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.

View More On Wikipedia.org
Recent contents Top users
  1. W

    Nonlinear first-order ODE?

    Homework Statement Let $$\frac{1}{2}\dot{r}^2=e+\frac{m}{r}-\frac{L^2}{2r^2}$$ where L is angular moment, and e is energy (so I guess I'll take as constants for now...) Homework Equations Not sure for now. The Attempt at a Solution So, if I let $$u=\frac{1}{r}$$ then my equation becomes...
  2. Cryo

    A Nonlinear susceptibility and group reps

    Dear All short explanation: I am trying to leverage my limited understanding of representation theory to explain (to myself) how many non-vanshing components of, for example, nonlinear optical susceptibility tensor ##\chi^{(2)}_{\alpha\beta\gamma}## can one have in a crystal with known point...
  3. M

    I How can you know if a numerical solution is correct?

    Hi PF, Suppose I numerically solve a nonlinear system of differential equations. How can I know if my solution is correct (if there is no known analytic solution)? What are the standard practices people do? I have a couple of ideas, but I want to know what people are already doing. Danke!
  4. I

    Nonlinear Optics: third-order susceptibility

    Hi. I've just learned about enumerating the second-order susceptibility (rather blindly) by 3^3 * (3*2*1) * 2 = 324. (tensor size * 3 frequency permutation * negative frequency) I'm guessing that for the third-order susceptibility would similarly yeild 3^4 * (4*3*2*1) * 2 = 3888? I couldn't...
  5. C

    MHB Regular perturbation nonlinear problem

    Hi all, I have this (nondimensionalised) system of ODEs that I am trying to analyse: \[ \begin{align} \frac{dr}{dt}= &\ - \left(\alpha+\frac{\epsilon}{2}\right)r + \left(1-\frac{\epsilon}{2}\right)\alpha p - \alpha^2\beta r p + \frac{\epsilon}{2} \\ \frac{dp}{dt}= &\...
  6. U

    A Taylor expansion for a nonlinear system and Picard Iterations

    Hello guys I struggle since yesterday with the following problem I am reading the book "Elements of applied bifurcation theory" by Kuznetsov . At one point he has the following Taylor expansion of a nonlinear system with respect to x=0 where ##x\in \mathbb(R)^n## $$\dot{x} = f(x) = \Lambda x +...
  7. SchroedingersLion

    Nonlinear Optics - Pockels effect

    Greetings, is anyone here familiar with nonlinear optics? I want to know wether the Pockels effect only occurs in optically anisotropic media or not. Of course, we need a medium with inversion symmetry ("non-centrosymmetric medium"), but I am not sure about the optical isotropy. In an...
  8. C

    A Exploring Scaling Symmetry in Differential Equations: A Case Study in Geometry

    A geometry problem I'm working on has boiled down to finding a function ##f(t)## such that $$f'' + \frac{2}{t}f' + \frac{f'^2}{\left( 1 - \frac{f}{t} \right) t } + \frac{f'f}{\left(1- \frac{f}{t} \right) t^2} = 0$$ It has two fairly simple solutions, namely ##f(t) = a## and ##f(t) =...
  9. T

    Solving a set of nonlinear quadratic equations

    I would like to solve this system, which is a sets of non linear quadratic equations, the system needed to be solved can be expressed in general as follow: ϒϒ'C – ϒα = B Where ϒ=(ϒ1,ϒ2,...ϒn)’ is a column vector and ϒ’ its transpose C=(c1,c2,…,cn)’ and B=(b1,b2,…bn)’ are a columns vector And...
  10. Y

    Interval of the maximum solution of a nonlinear equation

    Mentor note: Thread moved from the Technical Math section, so there is no template. @yamata1, , in the future, please post homework problems or exercises in the Homework & Coursework sections, not in the Technical Math sections. I have moved your post. Hello, I would like some help with an...
  11. H

    MATLAB Solving 2nd Order PDE System with Crank-Nicholson

    I have the following system of PDEs: \hat{\rho}\hat{c}_{th}\frac{\partial\hat{T}}{\partial\hat{x}}-\alpha_{1}\frac{\partial}{\partial\hat{x}}\left(\hat{k}(\hat{x})\frac{\partial\hat{T}}{\partial\hat{x}}\right)=\alpha_{1}\hat{\sigma}(\hat{x})\hat{E}...
  12. mertcan

    Computational High order nonlinear operator split method

    Hi, I really need to have sources related to high order operator split method for nonlinear pdf ode equations. Could you provide me with sources about that books files videos links...??
  13. W

