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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.
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
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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
1966:
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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 related
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823:
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1428:. 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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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
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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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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 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
3823:
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Mosconi, Francesco; Julou, Thomas; Desprat, Nicolas; Sinha, Deepak Kumar; Allemand, Jean-François; Vincent
Croquette; Bensimon, David (2008).
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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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2996:
Korenberg, Michael J.; Hunter, Ian W. (March 1996). "The identification of nonlinear biological systems: Volterra kernel approaches".
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553:, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler
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3774:
3686:
3385:
Billings S.A. "Nonlinear System
Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013
863:
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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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2696:– 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
126:
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1425:
603:). 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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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 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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1821:, which is always useful whether or not the resulting ordinary differential equation(s) is solvable.
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2505:, 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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1994:
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2702:– 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
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".
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106:
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Mathematical
Control Theory: Deterministic Finite Dimensional Systems. Second Edition
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554:
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302:
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A classic, extensively studied nonlinear problem is the dynamics of a frictionless
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17:
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2968:, The Frontiers Collection, Springer Berlin Heidelberg, 2007, pp. 181–276,
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1351:
1172:, 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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2723:– asymptotic periodic orbits to which destabilized fixed points are attracted.
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209:
165:
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2109:{\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
1467:, especially for autonomous equations. For example, the nonlinear equation
3025:
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 the
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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 with
4117:
3598:
853:, homogeneity does not follow from additivity. For example, an
568:
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, see
690:
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:
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2544:
2511:
2481:
2437:
2365:
2336:
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2268:
2202:
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2133:
2038:
1997:
1977:
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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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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).}
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2017:
1983:
1960:
1774:Change of variables into something easier to study
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576:of degree higher than one or in the argument of a
4012:List of nonlinear ordinary differential equations
3401:1985 24th IEEE Conference on Decision and Control
2468:{\displaystyle \sin(\theta )\approx \pi -\theta }
4017:List of nonlinear partial differential equations
1805:List of nonlinear partial differential equations
630:
4007:List of linear ordinary differential equations
4129:
3610:
486:
8:
3471:(fourth ed.). 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 states
2193:, called the small angle approximation, is
2119:which is an implicit solution involving an
1761:Examination of dissipative quantities (see
4136:
4122:
4114:
3803:
3636:
3617:
3603:
3595:
3430:David Tong: Lectures on Classical Dynamics
1400:and the relations that define the various
493:
479:
49:
3469:Nonlinear Ordinary Differential Equations
3370:
3196:
3115:
2917:"Explained: Linear and nonlinear systems"
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2004:
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525:in which the change of the output is not
2501:. The solution to this problem involves
1652:{\displaystyle {\frac {du}{dx}}+u^{2}=0}
3290:Campbell, David K. (25 November 2004).
2908:
1799:Nonlinear partial differential equation
1516:{\displaystyle {\frac {du}{dx}}=-u^{2}}
1424:is said to be nonlinear if it is not a
1014:is a linear map (as defined above) and
828:Additivity implies homogeneity for any
59:
52:
3246:
3049:"Some nonlinear challenges in biology"
2603:{\displaystyle \sin(\theta )\approx 1}
1682:and its derivatives. Note that if the
1971:where gravity points "downwards" and
1286:many methods have been designed; see
7:
4002:List of named differential equations
3397:"Finite Time Differential Equations"
2688:Types of nonlinear dynamic behaviors
2167:. For example, the linearization at
1840:Other methods include examining the
1832:. For example, the (very) nonlinear
3927:Method of undetermined coefficients
3708:Dependent and independent variables
3292:"Nonlinear physics: Fresh breather"
2494:{\displaystyle \theta \approx \pi }
1765:) analogous to conserved quantities
1729:, the problem would be linear (the
4292:Evolutionary developmental biology
1560:{\displaystyle u={\frac {1}{x+C}}}
1248:For a single equation of the form
1018:otherwise. The equation is called
240:Evolutionary developmental biology
25:
3467:Jordan, D. W.; Smith, P. (2007).
3448:and Anthony J. Pritchard (2005).
3163:Stephenson, C.; et., al. (2017).
2717:– self-reinforcing solitary waves
564:, which is a set of simultaneous
3824:Carathéodory's existence theorem
3586:) a Database of Physical Systems
3512:Advanced Engineering Mathematics
3226:. Berlin: Springer. p. 46.
