory's existence theorem
3575:) a Database of Physical Systems
3501:Advanced Engineering Mathematics
3215:. Berlin: Springer. p. 46.
2987:Annals of Biomedical Engineering
2782:Landau–Lifshitz–Gilbert equation
2308:{\displaystyle \theta \approx 0}
1401:Nonlinear differential equations
4415:Ordinary differential equations
3412:
3377:
3348:Journal of Symbolic Computation
3335:
3272:
2851:Aleksandr Mikhailovich Lyapunov
2772:Kadomtsev–Petviashvili equation
2723:Examples of nonlinear equations
2018:, which would eventually yield
1804:ordinary differential equations
1450:ordinary differential equations
1444:Ordinary differential equations
1395:nonlinear system identification
1352:systems of polynomial equations
396:Ordinary differential equations
4473:Partial differential equations
3250:
3202:
3145:
3084:
3029:
2978:
2946:
2922:
2898:
2806:Nonlinear Schrödinger equation
2580:
2574:
2554:{\displaystyle \theta =\pi /2}
2510:
2502:
2439:
2433:
2270:
2264:
2070:
2064:
1938:
1932:
1800:partial differential equations
1782:Partial differential equations
1452:are often exactly solvable by
1381:defines successive terms of a
1373:Nonlinear recurrence relations
1253:
1247:
1226:Nonlinear systems of equations
1181:
1175:
1144:
1138:
1089:
1083:
1053:
1047:
990:
984:
948:
942:
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859:
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734:
725:
719:
710:
698:
666:
660:
267:Partial differential equations
1:
2892:
1438:Dirichlet boundary conditions
633:
551:nonlinear system of equations
26:Nonlinearity (disambiguation)
3641:Notation for differentiation
2338:{\displaystyle \theta =\pi }
1859:Linearizations of a pendulum
1331:{\displaystyle x^{2}+x-1=0.}
33:Nonlinear Dynamics (journal)
7:
4223:Particle swarm optimization
3737:Exact differential equation
3560:Nonlinear Dynamics I: Chaos
3257:Nonlinear Dynamics I: Chaos
2963:10.1007/978-3-540-34153-6_7
2843:
2007:{\displaystyle d\theta /dt}
561:) appear as variables of a
116:Particle swarm optimization
10:
4623:
4468:Reaction-diffusion systems
4193:Self-organized criticality
3475:Khalil, Hassan K. (2001).
3062:10.1088/0951-7715/21/8/T03
2818:for unsaturated water flow
2777:Korteweg–de Vries equation
2730:Algebraic Riccati equation
2317:simple harmonic oscillator
1851:Illustration of a pendulum
1840:
1836:
1791:
1785:
1421:in fluid dynamics and the
1415:system of linear equations
613:Some authors use the term
263:Reaction–diffusion systems
92:Self-organized criticality
18:
4511:
4458:
4400:
4337:
4304:
4251:Artificial neural network
4236:
4161:
4140:
4047:Józef Maria Hoene-Wroński
4027:Gottfried Wilhelm Leibniz
4014:
3983:
3893:
3826:
3818:Cauchy–Kowalevski theorem
3795:
3788:
3750:
3689:
3628:
3621:
3361:10.1016/j.jsc.2008.03.004
2673:and approximate periods.
2517:{\displaystyle |\theta |}
2175:{\displaystyle \theta =0}
204:Artificial neural network
22:Non-linear editing system
4329:Evolutionary game theory
4256:Evolutionary computation
4188:Collective consciousness
3941:Finite difference method
3384:Vardia T. Haimo (1985).
1698:term were replaced with
1423:Lotka–Volterra equations
1363:differentiable functions
1108:is very general in that
455:Evolutionary game theory
345:Second-order cybernetics
209:Evolutionary computation
125:Collective consciousness
4347:Social network analysis
4286:Artificial intelligence
4218:Ant colony optimization
4178:Collective intelligence
3921:Variation of parameters
3911:Separation of variables
3808:Peano existence theorem
3803:Picard–Lindelöf theorem
3690:Attributes of variables
3571: – (in
3569:Nonlinear Model Library
3398:10.1109/CDC.1985.268832
2822:Self-balancing unicycle
2792:Navier–Stokes equations
2142:{\displaystyle C_{0}=2}
1973:{\displaystyle \theta }
1867:under the influence of
1823:Navier-Stokes equations
1808:separation of variables
1731:nonelementary integrals
1454:separation of variables
1434:superposition principle
1419:Navier–Stokes equations
1268:{\displaystyle f(x)=0,}
1218:, the result will be a
929:An equation written as
685:superposition principle
233:Artificial intelligence
147:Social network analysis
112:Ant colony optimization
84:Collective intelligence
4478:Dissipative structures
4319:Rational choice theory
4082:Carl David Tolmé Runge
3656:Differential-algebraic
3615:Differential equations
3392:. pp. 1729–1733.