    I A nonlinear recurrence relation

    Hi Physics Forums, I am stuck on the following nonlinear recurrence relation $$a_{n+1}a_n^2 = a_0,$$ for ##n\geq0##. Any ideas on how to defeat this innocent looking monster? I have re-edited the recurrence relation
  14. C

    I Nonlinear relation between coordinate time and proper time

    For Schwarzschild geomery $$ds^2=-(1-\frac{2GM}{r})dt^2+(1-\frac{2GM}{r})^{-1}dr^2+r^2d\Omega^2$$ For a Schwarzschild observer , the proper time and coordinate time are related by $$d\tau=(1-\frac{2GM}{r})^{1/2}dt$$ There is a often used relation between proper time and coordinate time $$d\tau...
  15. X

    Laser pumping: further increasing injection above threshold?

    Above the threshold, the stimulation emission becomes dominant for lasing. If increasing the pumping, what will be the change of spontaneous emission and its contribution to the output power and FWHM of the signal, based on the threshold carrier density clamping and the threshold gain clamping...
  16. A

    MHB Lemma 4.5.1 on page 77 in the book Averaging Methods in Nonlinear Dynamical Systems

    I need your advice on understanding a proof of a lemma from a book I am reading. I asked my question in overflow: https://mathoverflow.net/questions/299408/lemma-4-5-1-on-page-77-in-the-book-averaging-methods-in-nonlinear-dynamical-syst Does anyone understand the proof and can answer my...
  17. B

    A Nonlinear Schrodinger equation and linearity of Q.M.

    Hello all, you may already know that Q.M. is a linear theory however there is something called nonlinear Sch. eq. for example Gross-Pitaevskii equation. How can such a thing exist considering that Q.M. is a strictly linear theory. Cheers.
  18. Theia

    MHB Looking for a serie solution for a nonlinear ODE system

    Hi! \begin{cases} \dot{q} = a \left( 1 - q^2 \right) \\ \dot{a} = - \alpha - a^2 q\end{cases} \qquad \alpha \in (0, 1 ) I've looked into this ODE system about 7 months now, but I've not got anything promising how to write down the solution. I'm mostly interested in q-serie. (To those of you...
  19. H

    A Stability for a system of nonlinear ODEs

    Hi, I am looking at the following system of ODEs: \begin{eqnarray*} \dot{\omega}_{3}+\alpha\omega_{3} & = & \frac{\beta_{1}+\beta_{3}}{\rho_{0}}J_{3} \\ \dot{J_{3}}+2(\alpha_{2}-\alpha_{1})\beta_{2} & = & 0 \\ \dot{\beta}_{1}+\omega_{3}\beta_{2} & = & 0 \\...
  20. SemM

    A Solve a non-linear ODE of third order

    Hi, I tried to solve the following in Wolfram alpha: y''' + (1-x^2)y=0 y(0)=0 y'(0)=0 y''(0)=0 however, I got answer which cannot be reproduced (even at wolfram pages). I have tried ODE45 in MATLAB, but it only gives a plot. Is there any way to solve this analytically or numerically to give...
  21. Dor

    A Numerical solution for a two dimensional 3rd order nonlinear diff. eq.

    I'm a bit lost in all the numerous methods for solving differential equations and I would be very grateful if someone could point me to some direction. I want to solve the following boundary conditioned differential equation: $$a_1+a_2\nabla f(x,y)+a_3\nabla f(x,y)\cdot \nabla^2...
  22. SemM

    A Does Commutativity Affect Linearity?

    Hi, I have in a previous thread discussed the case where: \begin{equation} TT' = T'T \end{equation} and someone, said that this was a case of non-linear operators. Evidently, they commute, so their commutator is zero and therefore they can be measured at the same time. What makes them however...
  23. F

    A Optimization of a nonlinear system

    Can someone please tell me how to go about optimizing this system of equations? It is weird because the residuals are computed with ##A = B*X_1+C*X_2## but X_1 and X_2 are computed in a separate function ##[X_1,X_2]=f(k1,k2,H0,G0)##, and what I am optimizing is a parameter in that second...
  24. A

    MHB A question on matrix's eigenvalue problem from Eberhard Zeidler's first volume of Nonlinear Function

    The question is posted in the following post in MSE, I'll copy it here: https://math.stackexchange.com/questions/1407780/a-question-on-matrixs-eigenvalue-problem-from-eberhard-zeidlers-first-volume-o I have a question from Eberhard Zeidler's book on Non-Linear Functional Analysis, question...
  25. S

    What is 'phase space in chaos theory and nonlinear dynamics?

