# What is Non-linear: Definition and 339 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. ### Why is this not a general solution to this nonlinear DE?

a) We can use reduction of order $$p=y'\tag{1}$$ $$p'=y''\tag{2}$$ The DE becomes $$p'+p^2=0\tag{3}$$ $$\frac{1}{p^2}p'=-1\tag{4}$$ This last step contains the assumption that ##p^2=y'^2\neq 0##. $$-\left (\frac{1}{p(x)}-\frac{1}{p(x_i)}\right )=-(x-x_i)\tag{5}$$...
2. ### Non-linear isotropic dielectric capacitor

Question: Solution first part: Have I done it right? I don't know how to begin with second part since the dielectric is non-lineair, and most formulas like $$D=\epsilon E$$ and $$P= \epsilon_0 \xhi_e E$$, only apply for lineair dielectrics. What to do?
3. ### A Deriving Non-linear acoustic wave models, equilibrium state assumption

The standard derivation in obtaining a single wave equation involves making use of the heat equation with a Taylor expansion of the equation of state, then differentiating this equation and the continuity equation with respect to time, and combining with the divergence of the NS equation...
4. ### I Exploring the Impact of Non-Stretchy Mat Materials on Trampoline Physics

Trampolines are in effect coupled springs, with the mat being the much softer spring generally. E.g. On my Acon, when jumping about 1 meter, there is a max cone of depression about 60 cm deep and 1 meter across (1 meter point has a depression of only about 10 cm) At this same point the 160 or...
5. ### Find dissipative function for the non-linear force f=-bv^n

For a nonconservative force, What would be the dissipative function for a force f=-bvⁿ in Lagrangian (Where v is the velocity) [#qoute for a nonconservative force f=-bv The dissipative function is D=-(1/2)bv² ] Since ##f=\frac{\partial D}{\partial \dot x}## so the dissipative function should...
6. ### In search of a non-linear way to prepare students for this unique exam

Suppose there is an exam of maths, for a particular service, whose syllabus is of Bachelor of Science: Maths level, i.e. the syllabus includes (exhaustively) Linear Algebra, Abstract Algebra, Calculus, Vector Calculus, Real Analysis, Complex Analysis, Ordinary Differential Equations, Partial...
7. ### Solving a 2nd order non-linear DE by dimensional analysis/observation

The top most 2nd order non-linear DE is the one that has to be solved. Below is the solution. This problem is from Morin's Classical Mechanics. May I know how he could guess that r = Agt^2? Firstly, why must g tilda be a variable within r? I do not understand what he meant by 'parameter'...
8. ### I Fit a non-linear function to this time series

I have an experimantally obtained time series: n_test(t) with about 5500 data points. Now I assume that this n_test(t) should follow the following equation: n(t) = n_max - (n_max - n_start)*exp(-t/tau). How can I find the values for n_start, n_max and tau so as to find the best fit to the...
9. ### A Non-Linear Theory: Summation Meaningful in Einstein Gravitation?

In the famous book, Gravitation, by Misner, Thorne and Wheeler, it talks about the stress-energy tensor of a swarm of particles (p.138). The total stress-energy is summed up from all categories of particles. Is summation meaningful in the non-linear theory of Einstein gravitation? Thanks.
10. ### How to distinguish between linear and non-linear circuits?

Many circuit analysis techniques only apply to linear circuits. I don’t quite understand how to distinguish between linear and non-linear circuits. I understand the mathematical concept of linearity. I understand why components like resistors, capacitors and inductors are linear. I don’t quite...
11. ### I Why is this differential equation non-linear?

Hi, Could you please have a look on the attachment? Question 1: Why is this differential equation non-linear? Is it u=\overset{\cdot }{m} which makes it non-linear? I think one can consider x_{3} , k, and g to be constants. If it is really u=\overset{\cdot }{m} which makes it non-linear then...
12. ### I How to propagate errors through a regression & non-linear model?