2955:The Nonlinear Universe
2832:Van der Pol oscillator
2762:Ginzburg–Landau theory
2659:
2593:
2555:
2518:
2484:
2458:
2411:
2339:
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2176:
2143:
2116:(nonelementary unless
2114:nonelementary integral
2099:
2008:
1974:
1951:
1860:
1852:
1843:Pendulum (mathematics)
1819:boundary value problem
1712:
1692:
1665:
1642:
1579:
1550:
1506:
1411:differential equations
1332:
1277:Root-finding algorithm
1269:
1212:
1188:
1151:
1122:
1102:
1101:{\displaystyle f(x)=C}
1060:
1031:
997:
961:
960:{\displaystyle f(x)=C}
920:
808:
751:
673:
631:
559:differential equations
446:Rational choice theory
271:Dissipative structures
24:. For other uses, see
4291:Evolutionary robotics
4208:Agent-based modelling
4067:Augustin-Louis Cauchy
4062:Joseph-Louis Lagrange
3956:Finite element method
3946:Crank–Nicolson method
3880:Numerical integration
3859:Exponential stability
3751:Relation to processes
3636:Differential operator
2953:"Nonlinear Biology",
2660:
2594:
2556:
2519:
2485:
2459:
2412:
2340:
2310:
2284:
2243:
2177:
2144:
2100:
2009:
1975:
1952:
1858:
1850:
1713:
1693:
1691:{\displaystyle u^{2}}
1666:
1643:
1580:
1551:
1507:
1365:, the main method is
1333:
1270:
1220:differential equation
1213:
1189:
1157:can literally be any
1152:
1123:
1103:
1061:
1032:
998:
962:
921:
809:
752:
674:
619:
374:Theory of computation
237:Evolutionary robotics
104:Agent-based modelling
4450:Coupled map lattices
4410:Time series analysis
4352:Small-world networks
3961:Finite volume method
3885:Dirac delta function
3854:Asymptotic stability
3796:Existence/uniqueness
3661:Integro-differential
3564:MIT's OpenCourseWare
3435:Diederich Hinrichsen
3268:MIT's OpenCourseWare
2934:www.birmingham.ac.uk
2827:Sine-Gordon equation
2606:
2565:
2531:
2498:
2492:hyperbolic sinusoids
2468:
2424:
2352:
2323:
2293:
2255:
2189:
2160:
2120:
2025:
1984:
1964:
1886:
1873:Lagrangian mechanics
1742:conserved quantities
1702:
1675:
1655:
1596:
1578:{\displaystyle u=0,}
1560:
1519:
1463:
1430:linearly independent
1391:Hofstadter sequences
1297:
1279:. In the case where
1241:
1202:
1187:{\displaystyle f(x)}
1169:
1150:{\displaystyle f(x)}
1132:
1112:
1077:
1068:homogeneous function
1059:{\displaystyle f(x)}
1041:
1015:
996:{\displaystyle f(x)}
978:
936:
853:
826:continuous functions
763:
691:
672:{\displaystyle f(x)}
654:
425:Coupled map lattices
391:Time series analysis
151:Small-world networks
4607:Concepts in physics
4430:Population dynamics
4324:Bounded rationality
4266:Genetic programming
3971:Perturbation theory
3951:Runge–Kutta methods
3931:Integral transforms
3864:Rate of convergence
3760:(discrete analogue)
3441:. Springer Verlag.
3342:Lazard, D. (2009).
3297:2004Natur.432..455C
3170:2017NatSR...741621S
3115:2008Chaos..18c3118G
3054:2008Nonli..21..131M
1879:nonlinear equation
1746:Hamiltonian systems
1740:Examination of any
1379:recurrence relation
1345:real-root isolation
1290:polynomial equation
1030:{\displaystyle C=0}
450:Bounded rationality
408:Population dynamics
317:Conversation theory
217:Genetic programming
142:Scale-free networks
74:Collective behavior
4576:Computation theory
4571:Information theory
4526:Operationalization
4402:Nonlinear dynamics
4314:Prisoner's dilemma
4261:Genetic algorithms
4092:Sofya Kovalevskaya
3926:Integrating factor
3849:Lyapunov stability
3769:Stochastic partial
3262:2008-02-12 at the
2999:10.1007/bf02667354
2757:General relativity
2752:Colebrook equation
2747:Boltzmann equation
2743:for optimal policy
2655:
2589:
2551:
2514:
2480:
2454:
2407:
2335:
2305:
2279:
2238:
2172:
2139:
2095:
2016:integrating factor
2004:
1970:
1947:
1861:
1853:
1767:Bifurcation theory
1757:Linearization via
1708:
1688:
1661:
1638:
1575:
1546:
1502:
1328:
1265:
1208:
1184:
1147:
1118:
1098:
1056:
1027:
993:
957:
916:
804:
803:
747:
746:
669:
571:linear combination
441:Prisoner's dilemma
386:Nonlinear dynamics
341:Operationalization
337:Information theory
213:Genetic algorithms
4602:Dynamical systems
4597:Nonlinear systems
4584:
4583:
4488:Cellular automata
4460:Pattern formation
4392:Adaptive networks
4183:Collective action
4153:Self-organization
4100:
4099:
3979:
3978:
3784:
3783:
3534:978-0-387-98489-6
3512:978-0-471-15496-9
3486:978-0-13-067389-3
3479:. Prentice Hall.