    The term 'phase space' is often used in the study of nonlinear dynamics.What is it.
  26. D

    Nonlinear Equations with Four Variables: Solving for All Solutions

    Homework Statement Find all solutions of: a + c = 5 b + d + ac = 5 ad+ bc = 5 bd = -6 Homework EquationsThe Attempt at a Solution I tried subsitution and end up with a cubic equations. I am pretty sure it can be done easier.
  27. maistral

    A Nonlinear regression in two or more independent variables

    Hi. I wanted to learn more on this topic, but it seems all the available resources in the internet points to using R, SPSS, MINITAB or EXCEL. Is there an established numerical method for such cases? I am aware of the Levenberg-Marquardt, Gauss-Newton and such methods for nonlinear regression on...
  28. M

    Numerical Nonlinear Lifting Line Theory in MA

    1. Homework Statement Hello all. It is not a homework actually. I just didn't know at which forum I should post. I am working on a MATLAB code solving the finite wing properties iteratively by using the Anderson's Numerical Lifting Line Method. However, I got some wrong results. The...
  29. D

    A Bright, dark soliton for cubic-quintic nonlinear Schrodinger

    For a given stationary cubic-quintic nonlinear Schrodinger equation, EU=-U_XX+G1|U|^2U+G2 |U|^4 U, where X=X(t,x). There are bright and dark solitons. In many references, it is found that there is typo or mistake in dark soliton by substituting their soliton solution to this above eqaution. The...
  30. e2m2a

    I Test for inconsistency of system of nonlinear equations

    Is there a quick test to determine if a system of nonlinear equations is inconsistent. For example, suppose there is a system of equations such as: 3x cubed + 2y cubed = z cubed 2x cubed + 5y cubed = z cubed Since these two equations are clearly not dependent, could we say that since...
  31. O

    Switched Reluctance Motors: Nonlinear Loads?

    Homework Statement Would a switched reluctance motor be described as a non linear load? Since it is switched on and off rapidly to turn the rotor, does that make it non linear?
  32. M

    Nonlinear Schrodinger Equation Dispersion Relation

    The Nonlinear Schrodinger Equation (NSE) is presented as: $$i\frac{∂A}{∂z} = \frac{1}{2}β_2\frac{∂^2A}{∂t^2}-\gamma|A^2|A$$ The steady state solution $$A(z)$$ Can be derived as an Ansatz given by: $$ A(z) = \rho(z)e^{i\phi(z)}$$ By substituting and solving the ODE, the steady state...
  33. tl_ccc

    Nonlinear contact convergence problem in ANSYS Workbench?

    I am using the static structural module of ANSYS workbench to do a simulation. In my model, there is a gear and a spring which presses against the gear, moves along it and pushes it to turn counterclockwise. These two objects are in frictional contact. In my calculation, I always have the...
  34. H

    I Linear and nonlinear physical theories

    Classical physics is a nonlinear theory, but how is it that? Why is it nonlinear? Also quantum mechanics is a linear theory so that the sum of the solutions of the schrödinger equation is itself a solution. But I'm not sure I grasp this completely. Why is quantum mechanics linear while...
  35. Anders Johnsen

    Nonlinear material problem in ANSYS Workbench

    I'm currently conduction a nonlinear analysis of a partially buried, cylindrically shaped sump-system. I've decided to best simulate the stresses on the sump by modelling a shallow cylinder with gravel-properties around it and a cube with soil-properties surrounding both sump and gravel. As I'm...
  36. aphirst

    Failure of Optimisation for Nonlinear Equation Systems

    I wasn't sure into which category I should post this, so feel free to move it into a more appropriate place. As part of my work I'm solving a system of nonlinear equations, of a usual form: $$\vec{F}(\vec{X})=\begin{pmatrix}F_1(X_1, X_2, \cdots X_N) \\ F_2(\cdots) \\ \vdots \\...
  37. D

    Solving a system of nonlinear equations

    This is actually not a homework problem, but a problem I'm encountering while working on a little project and I'm not sure if it's even solvable or if it makes sense what I'm doing 1. Homework Statement First, I have the equation $$p_{ij} = \frac{1}{2}\left( \tanh{(-\frac{\theta_i +...
  38. R

    MHB Self-similarity of nonlinear PDE

    Hi everyone, I am a student in Mechanical Engineering and I am currently working on an assignment where I am exploiting the possibility of self-similarity for a PDE of a given problem. The PDE in my assignment consists of two independent variables (x for space and t for time), and one dependent...
  39. A

    B First Order Non-Linear ODE (what method to use?)