2998:Annals of Biomedical Engineering
2793:Landau–Lifshitz–Gilbert equation
2319:{\displaystyle \theta \approx 0}
1412:Nonlinear differential equations
4426:Ordinary differential equations
3359:Journal of Symbolic Computation
2862:Aleksandr Mikhailovich Lyapunov
2783:Kadomtsev–Petviashvili equation
2734:Examples of nonlinear equations
2029:, which would eventually yield
1815:ordinary differential equations
1461:ordinary differential equations
1455:Ordinary differential equations
1406:nonlinear system identification
1363:systems of polynomial equations
407:Ordinary differential equations
4484:Partial differential equations
2817:Nonlinear Schrödinger equation
2591:
2585:
2565:{\displaystyle \theta =\pi /2}
2521:
2513:
2450:
2444:
2281:
2275:
2081:
2075:
1949:
1943:
1811:partial differential equations
1793:Partial differential equations
1463:are often exactly solvable by
1392:defines successive terms of a
1384:Nonlinear recurrence relations
1264:
1258:
1237:Nonlinear systems of equations
1192:
1186:
1155:
1149:
1100:
1094:
1064:
1058:
1001:
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918:
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888:
870:
808:
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790:
781:
751:
745:
736:
730:
721:
709:
677:
671:
278:Partial differential equations
1:
1449:Dirichlet boundary conditions
562:nonlinear system of equations
37:Nonlinearity (disambiguation)
3652:Notation for differentiation
2349:{\displaystyle \theta =\pi }
1870:Linearizations of a pendulum
1342:{\displaystyle x^{2}+x-1=0.}
44:Nonlinear Dynamics (journal)
4234:Particle swarm optimization
3748:Exact differential equation
3571:Nonlinear Dynamics I: Chaos
3268:Nonlinear Dynamics I: Chaos
2974:10.1007/978-3-540-34153-6_7
2018:{\displaystyle d\theta /dt}
572:) appear as variables of a
127:Particle swarm optimization
4634:
4479:Reaction-diffusion systems
4204:Self-organized criticality
3486:Khalil, Hassan K. (2001).
3073:10.1088/0951-7715/21/8/T03
2829:for unsaturated water flow
2788:Korteweg–de Vries equation
2741:Algebraic Riccati equation
2328:simple harmonic oscillator
1862:Illustration of a pendulum
1851:
1802:
1796:
1432:in fluid dynamics and the
1426:system of linear equations
624:Some authors use the term
274:Reaction–diffusion systems
103:Self-organized criticality
29:
4262:Artificial neural network
4058:Józef Maria Hoene-Wroński
4038:Gottfried Wilhelm Leibniz
3829:Cauchy–Kowalevski theorem
3372:10.1016/j.jsc.2008.03.004
2684:and approximate periods.
2528:{\displaystyle |\theta |}
2186:{\displaystyle \theta =0}
215:Artificial neural network
33:Non-linear editing system
4340:Evolutionary game theory
4267:Evolutionary computation
4199:Collective consciousness
3952:Finite difference method
3395:Vardia T. Haimo (1985).
1709:term were replaced with
1434:Lotka–Volterra equations
1374:differentiable functions
1119:is very general in that
466:Evolutionary game theory
356:Second-order cybernetics
220:Evolutionary computation
136:Collective consciousness
4358:Social network analysis
4297:Artificial intelligence
4229:Ant colony optimization
4189:Collective intelligence
3932:Variation of parameters
3922:Separation of variables
3819:Peano existence theorem
3814:Picard–Lindelöf theorem
3701:Attributes of variables
3582: – (in
3580:Nonlinear Model Library
3409:10.1109/CDC.1985.268832
2833:Self-balancing unicycle
2803:Navier–Stokes equations
2153:{\displaystyle C_{0}=2}
1984:{\displaystyle \theta }
1878:under the influence of
1834:Navier-Stokes equations
1819:separation of variables
1742:nonelementary integrals
1465:separation of variables
1445:superposition principle
1430:Navier–Stokes equations
1279:{\displaystyle f(x)=0,}
1229:, the result will be a
940:An equation written as
696:superposition principle
244:Artificial intelligence
158:Social network analysis
123:Ant colony optimization
95:Collective intelligence
4489:Dissipative structures
4330:Rational choice theory
4093:Carl David Tolmé Runge
3667:Differential-algebraic
3626:Differential equations
3403:. pp. 1729–1733.
