What is Nonlinear: Definition and 624 Discussions

In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function 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 linear combination of the unknown variables or functions 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 linear 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.
As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities 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 random 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.
Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:

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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  1. Adri

    I How to Solve a Nonlinear PDE with Sinh Function?

    The PDE is $$ \frac{1}{a^2 x^2} (u_y)^2 - (u_x)^2 =1$$ I know the solution, its ## u=x senh(ay) ##, but I dont know how I can get it. I've tried variable separation and method of characteristics but they dont seem to work.
  2. Z

    Linearizing a circuit with a nonlinear element

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  3. Z

    Plot ##i_A## vs ##v_A## characteristics for this nonlinear network

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  4. al4n

    B How to find an integer solution to a nonlinear equation?

    given something like: an = c where c is given and a, n, and c are only allowed to be integers. how would one find the value of say n or a?
  5. Euge

    POTW A Nonlinear Elliptic PDE on a Bounded Domain

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  6. F

    I Exploring Nonlinear Least Squares for Regression Analysis

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  7. patric44

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  8. R

    A Solve a nonlinear matrix equation

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  9. F

    I Linear Models vs Nonlinear Models

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  10. W

    A Solving nonlinear singular differential equations

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  11. W

    I Nonlinear differential equation

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  12. J

    Cam forces acting on a linear motion bearing with nonlinear forces

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  13. M

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  14. S

    Truss behaviour, load-deformation curve

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  15. person123

    I Solving a System of Nonlinear Equations Symbolically

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  16. CptXray

    I Coherent state evolution - nonlinear Hamiltonian

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  17. Isaac0427

    I An interesting Nonlinear Differential Equation

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  18. T

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  19. S

    Does the ruby laser (the first invented laser) have nonlinear behavior?

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  20. Haorong Wu

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  21. BWV

    I Deep Learning the new key to nonlinear PDEs?

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  22. S

    MHB Second-Order Nonlinear Differential Equation

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  23. J

    A Nonlinear Wave Equation (Nonlinear Helmholtz)

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  24. patric44

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  25. Kaguro

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  26. Julio1

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  27. F

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  28. F

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  29. J

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  30. J

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  31. J

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  32. B

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  33. Arman777

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  34. MathematicalPhysicist

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  35. H

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  36. A

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  37. W

    I Number of Solutions of a Nonlinear System

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  38. M

    MHB Mapping linear spaces to nonlinear ones.

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  39. M

    Solve simple nonlinear equations in the form [A]x=b

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  40. stockzahn

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  41. T

    I Nonlinear Second Order ODE: Can We Find an Analytical Solution?

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  42. bob14

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  43. S

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  44. Z

    I Non-Linear Evaporation: Explained for Beginners

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  45. alan123hk

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  46. Prez Cannady

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  47. R

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  48. Chromatic_Universe

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  49. E

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