3477:Nonlinear Systems
3467:978-0-19-920824-1
3291:(7016): 455–456.
3178:10.1038/srep41621
3123:10.1063/1.2964200
2866:Initial condition
2816:Richards equation
2767:Ishimori equation
2716:Self-oscillations
2641:
2387:
2224:
2154:Taylor expansions
2110:elliptic integral
2074:
2073:
1921:
1752:Lyapunov function
1733:are encountered.
1720:exponential decay
1711:{\displaystyle u}
1664:{\displaystyle u}
1617:
1544:
1484:
1211:{\displaystyle x}
1121:{\displaystyle x}
615:nonlinear science
540:dynamical systems
534:, and many other
508:non-linear system
492:
491:
279:Cellular automata
253:Pattern formation
184:Adaptive networks
88:Collective action
57:Self-organization
4614:
4498:Self-replication
4387:Dynamic networks
4276:Machine learning
4203:Phase transition
4127:
4120:
4113:
4104:
4103:
4077:Phyllis Nicolson
4057:Rudolf Lipschitz
3894:Solution methods
3869:Series solutions
3793:
3792:
3626:
3625:
3608:
3601:
3594:
3585:
3584:
3538:
3516:
3504:
3490:
3471:
3452:
3421:
3416:
3410:
3409:
3381:
3375:
3372:
3366:
3365:
3363:
3339:
3333:
3332:
3276:
3270:
3254:
3248:
3247:
3241:
3233:
3231:
3229:
3206:
3200:
3199:
3189:
3149:
3143:
3142:
3108:
3088:
3082:
3081:
3033:
3027:
3026:
2982:
2976:
2975:
2950:
2944:
2943:
2941:
2940:
2926:
2920:
2919:
2917:
2916:
2902:
2856:Dynamical system
2811:Power-flow study
2801:Nonlinear optics
2787:Liénard equation
2741:Bellman equation
2664:
2662:
2661:
2656:
2642:
2640:
2639:
2638:
2625:
2621:
2620:
2610:
2598:
2596:
2595:
2590:
2560:
2558:
2557:
2552:
2547:
2523:
2521:
2520:
2515:
2513:
2505:
2489:
2487:
2486:
2481:
2463:
2461:
2460:
2455:
2416:
2414:
2413:
2408:
2388:
2386:
2385:
2384:
2371:
2367:
2366:
2356:
2344:
2342:
2341:
2336:
2314:
2312:
2311:
2306:
2288:
2286:
2285:
2280:
2247:
2245:
2244:
2239:
2225:
2223:
2222:
2221:
2208:
2204:
2203:
2193:
2181:
2179:
2178:
2173:
2148:
2146:
2145:
2140:
2132:
2131:
2104:
2102:
2101:
2096:
2094:
2093:
2075:
2051:
2050:
2041:
2040:
2032:
2013:
2011:
2010:
2005:
1997:
1979:
1977:
1976:
1971:
1956:
1954:
1953:
1948:
1922:
1920:
1919:
1918:
1905:
1901:
1900:
1890:
1759:Taylor expansion
1744:, especially in
1717:
1715:
1714:
1709:
1697:
1695:
1694:
1689:
1687:
1686:
1670:
1668:
1667:
1662:
1647:
1645:
1644:
1639:
1631:
1630:
1618:
1616:
1608:
1600:
1584:
1582:
1581:
1576:
1555:
1553:
1552:
1547:
1545:
1543:
1529:
1511:
1509:
1508:
1503:
1501:
1500:
1485:
1483:
1475:
1467:
1337:
1335:
1334:
1329:
1309:
1308:
1282:
1274:
1272:
1271:
1266:
1217:
1215:
1214:
1209:
1198:with respect to
1193:
1191:
1190:
1185:
1156:
1154:
1153:
1148:
1127:
1125:
1124:
1119:
1107:
1105:
1104:
1099:
1065:
1063:
1062:
1057:
1036:
1034:
1033:
1028:
1002:
1000:
999:
994:
966:
964:
963:
958:
925:
923:
922:
917:
813:
811:
810:
805:
756:
754:
753:
748:
678:
676:
675:
670:
629:
504:nonlinear system
484:
477:
470:
361:Systems thinking
287:Self-replication
225:Machine learning
179:Dynamic networks
100:Phase transition
39:
38:
36:
29:
4622:
4621:
4617:
4616:
4615:
4613:
4612:
4611:
4587:
4586:
4585:
4580:
4546:System dynamics
4507:
4493:Spatial ecology
4454:
4396:
4382:Systems biology
4333:
4300:
4271:Artificial life
4241:
4232:
4228:Swarm behaviour
4213:Synchronization
4173:Social dynamics
4164:
4157:
4136:
4134:Complex systems
4131:
4101:
4096:
4037:Jacob Bernoulli
4010:
3975:
3966:Galerkin method
3889:
3827:Solution topics
3822:
3780:
3746:
3685:
3617:
3612:
3546:
3541:
3535:
3521:Sontag, Eduardo
3513:
3495:Kreyszig, Erwin
3487:
3468:
3449:
3429:
3427:Further reading
3424:
3417:
3413:
3382:
3378:
3373:
3369:
3340:
3336:
3305:10.1038/432455a
3277:
3273:
3264:Wayback Machine
3255:
3251:
3235:
3234:
3227:
3225:
3223:
3207:
3203:
3150:
3146:
3089:
3085:
3034:
3030:
2983:
2979:
2973:
2952:
2951:
2947:
2938:
2936:
2928:
2927:
2923:
2914:
2912:
2904:
2903:
2899:
2895:
2890:
2886:Volterra series
2846:
2841:
2837:Vlasov equation
2725:
2683:Amplitude death
2679:
2671:phase portraits
2634:
2630:
2626:
2616:
2612:
2611:
2609:
2607:
2604:
2603:
2566:
2563:
2562:
2561:, around which
2543:
2532:
2529:
2528:
2509:
2501:
2499:
2496:
2495:
2469:
2466:
2465:
2425:
2422:
2421:
2380:
2376:
2372:
2362:
2358:
2357:
2355:
2353:
2350:
2349:
2324:
2321:
2320:
2294:
2291:
2290:
2256:
2253:
2252:
2217:
2213:
2209:
2199:
2195:
2194:
2192:
2190:
2187:
2186:
2161:
2158:
2157:
2127:
2123:
2121:
2118:
2117:
2089:
2085:
2046:
2042:
2033:
2031:
2026:
2023:
2022:
1993:
1985:
1982:
1981:
1965:
1962:
1961:
1914:
1910:
1906:
1896:
1892:
1891:
1889:
1887:
1884:
1883:
1845:
1839:
1831:characteristics
1796:
1790:
1784:
1703:
1700:
1699:
1682:
1678:
1676:
1673:
1672:
1656:
1653:
1652:
1626:
1622:
1609:
1601:
1599:
1597:
1594:
1593:
1561:
1558:
1557:
1533:
1528:
1520:
1517:
1516:
1496:
1492:
1476:
1468:
1466:
1464:
1461:
1460:
1446:
1403:
1375:
1367:Newton's method
1304:
1300:
1298:
1295:
1294:
1280:
1242:
1239:
1238:
1232:linear equation
1228:
1203:
1200:
1199:
1196:differentiation
1170:
1167:
1166:
1163:boundary values
1133:
1130:
1129:
1113:
1110:
1109:
1078:
1075:
1074:
1073:The definition
1042:
1039:
1038:
1016:
1013:
1012:
979:
976:
975:
937:
934:
933:
854:
851:
850:
764:
761:
760:
692:
689:
688:
655:
652:
651:
648:linear function
636:
630:
625:
573:of the unknown
488:
459:
458:
457:
452:
448:
443:
438:
428:
427:
422:
418:
414:
410:
406:
402:
398:
393:
388:
378:
377:
376:
371:
367:
363:
359:
357:Systems science
355:
353:System dynamics
351:
347:
343:
339:
335:
331:
327:
323:
319:
314:
309:
295:
294:
289:
285:
283:Spatial ecology
281:
277:
273:
269:
265:
260:
255:
245:
244:
239:
235:
231:
227:
223:
221:Artificial life
219:
215:
211:
206:
201:
187:
186:
181:
177:
175:Systems biology
173:
169:
165:
161:
157:
153:
149:
144:
139:
129:
128:
127:
122:
120:Swarm behaviour
118:
114:
110:
108:Synchronization
106:
102:
98:
94:
90:
86:
81:
79:Social dynamics
76:
66:
65:
64:
59:
43:Complex systems
37:
30:
19:
17:
12:
11:
5:
4620:
4610:
4609:
4604:
4599:
4582:
4581:
4579:
4578:
4573:
4568:
4563:
4558:
4553:
4548:
4543:
4538:
4536:Self-reference
4533:
4528:
4523:
4517:
4515:
4513:Systems theory
4509:
4508:
4506:
4505:
4500:
4495:
4490:
4485:
4480:
4475:
4470:
4464:
4462:
4456:
4455:
4453:
4452:
4447:
4442:
4440:Multistability
4437:
4432:
4427:
4422:
4417:
4412:
4406:
4404:
4398:
4397:
4395:
4394:
4389:
4384:
4379:
4374:
4369:
4364:
4359:
4354:
4349:
4343:
4341:
4335:
4334:
4332:
4331:
4326:
4321:
4316:
4310:
4308:
4302:
4301:
4299:
4298:
4293:
4288:
4283:
4278:
4273:
4268:
4263:
4258:
4253:
4247:
4245:
4234:
4233:
4231:
4230:
4225:
4220:
4215:
4210:
4205:
4200:
4198:Herd mentality
4195:
4190:
4185:
4180:
4175:
4169:
4167:
4159:
4158:
4156:
4155:
4150:
4144:
4142:
4138:
4137:
4130:
4129:
4122:
4115:
4107:
4098:
4097:
4095:
4094:
4089:
4084:
4079:
4074:
4069:
4064:
4059:
4054:
4052:Ernst Lindelöf
4049:
4044:
4039:
4034:
4032:Leonhard Euler
4029:
4024:
4018:
4016:
4015:Mathematicians
4012:
4011:
4009:
4008:
4003:
3998:
3993:
3987:
3985:
3981:
3980:
3977:
3976:
3974:
3973:
3968:
3963:
3958:
3953:
3948:
3943:
3938:
3933:
3928:
3923:
3918:
3913:
3908:
3903:
3897:
3895:
3891:
3890:
3888:
3887:
3882:
3877:
3871:
3866:
3861:
3856:
3851:
3846:
3841:
3839:Phase portrait
3836:
3830:
3828:
3824:
3823:
3821:
3820:
3815:
3810:
3805:
3799:
3797:
3790:
3786:
3785:
3782:
3781:
3779:
3778:
3773:
3772:
3771:
3761:
3754:
3752:
3748:
3747:
3745:
3744:
3742:On jet bundles
3739:
3734:
3729:
3724:
3719:
3714:
3709:
3707:Nonhomogeneous
3704:
3699:
3693:
3691:
3687:
3686:
3684:
3683:
3678:
3673:
3668:
3663:
3658:
3653:
3648:
3643:
3638:
3632:
3630:
3623:
3622:Classification
3619:
3618:
3611:
3610:
3603:
3596:
3588:
3582:
3581:
3576:
3566:
3557:
3552:
3545:
3544:External links
3542:
3540:
3539:
3533:
3517:
3511:
3491:
3485:
3472:
3466:
3453:
3447:
3430:
3428:
3425:
3423:
3422:
3411:
3376:
3367:
3354:(3): 222–231.