2966:The Nonlinear Universe
2843:Van der Pol oscillator
2773:Ginzburg–Landau theory
2670:
2604:
2566:
2529:
2495:
2469:
2422:
2350:
2320:
2294:
2253:
2187:
2154:
2127:(nonelementary unless
2125:nonelementary integral
2110:
2019:
1985:
1962:
1871:
1863:
1854:Pendulum (mathematics)
1830:boundary value problem
1723:
1703:
1676:
1653:
1590:
1561:
1517:
1422:differential equations
1343:
1288:Root-finding algorithm
1280:
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 equations
457:Rational choice theory
282:Dissipative structures
35:. For other uses, see
4302:Evolutionary robotics
4219:Agent-based modelling
4078:Augustin-Louis Cauchy
4073:Joseph-Louis Lagrange
3967:Finite element method
3957:Crank–Nicolson method
3891:Numerical integration
3870:Exponential stability
3762:Relation to processes
3647:Differential operator
2964:"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 is
1344:
1281:
1231:differential equation
1224:
1200:
1168:can literally be any
1163:
1134:
1114:
1072:
1043:
1009:
973:
932:
820:
763:
685:
385:Theory of computation
248:Evolutionary robotics
115:Agent-based modelling
4461:Coupled map lattices
4421:Time series analysis
4363:Small-world networks
3972:Finite volume method
3896:Dirac delta function
3865:Asymptotic stability
3807:Existence/uniqueness
3672:Integro-differential
3575:MIT's OpenCourseWare
3446:Diederich Hinrichsen
3279:MIT's OpenCourseWare
2945:www.birmingham.ac.uk
2838:Sine-Gordon equation
2617:
2576:
2542:
2509:
2503:hyperbolic sinusoids
2479:
2435:
2363:
2334:
2304:
2266:
2200:
2171:
2131:
2036:
1995:
1975:
1897:
1884:Lagrangian mechanics
1753:conserved quantities
1713:
1686:
1666:
1607:
1589:{\displaystyle u=0,}
1571:
1530:
1474:
1441:linearly independent
1402:Hofstadter sequences
1308:
1290:. In the case where
1252:
1213:
1198:{\displaystyle f(x)}
1180:
1161:{\displaystyle f(x)}
1143:
1123:
1088:
1079:homogeneous function
1070:{\displaystyle f(x)}
1052:
1026:
1007:{\displaystyle f(x)}
989:
947:
864:
837:continuous functions
774:
702:
683:{\displaystyle f(x)}
665:
436:Coupled map lattices
402:Time series analysis
162:Small-world networks
4618:Concepts in physics
4441:Population dynamics
4335:Bounded rationality
4277:Genetic programming
3982:Perturbation theory
3962:Runge–Kutta methods
3942:Integral transforms
3875:Rate of convergence
3771:(discrete analogue)
3452:. Springer Verlag.
3353:Lazard, D. (2009).
3308:2004Natur.432..455C
3181:2017NatSR...741621S
3126:2008Chaos..18c3118G
3065:2008Nonli..21..131M
1890:nonlinear equation
1757:Hamiltonian systems
1751:Examination of any
1390:recurrence relation
1356:real-root isolation
1301:polynomial equation
1041:{\displaystyle C=0}
461:Bounded rationality
419:Population dynamics
328:Conversation theory
228:Genetic programming
153:Scale-free networks
85:Collective behavior
4587:Computation theory
4582:Information theory
4537:Operationalization
4413:Nonlinear dynamics
4325:Prisoner's dilemma
4272:Genetic algorithms
4103:Sofya Kovalevskaya
3937:Integrating factor
3860:Lyapunov stability
3780:Stochastic partial
3273:2008-02-12 at the
3010:10.1007/bf02667354
2768:General relativity
2763:Colebrook equation
2758:Boltzmann equation
2754:for optimal policy
2666:
2600:
2562:
2525:
2491:
2465:
2418:
2346:
2316:
2290:
2249:
2183:
2150:
2106:
2027:integrating factor
2015:
1981:
1958:
1872:
1864:
1778:Bifurcation theory
1768:Linearization via
1719:
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 combination
452:Prisoner's dilemma
397:Nonlinear dynamics
352:Operationalization
348:Information theory
224:Genetic algorithms
4613:Dynamical systems
4608:Nonlinear systems
4595:
4594:
4499:Cellular automata
4471:Pattern formation
4403:Adaptive networks
4194:Collective action
4164:Self-organization
4111:
4110:
3990:
3989:
3795:
3794:
3545:978-0-387-98489-6
3523:978-0-471-15496-9
3497:978-0-13-067389-3
3490:. Prentice Hall.