    Hi, The problem is to solve: dy/dx = −[2x + ln(y)]*(y/x) Attempt: I have tried to see if it is exact, I found it not to be, I can't easily find a function to multiply by to make it exact either (unless I am missing something obvious). It clearly isn't seperable, nor is it homogenous (I know...
  40. S

    I Solve Nonlinear DE: Friedmann Eqns for H 0-10^7

    From cosmology, the friedmann equations are given by, ##H^2 = (\frac{\dot a}{a})^2 = \frac{8\pi G}{3} \rho \, , \quad \frac{\ddot a}{a} = -\frac{4\pi G}{3}(\rho+3p) \, , \quad## where ##\rho = \frac{1}{2}(\dot \phi^2 + \phi^2)## and ##p = \frac{1}{2}(\dot \phi^2 - \phi^2)## To get ##\dot H##...
  41. S

    A Squeezed Light & NonLinear Optics

    We're hearing things in the news these days about Squeezed Light, and how it can be used to improve everything from LIGO detectors, to positional sensors, to Quantum Computing. What is Squeezed Light, what useful applications is it being investigated for, and how does it provide this extra...
  42. Rahul Shenoy

    Parameter sensitivity analysis for nonlinear system

    I am working on a problem, where I have arrived at the following nonlinear state space equation: dx1/dt = x2; dx2/dt = c11 x1 + c21x2 + c31x3 + c41x4 + c51x1x22; dx3/dt = x4; dx4/dt = c12 x1 + c22x2 + c32x3 + c42x4 + c52x1x22; c11, c21, c31, c41, c51, c12, c22, c32, c42, c52 are all a function...
  43. D

    A Nonlinear regression which can be partially reduced to linear regression

    I encountered several times the following problem: Say I have a variable y dependent in a nonlinear way on m parameters ##\{x_i\}##, with ##i \in \{1,m\}##. However there is a linear relation between n>m functions ##f_j\in{x_i}##, i.e., ##y=\sum_j z_j f_j##. So I can get a solution of my problem...
  44. M

    A Nonlinear differential equation

    ρCp (∂T/∂t) + k (∂2T/∂x2) = exp(-σt2)exp(-λx2)φo i have this equation... i was thinking of taylor series expansion to solve it and make it easier... any ideas on how to solve?
  45. E

    How to Maximize a Nonlinear Function with Limited Variables?

    Hi to everyone, I'm optimizing a nonlinear function but I'm struggling to achieve it. The function is the following: X and i are relationed so i doesn't go to infinite. Do you have any idea how to maximize this function? Thanks in advance, Eric
  46. T

    Python Runge Kutta for nonlinear system of equation

    I am applying a 4th order Runge-Kutta code to solve the following: \begin{equation} \frac {\partial y_1}{\partial t} = y_2 y_3 - C_1 y_1 \end{equation} \begin{equation} \frac {\partial y_2}{\partial t} = y_3y_1 - C_2 y_2 \end{equation} \begin{equation} \frac {\partial y_3}{\partial z} = y_4...
  47. P

    Is the system linear or nonlinear

    Homework Statement 3y(t)+2=x(t) Homework Equations k1y1(t) + k2y2(t) + 2(k1+k2) = k1x1(t)+k2x2(t) The Attempt at a Solution I know the system is non linear but I cannot explain why. It has something to do with 2(k1+k2) but I am unsure.
  48. J

    Design of nonlinear spring from load deflection graph

    Hi Could anybody guide me to a software that can be used to design a non linear spring from load deflection characteristics. need it for both extension and compression spring. thanks noor
  49. baby_1

    Autocorrelation function of the output of the nonlinear device

    Hello for an input signal with a noise we have and for obtain Power spectural density we use autocorrelation function where hkm is but I need to know what is autocorrelation function for different inputs with different frequencies? such as Any help will appericate
  50. Chandler

    Other Intermediate Optics (including nonlinear optics)?

    Hey everyone. I'm currently in a new research lab that focuses on optics. One thing I'm currently tasked with is handling the femtosecond laser we have. However, to do this, I need a stronger background in optics than I currently have (which is a few years of undergrad optics, some quantum...
Back
Top