3334:
3271:
3249:
3222:978-3642202292
3221:
3201:
3144:
3083:
3028:
2993:(2): 250–268.
2977:
2971:
2945:
2921:
2896:
2894:
2891:
2889:
2888:
2883:
2881:Vector soliton
2878:
2873:
2868:
2863:
2858:
2853:
2847:
2845:
2842:
2840:
2839:
2834:
2829:
2824:
2819:
2813:
2808:
2803:
2798:
2796:fluid dynamics
2789:
2784:
2779:
2774:
2769:
2764:
2759:
2754:
2749:
2744:
2738:
2732:
2726:
2724:
2721:
2720:
2719:
2713:
2707:
2701:
2698:Multistability
2695:
2686:
2678:
2675:
2666:
2665:
2654:
2651:
2648:
2645:
2637:
2633:
2629:
2624:
2619:
2615:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2550:
2546:
2542:
2539:
2536:
2512:
2508:
2504:
2479:
2476:
2473:
2453:
2450:
2447:
2444:
2441:
2438:
2435:
2432:
2429:
2418:
2417:
2406:
2403:
2400:
2397:
2394:
2391:
2383:
2379:
2375:
2370:
2365:
2361:
2334:
2331:
2328:
2304:
2301:
2298:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2249:
2248:
2237:
2234:
2231:
2228:
2220:
2216:
2212:
2207:
2202:
2198:
2171:
2168:
2165:
2138:
2135:
2130:
2126:
2106:
2105:
2092:
2088:
2084:
2081:
2078:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2049:
2045:
2039:
2036:
2030:
2003:
2000:
1996:
1992:
1989:
1969:
1958:
1957:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1917:
1913:
1909:
1904:
1899:
1895:
1841:Main article:
1838:
1835:
1815:scale analysis
1786:Main article:
1783:
1780:
1779:
1778:
1775:
1769:
1764:
1761:
1755:
1748:
1707:
1685:
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1660:
1649:
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1634:
1629:
1625:
1621:
1615:
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1574:
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1542:
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1513:
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1499:
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1479:
1474:
1471:
1445:
1442:
1402:
1399:
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1327:
1324:
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1318:
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1312:
1307:
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1264:
1261:
1258:
1255:
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1227:
1224:
1207:
1183:
1180:
1177:
1174:
1146:
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1140:
1137:
1117:
1097:
1094:
1091:
1088:
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1082:
1055:
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1026:
1023:
1020:
992:
989:
986:
983:
968:
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956:
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947:
944:
941:
927:
926:
915:
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909:
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900:
897:
894:
891:
888:
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882:
879:
876:
873:
870:
867:
864:
861:
858:
844:antilinear map
815:
814:
802:
799:
796:
793:
790:
787:
784:
781:
778:
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769:
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736:
733:
730:
727:
724:
721:
718:
715:
712:
709:
706:
703:
700:
697:
683:Additivity or
668:
665:
662:
659:
635:
632:
627:Stanisław Ulam
623:
544:linear systems
532:mathematicians
490:
489:
487:
486:
479:
472:
464:
461:
460:
439:
434:
433:
430:
429:
416:Multistability
389:
384:
383:
380:
379:
349:Self-reference
310:
303:Systems theory
301:
300:
297:
296:
256:
251:
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246:
202:
193:
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188:
140:
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96:Herd mentality
77:
72:
71:
68:
67:
60:
55:
54:
51:
50:
46:
45:
15:
9:
6:
4:
3:
2:
4619:
4608:
4605:
4603:
4600:
4598:
4595:
4594:
4592:
4577:
4574:
4572:
4569:
4567:
4564:
4562:
4559:
4557:
4554:
4552:
4549:
4547:
4544:
4542:
4541:Goal-oriented