3488:Nonlinear Systems
3478:978-0-19-920824-1
3302:(7016): 455–456.
3189:10.1038/srep41621
3134:10.1063/1.2964200
2877:Initial condition
2827:Richards equation
2778:Ishimori equation
2727:Self-oscillations
2652:
2398:
2235:
2165:Taylor expansions
2121:elliptic integral
2085:
2084:
1932:
1763:Lyapunov function
1744:are encountered.
1731:exponential decay
1722:{\displaystyle u}
1675:{\displaystyle u}
1628:
1555:
1495:
1222:{\displaystyle x}
1132:{\displaystyle x}
626:nonlinear science
551:dynamical systems
545:, and many other
519:non-linear system
503:
502:
290:Cellular automata
264:Pattern formation
195:Adaptive networks
99:Collective action
68:Self-organization
18:Non-linear system
16:(Redirected from
4625:
4509:Self-replication
4398:Dynamic networks
4287:Machine learning
4214:Phase transition
4138:
4131:
4124:
4115:
4088:Phyllis Nicolson
4068:Rudolf Lipschitz
3905:Solution methods
3880:Series solutions
3804:
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 system
2822:Power-flow study
2812:Nonlinear optics
2798:Liénard equation
2752:Bellman equation
2675:
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 expansion
1755:, especially in
1728:
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 to
1204:
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 system
495:
488:
481:
372:Systems thinking
298:Self-replication
236:Machine learning
190:Dynamic networks
111:Phase transition
50:
47:
40:
21:
4633:
4632:
4628:
4627:
4626:
4624:
4623:
4622:
4598:
4597:
4596:
4591:
4557:System dynamics
4518:
4504:Spatial ecology
4465:
4407:
4393:Systems biology
4344:
4311:
4282:Artificial life
4252:
4243:
4239:Swarm behaviour
4224:Synchronization
4184:Social dynamics
4175:
4168:
4147:
4145:Complex systems
4142:
4112:
4107:
4048:Jacob Bernoulli
4021:
3986:
3977:Galerkin method
3900:
3838:Solution topics
3833:
3791:
3757:
3696:
3628:
3623:
3557:
3552:
3546:
3532:Sontag, Eduardo
3530:
3524:
3506:Kreyszig, Erwin
3504:
3498:
3485:
3479:
3466:
3460:
3444:
3440:
3438:Further reading
3435:
3428:
3424:
3394:
3393:
3389:
3384:
3380:
3352:
3351:
3347:
3316:10.1038/432455a
3289:
3288:
3284:
3275:Wayback Machine
3266:
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 series
2857:
2852:
2848:Vlasov equation
2736:
2694:Amplitude death
2690:
2682:phase portraits
2641:
2637:
2623:
2622:
2615:
2614:
2574:
2573:
2572:, around which
2540:
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:characteristics
1807:
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 method
1311:
1306:
1305:
1291:
1250:
1249:
1243:linear equation
1239:
1211:
1210:
1207:differentiation
1178:
1177:
1174:boundary values
1141:
1140:
1121:
1120:
1086:
1085:
1084:The definition
1050:
1049:
1024:
1023:
987:
986:
945:
944:
862:
861:
772:
771:
700:
699:
663:
662:
659:linear function
647:
641:
636:
584:of the unknown
499:
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 science
366:
364:System dynamics
362:
358:
354:
350:
346:
342:
338:
334:
330:
325:
320:
306:
305:
300:
296:
294:Spatial ecology
292:
288:
284:
280:
276:
271:
266:
256:
255:
250:
246:
242:
238:
234:
232:Artificial life
230:
226:
222:
217:
212:
198:
197:
192:
188:
186:Systems biology
184:
180:
176:
172:
168:
164:
160:
155:
150:
140:
139:
138:
133:
131:Swarm behaviour
129:
125:
121:
119:Synchronization
117:
113:
109:
105:
101:
97:
92:
90:Social dynamics
87:
77:
76:
75:
70:
54:Complex systems
48:
41:
30:
28:
23:
22:
15:
12:
11:
5:
4631:
4629:
4621:
4620:
4615:
4610:
4600:
4599:
4593:
4592:
4590:
4589:
4584:
4579:
4574:
4569:
4564:
4559:
4554:
4549:
4547:Self-reference
4544:
4539:
4534:
4528:
4526:
4524:Systems theory
4520:
4519:
4517:
4516:
4511:
4506:
4501:
4496:
4491:
4486:
4481:
4475:
4473:
4467:
4466:
4464:
4463:
4458:
4453:
4451:Multistability
4448:
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 mentality
4206:
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 Lindelöf
4060:
4055:
4050:
4045:
4043:Leonhard Euler
4040:
4035:
4029:
4027:
4026:Mathematicians
4023:
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 portrait
3847:
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 bundles
3750:
3745:
3740:
3735:
3730:
3725:
3720:
3718:Nonhomogeneous
3715:
3710:
3704:
3702:
3698:
3697:
3695:
3694:
3689:
3684:
3679:
3674:
3669:
3664:
3659:
3654:
3649:
3643:
3641:
3634:
3633:Classification
3630:
3629:
3624:
3622:
3621:
3614:
3607:
3599:
3593:
3592:
3587:
3577:
3568:
3563:
3556:
3555:External links
3553:
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:
3233:978-3642202292
3232:
3212:
3155:
3094:
3039:
3004:(2): 250–268.