Hi, I was working on a predictive linear regression model and was hoping to obtain some bounds to represent the uncertainty present in the model. Question: I suppose this boils down into two separate components: 1. What is a good measure of uncertainty from a linear regression model? MSE, or...
13. ### A Eigenvalues of block matrix/Related non-linear eigenvalue problem

I have a matrix M which in block form is defined as follows: \begin{pmatrix} A (\equiv I + 3\alpha J) & B (\equiv -\alpha J) \\ I & 0 \end{pmatrix} where J is an n-by-n complex matrix, I is the identity and \alpha \in (0,1] is a parameter. The problem is to determine whether the eigenvalues of...
14. ### A Improper boundary in non-linear ODE (pseudospectral methods)

Hello, I am trying to compute some non-linear equations with pseudospectral/collocation methods. Basically I am expanding the solution as $$y(x)=\sum_{n=0}^{N-1} a_n T_n(x),$$ Being the basis an Chebyshev polynomial with the mapping x in [0,inf]. Then we put this into a general...
15. ### I Non-linear ODE: initial conditions

Say you have the set of coupled, non-linear ODEs as derived in this thread, it has two unknowns ##N(t)## and ##\theta(t)##: $$N - mg = - m\frac{L}{2}\left(\dot{\theta}^2\cos(\theta) + \ddot{\theta}\sin(\theta)\right)$$ $$\frac{L}{2}N\sin(\theta) = \frac{1}{12}ml^2\ddot{\theta}$$ What freedom...
16. ### B Non-linear Behavior, Chaos to Order

I often hear about non-linear or chaos having order, and have difficulty grasping the concept. For example, you have weather disorder, how can it bring order? or you have Covid chaos, how can it bring order? Can you give some examples of the concept? What is fact, myth, and misconception...
17. C

### Mathematical Engineer, Electrical Engineer, & Author

My History -------------- I attended Oregon State U. and majored 3 years in Electrical Engineering. Then I switched to a Math major for my final years and graduated with a B.S. in Math (1967). Developed several Apps for Engineers & Scientists. My first job was with Lockheed Aircraft Co...
18. ### I How to solve a small system of non-linear equations?

If we have the following relations between x, y, and z: M_{xy}=\frac{2xy}{x+y} M_{xz}=\frac{2xz}{x+z} M_{yz}=\frac{2yz}{y+z} where M_{xy}, M_{xz}, and M_{yz} are known constants, what technique can be used to determine the values of x, y, and z?
19. ### Is non-linear Tension Force possible?

I seem to remember from my school days that Tension Force can only be linear. Is this true? In 1 (in the graphic) the tension will follow the line of the rope In 2 there is an unbendable, unbreakable, steel cable formed into an arch. a ] In 3 which direction will the Tension be? b ] In 3 if...
20. ### A Force to fix sigma8 or A_s for Forecasts

Hello, in the context of Forecasts with Fisher's formalism, I make vary cosmological parameters to compute the elements of the Fisher matrix. First, I generate with CAMB code a linear power spectrum. Then, from this, I am computing ##\sigma_{8,\text{linear}}##. Secondly, Before relaunching...
21. ### Understanding and Tuning Non-Linear PID Controllers for UAVs

Hello, I'm working on a project controlling a UAV (quadcopter) I'm trying to understand non-linear PID controllers. I know that a linear PID is given by: and a non-linear PID the konstant terms are replaced with functions My question is: how to i find these function and tune the paremeters, i...
22. ### Engineering A circuit with non-linear devices

I honestly don't know how to quite even begin this problem. Looking at Fig 3-2, the slopes of the graphs are 1/R, and hence where the slopes are 0, we have infinite resistance, in which case current wouldn't flow through that resistor and hence simplify the circuit. So I was trying to find...
23. ### A Computing sig8 for correction in non-linear regime

I need to apply, with CAMB code, a correction on ##\sigma_{8}## between linear and non-linear regime to keep it fixed (I make change the values of cosmological parameters at each iteration). I have to compute ##\sigma_{8}## from the ##P_{k}## and found the following relation (I put also the text...
24. ### I How can we identify non-linear singular differential equation

i am doing research to make criteria by which i can identify easily linear and non-linear and also identify its singular or not by doing simple test.please help me in this regard.
25. ### MHB Solving 1st order non-linear ODE