4539:
4537:
4534:
4532:
4529:
4527:
4524:
4522:
4519:
4518:
4516:
4514:
4510:
4504:
4503:Geomorphology
4501:
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4408:
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4380:
4378:
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4360:
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4340:
4336:
4330:
4327:
4325:
4322:
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4317:
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4311:
4309:
4307:
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4297:
4294:
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4277:
4274:
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4267:
4264:
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4235:
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4224:
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4209:
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4139:
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4128:
4123:
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4109:
4108:
4105:
4093:
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4065:
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4035:
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4028:
4025:
4023:
4020:
4019:
4017:
4013:
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4002:
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3997:
3994:
3992:
3989:
3988:
3986:
3982:
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3969:
3967:
3964:
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3954:
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3924:
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3829:
3825:
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3809:
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3800:
3798:
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3791:
3787:
3777:
3774:
3770:
3767:
3766:
3765:
3762:
3759:
3756:
3755:
3753:
3749:
3743:
3740:
3738:
3735:
3733:
3730:
3728:
3725:
3723:
3720:
3718:
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3713:
3710:
3708:
3705:
3703:
3700:
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3695:
3694:
3692:
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3682:
3679:
3677:
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3672:
3669:
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3664:
3662:
3659:
3657:
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3652:
3649:
3647:
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3642:
3639:
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3634:
3633:
3631:
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3624:
3620:
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3602:
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3590:
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3586:
3580:
3577:
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3561:
3558:
3556:
3553:
3551:
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3536:
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3518:
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3488:
3482:
3478:
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3469:
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3454:
3450:
3448:9783540441250
3444:
3440:
3436:
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3420:
3415:
3407:
3403:
3399:
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3391:
3387:
3380:
3371:
3362:
3357:
3353:
3349:
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3338:
3330:
3326:
3322:
3318:
3314:
3310:
3306:
3302:
3298:
3294:
3290:
3286:
3282:
3275:
3269:
3265:
3261:
3258:
3253:
3245:
3239:
3224:
3218:
3214:
3213:
3205:
3197:
3193:
3188:
3183:
3179:
3175:
3171:
3167:
3163:
3159:
3155:
3148:
3140:
3136:
3132:
3128:
3124:
3120:
3116:
3112:
3107:
3102:
3099:(3): 033118.