2988:
2982:
2956:
2932:
2907:
2905:
2902:
2900:
2899:
2894:
2892:Vector soliton
2889:
2884:
2879:
2874:
2869:
2864:
2858:
2856:
2853:
2851:
2850:
2845:
2840:
2835:
2830:
2824:
2819:
2814:
2809:
2807:fluid dynamics
2800:
2795:
2790:
2785:
2780:
2775:
2770:
2765:
2760:
2755:
2749:
2743:
2737:
2735:
2732:
2731:
2730:
2724:
2718:
2712:
2709:Multistability
2706:
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 analysis
1797:Main article:
1794:
1791:
1790:
1789:
1786:
1780:
1775:
1772:
1766:
1759:
1718:
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1275:
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1188:
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1157:
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1128:
1108:
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1099:
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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 map
826:
825:
813:
810:
807:
804:
801:
798:
795:
792:
789:
786:
783:
780:
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756:
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747:
744:
741:
738:
735:
732:
729:
726:
723:
720:
717:
714:
711:
708:
694:Additivity or
679:
676:
673:
670:
646:
643:
638:Stanisław Ulam
634:
555:linear systems
543:mathematicians
501:
500:
498:
497:
490:
483:
475:
472:
471:
450:
445:
444:
441:
440:
427:Multistability
400:
395:
394:
391:
390:
360:Self-reference
321:
314:Systems theory
312:
311:
308:
307:
267:
262:
261:
258:
257:
213:
204:
203:
200:
199:
151:
146:
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141:
107:Herd mentality
88:
83:
82:
79:
78:
71:
66:
65:
62:
61:
57:
56:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4630:
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4611:
4609:
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4605:
4603:
4588:
4585:
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4580:
4578:
4575:
4573:
4570:
4568:
4565:
4563:
4560:
4558:
4555:
4553:
4552:Goal-oriented
4550:
4548:
4545:
4543:
4540:
4538:
4535:
4533:
4530:
4529:
4527:
4525:
4521:
4515:
4514:Geomorphology
4512:
4510:
4507:
4505:
4502:
4500:
4497:
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4492:
4490:
4487:
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4364:
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4320:
4318:
4314:
4308:
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4300:
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4288:
4285:
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4280:
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4265:
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4260:
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4232:
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4227:
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4207:
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4018:
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4008:
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3999:
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3993:
3983:
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3760:
3754:
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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:
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3648:
3645:
3644:
3642:
3638:
3635:
3631:
3627:
3620:
3615:
3613:
3608:
3606:
3601:
3600:
3597:
3591:
3588:
3585:
3581:
3578:
3576:
3572:
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3564:
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3559:
3558:
3554:
3547:
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3537:
3533:
3529:
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3514:
3513:
3507:
3503:
3499:
3493:
3489:
3484:
3480:
3474:
3470:
3465:
3461:
3459:9783540441250
3455:
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:
2983:9783540341529
2979:
2975:
2971:
2967:
2960:
2957:
2946:
2942:
2936:
2933:
2922:
2918:
2912:
2909:
2903:
2898:
2895:
2893:
2890:
2888:
2887:Mode coupling
2885:
2883:
2882:Linear system
2880:
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 beam
2744:
2742:
2739:
2738:
2733:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2705:
2701:
2698:
2695:
2692:
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2687:
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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:dimensionless
1885:
1881:
1877:
1868:
1860:
1855:
1847:
1845:
1843:
1838:
1835:
1831:
1827:
1822:
1820:
1817:, as seen in
1816:
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:Homogeneity:
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:singularities
610:
606:
602:
601:linearization
597:
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-oriented
337:
333:
329:
324:
319:
315:
310:
309:
304:
303:Geomorphology
301:
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:
4378:Graph theory
4307:Evolvability
4098:Martin Kutta
4053:Émile Picard
4033:Isaac Newton
3947:Euler method
3917:Substitution
3538:. Springer.
3535:
3511:
3487:
3468:
3449:
3425:
3400:
3390:
3381:
3362:
3358:
3348:
3299:
3295:
3285:
3263:
3237:. Retrieved
3222:
3215:
3172:
3168:
3158:
3107:
3103:
3097:
3056:
3053:Nonlinearity
3052:
3042:
3001:
2997:
2991:
2965:
2959:
2948:. Retrieved
2944:
2935:
2924:. Retrieved
2920:
2911:
2721:Limit cycles
2678:
2537:
2430:
2326:. This is a
2261:
2162:
2118:
1970:
1873:
1839:
1823:
1808:
1783:Perturbation
1746:
1735:
1661:
1597:
1525:
1459:First order
1458:
1438:
1436:in biology.
1415:
1398:logistic map
1388:A nonlinear
1387:
1371:
1369:algorithms.
1367:Gröbner base
1360:
1299:
1298:, one has a
1247:
1240:
1083:
1019:
1015:
982:
980:
939:
850:
843:
832:
827:
658:
648:
631:
625:
623:
598:
593:
561:
559:
527:proportional
518:
514:
504:
455:
405:
396:
326:
272:
253:Evolvability
218:
174:Graph theory
156:
93:
4577:Autopoiesis
4572:Cybernetics
4562:Sensemaking
4532:Homeostasis
4494:Percolation
4456:Bifurcation
4431:Phase space
4317:Game theory
4174:Collective
3855:Phase space
3713:Homogeneous
3059:(8): T131.
1738:closed-form
1354:roots; see
1020:homogeneous
835:, and, for
651:mathematics
507:mathematics
447:Game theory
431:Bifurcation
411:Phase space
376:Sensemaking
344:Homeostasis
323:Autopoiesis
318:cybernetics
286:Percolation
4602:Categories
4388:Robustness
4368:Centrality
4254:adaptation
4152:Background
4083:John Crank
3912:Inspection
3775:Stochastic
3769:Difference
3743:Autonomous
3687:Non-linear
3677:Fractional
3640:Operations
3239:20 January
2950:2018-06-30
2926:2018-06-30
2904:References
1803:See also:
1733:problem).
1296:polynomial
981:is called
839:, for any
655:linear map
645:Definition
574:polynomial
547:scientists
539:physicists
535:biologists
415:Attractors
210:adaptation
182:Robustness
166:Centrality
4436:Attractor
4249:Evolution
4159:Emergence
3887:solutions
3845:Wronskian
3800:Solutions
3728:Decoupled
3692:Holonomic
3516:. Wiley.
3324:0028-0836
3249:cite book
3175:: 41621.
3117:0803.2252
3089:119808230
3081:0951-7715
3018:0090-6964
2704:aperiodic
2634:θ
2595:≈
2589:θ
2583:
2552:π
2546:θ
2518:θ
2489:π
2486:≈
2483:θ
2463:θ
2460:−
2457:π
2454:≈
2448:θ
2442:
2410:θ
2407:−
2404:π
2380:θ
2344:π
2338:θ
2311:≈
2308:θ
2288:θ
2285:≈
2279:θ
2273:
2241:θ
2217:θ
2175:θ
2079:θ
2073:
2049:θ
2040:∫
2002:θ
1979:θ
1947:θ
1941:
1914:θ
1501:−
1328:−
1205:contains
1016:nonlinear
913:β
895:α
883:β
874:α
797:α
785:α
590:functions
586:variables
566:equations
531:engineers
206:Evolution
73:Emergence
4542:Feedback
4350:Networks
4176:behavior
3995:Examples
3885:Integral
3657:Ordinary
3534:(1998).
3508:(1998).
3417:45426376
3332:15565139
3271:Archived
3207:28155863
3169:Sci. Rep
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578:function
336:Feedback
269:Fractals
148:Networks
4567:Entropy
4383:Scaling
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