3098:
3094:
3087:
3079:
3075:
3071:
3067:
3063:
3059:
3055:
3051:
3047:
3043:
3039:
3032:
3024:
3020:
3016:
3012:
3008:
3004:
3000:
2996:
2992:
2988:
2981:
2974:
2972:9783540341529
2968:
2964:
2960:
2956:
2949:
2935:
2931:
2925:
2911:
2907:
2901:
2897:
2887:
2884:
2882:
2879:
2877:
2876:Mode coupling
2874:
2872:
2871:Linear system
2869:
2867:
2864:
2862:
2859:
2857:
2854:
2852:
2849:
2848:
2838:
2835:
2833:
2830:
2828:
2825:
2823:
2820:
2817:
2814:
2812:
2809:
2807:
2804:
2802:
2799:
2797:
2793:
2790:
2788:
2785:
2783:
2780:
2778:
2775:
2773:
2770:
2768:
2765:
2763:
2760:
2758:
2755:
2753:
2750:
2748:
2745:
2742:
2739:
2736:
2735:Ball and beam
2733:
2731:
2728:
2727:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2694:
2690:
2687:
2684:
2681:
2680:
2674:
2672:
2652:
2649:
2646:
2643:
2635:
2631:
2627:
2622:
2617:
2613:
2602:
2601:
2600:
2586:
2583:
2577:
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2568:
2548:
2544:
2540:
2537:
2534:
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2506:
2493:
2477:
2474:
2471:
2451:
2448:
2445:
2442:
2436:
2430:
2427:
2404:
2401:
2398:
2395:
2392:
2389:
2381:
2377:
2373:
2368:
2363:
2359:
2348:
2347:
2346:
2332:
2329:
2326:
2318:
2302:
2299:
2296:
2276:
2273:
2267:
2261:
2258:
2235:
2232:
2229:
2226:
2218:
2214:
2210:
2205:
2200:
2196:
2185:
2184:
2183:
2169:
2166:
2163:
2155:
2150:
2136:
2133:
2128:
2124:
2115:
2111:
2090:
2086:
2082:
2079:
2076:
2067:
2061:
2058:
2055:
2052:
2047:
2043:
2037:
2034:
2028:
2021:
2020:
2019:
2017:
2001:
1998:
1994:
1990:
1987:
1967:
1944:
1941:
1935:
1929:
1926:
1923:
1915:
1911:
1907:
1902:
1897:
1893:
1882:
1881:
1880:
1878:
1877:dimensionless
1874:
1870:
1866:
1857:
1849:
1844:
1834:
1832:
1827:
1824:
1820:
1816:
1811:
1809:
1806:, as seen in
1805:
1801:
1795:
1789:
1776:
1773:
1770:
1768:
1765:
1762:
1760:
1756:
1753:
1749:
1747:
1743:
1739:
1738:
1737:
1734:
1732:
1728:
1723:
1721:
1705:
1683:
1679:
1658:
1635:
1632:
1627:
1623:
1619:
1613:
1610:
1605:
1602:
1592:
1591:
1590:
1588:
1572:
1569:
1566:
1563:
1540:
1537:
1534:
1530:
1525:
1522:
1497:
1493:
1489:
1486:
1480:
1477:
1472:
1469:
1459:
1458:
1457:
1455:
1451:
1441:
1439:
1435:
1431:
1426:
1424:
1420:
1416:
1412:
1408:
1398:
1396:
1392:
1388:
1384:
1380:
1370:
1368:
1364:
1359:
1357:
1353:
1348:
1346:
1342:
1325:
1322:
1319:
1316:
1313:
1310:
1305:
1301:
1292:
1291:
1286:
1278:
1262:
1259:
1256:
1250:
1244:
1235:
1233:
1223:
1221:
1205:
1197:
1178:
1172:
1164:
1160:
1141:
1135:
1115:
1095:
1092:
1086:
1080:
1071:
1069:
1050:
1044:
1024:
1021:
1018:
1010:
1006:
987:
981:
973:
954:
951:
945:
939:
932:
931:
930:
910:
904:
901:
898:
892:
886:
883:
880:
874:
871:
868:
865:
862:
856:
849:
848:
847:
845:
841:
838:
834:
831:
827:
823:
820:
800:
794:
788:
785:
782:
776:
773:
767:
759:Homogeneity:
758:
743:
737:
731:
728:
722:
716:
713:
707:
704:
701:
695:
686:
682:
681:
680:
663:
657:
649:
645:
641:
628:
622:
618:
616:
611:
608:
603:
602:singularities
599:
595:
591:
590:linearization
586:
584:
580:
576:
572:
568:
564:
560:
556:
552:
547:
545:
541:
537:
533:
529:
525:
521:
517:
513:
509:
505:
501:
497:
485:
480:
478:
473:
471:
466:
465:
463:
462:
456:
453:
451:
447:
442:
437:
432:
431:
426:
423:
421:
417:
413:
409:
405:
401:
397:
392:
387:
382:
381:
375:
372:
370:
366:
362:
358:
354:
350:
346:
342:
338:
334:
330:
329:Goal-oriented
326:
322:
318:
313:
308:
304:
299:
298:
293:
292:Geomorphology
290:
288:
284:
280:
276:
272:
268:
264:
259:
254:
249:
248:
243:
240:
238:
234:
230:
226:
222:
218:
214:
210:
205:
200:
196:
191:
190:
185:
182:
180:
176:
172:
168:
164:
160:
156:
152:
148:
143:
138:
133:
132:
126:
123:
121:
117:
113:
109:
105:
101:
97:
93:
89:
85:
80:
75:
70:
69:
63:
58:
53:
52:
48:
47:
44:
41:
40:
34:
27:
23:
4367:Graph theory
4296:Evolvability
4087:Martin Kutta
4042:Émile Picard
4022:Isaac Newton
3936:Euler method
3906:Substitution
3527:. Springer.
3524:
3500:
3476:
3457:
3438:
3414:
3389:
3379:
3370:
3351:
3347:
3337:
3288:
3284:
3274:
3252:
3226:. Retrieved
3211:
3204:
3161:
3157:
3147:
3096:
3092:
3086:
3045:
3042:Nonlinearity
3041:
3031:
2990:
2986:
2980:
2954:
2948:
2937:. Retrieved
2933:
2924:
2913:. Retrieved
2909:
2900:
2710:Limit cycles
2667:
2526:
2419:
2315:. This is a
2250:
2151:
2107:
1959:
1862:
1828:
1812:
1797:
1772:Perturbation
1735:
1724:
1650:
1586:
1514:
1448:First order
1447:
1427:
1425:in biology.
1404:
1387:logistic map
1377:A nonlinear
1376:
1360:
1358:algorithms.
1356:Gröbner base
1349:
1288:
1287:, one has a
1236:
1229:
1072:
1008:
1004:
971:
969:
928:
839:
832:
821:
816:
647:
637:
620:
614:
612:
587:
582:
550:
548:
516:proportional
507:
503:
493:
444:
394:
385:
315:
261:
242:Evolvability
207:
163:Graph theory
145:
82:
4566:Autopoiesis
4561:Cybernetics
4551:Sensemaking
4521:Homeostasis
4483:Percolation
4445:Bifurcation
4420:Phase space
4306:Game theory
4163:Collective
3844:Phase space
3702:Homogeneous
3048:(8): T131.
1727:closed-form
1343:roots; see
1009:homogeneous
824:, and, for
640:mathematics
496:mathematics
436:Game theory
420:Bifurcation
400:Phase space
365:Sensemaking
333:Homeostasis
312:Autopoiesis
307:cybernetics
275:Percolation
4591:Categories
4377:Robustness
4357:Centrality
4243:adaptation
4141:Background
4072:John Crank
3901:Inspection
3764:Stochastic
3758:Difference
3732:Autonomous
3676:Non-linear
3666:Fractional
3629:Operations
3228:20 January
2939:2018-06-30
2915:2018-06-30
2893:References
1792:See also:
1722:problem).
1285:polynomial
970:is called
828:, for any
644:linear map
634:Definition
563:polynomial
536:scientists
528:physicists
524:biologists
404:Attractors
199:adaptation
171:Robustness
155:Centrality
4425:Attractor
4238:Evolution
4148:Emergence
3876:solutions
3834:Wronskian
3789:Solutions
3717:Decoupled
3681:Holonomic
3505:. Wiley.
3313:0028-0836
3238:cite book
3164:: 41621.
3106:0803.2252
3078:119808230
3070:0951-7715
3007:0090-6964
2693:aperiodic
2623:θ
2584:≈
2578:θ
2572:
2541:π
2535:θ
2507:θ
2478:π
2475:≈
2472:θ
2452:θ
2449:−
2446:π
2443:≈
2437:θ
2431:
2399:θ
2396:−
2393:π
2369:θ
2333:π
2327:θ
2300:≈
2297:θ
2277:θ
2274:≈
2268:θ
2262:
2230:θ
2206:θ
2164:θ
2068:θ
2062:
2038:θ
2029:∫
1991:θ
1968:θ
1936:θ
1930:
1903:θ
1490:−
1317:−
1194:contains
1005:nonlinear
902:β
884:α
872:β
863:α
786:α
774:α
579:functions
575:variables
555:equations
520:engineers
195:Evolution
62:Emergence
4531:Feedback
4339:Networks
4165:behavior
3984:Examples
3874:Integral
3646:Ordinary
3523:(1998).
3497:(1998).
3406:45426376
3321:15565139
3260:Archived
3196:28155863
3158:Sci. Rep
3139:18345817
3131:19045456
3023:20643206
2910:MIT News
2861:Feedback
2844:See also
2704:Solitons
1871:. Using
1865:pendulum
1383:sequence
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