I want to solve $\d{y}{x}=\frac{3*(2x-7y)+6}{2*(2x-7y)-3}.$ I don't know its step by step solution. But using some trick of solving ordinary differential equation (which I saw on the Internet), I got the following solution:- $-\frac{17}{21}*(3x-2y)+ln(119y-34x-48)=C$. Now how to solve this...
26. ### Design a non-linear conical compression spring

What is the method to design a Non - Telescopic, Non - Linear, conical compression spring from a Force - Deflection characteristics
27. ### Converting a nonlinear eqn of motion to a state-space model

Homework Statement equations above are descriptive of a system with two configuration variables, q1 and q2. inputs are tau1 and tau2. d and c values are given. the question is about conversion of above equations to a state-space equation where the state-variables are x1 = q1_dot, x2 = q1_2dot...
28. ### I Nonlinear Pendulum: Calculating Angular Displacement

If I calculate the time period of a non linear pendulum using elliptical integral equation, then how can I find out the angular displacement.
29. ### I Can this type of transformation be non-linear?

I've finally worked out a derivation of the Lorentz transformation that doesn't use the now out of favor i^2=-1, but it still has one weak spot: it assumes that the transformation is linear. It seems quite reasonable to me that it would be linear since it has to graph straight lines on to...
30. ### Non-linear second-order ODE to Fuchsian equation

Homework Statement z\frac{d^2z}{dw^2}+\left(\frac{dz}{dw}\right)^2+\frac{\left(2w^2-1\right)}{w^3}z\frac{dz}{dw}+\frac{z^2}{2w^4}=0 (a) Use z=\sqrt y to linearize the equation. (b) Use t=\frac{1}{w} to make singularities regular. (c) Solve the equation. (d) Is the last equation obtained a...
31. ### I Converting Non-Linear Functions to Linear: Exploring the Possibility

Hello everyone, I've always had this question in my mind: Can we convert the non-linear function into a system of linear functions? I don't know if this is actually something exist in math (I searched a little bit to be honest), but I'm really interested in this question because it would make...
32. ### Rotordynamics: unbalance loads due to displacement - FEA

Hi, Im wondering if / how commercial FEA codes take into account additional unbalance forces due to the off-axis displacement of masses. For instance, if I am modeling a jeffcott rotor with an unbalance, I would traditionally represent this unbalance with a force that increases with speed^2...
33. ### Solving a Second Order Non-Linear PDE with Undetermined Coefficients

Homework Statement ##\frac{d^2y}{dx^2}=2xy\frac{dy}{dx}##Homework Equations This is second order non-linear pde of the 'form' ## f(y'',y',y,x) ## . I have read that there are 2 simplified versions of a second order non-linear pde that can be solved easily and these are 1) when there is no y...
34. ### A Is non-linear quantum mechanics (even) plausible?

The question is prompted by a claim raised in another thread that "There is good reason (theoretically) to believe that linearity fails at high enough energies." I've put this with an A prefix because it is going to be about some damned difficult maths, I know. But please try to avoid the...
35. ### I Minimizing a non-linear ODE in f, f_t

I'm working on a physics "potential" problem and trying to create an alternate function to describe the potential energy. I'm having trouble figuring out how to solve a nonlinear ODE, or even a limiting boundary for minimizing a drop off shape function. I was able to reduce my problem to the...
36. ### 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...
37. M

### MATLAB Suggestions for a non-linear fit model in Matlab

Good afternoon, I have a problem which I haven't solved yet, regarding a non-linear fit to a set of experimental data. I tried to solve it in Matlab, which I handle a little bit. I have a sensor, which has been designed to have 4 different filters in front of it. By making a sweep with the...
38. ### Is Plate Capacitor Gap a "Non-linear" Medium?

Could the space (air gap) between two DC charged parallel plates be considered to be a "non-linear" medium with respect to an EMF radiated by a coil contained within that space?
39. ### How to model a non-linear pendulum with air resistance?

Homework Statement I have found a differential equation that models a non-linear pendulum with air resistance, and now I have data. I've looked at the following site for guidance on how to analyse the data. It compares the motion of a damped spring, and compares it to the motion of a damped...
40. ### Motion of a non-linear pendulum with air resistance

Homework Statement I need to come up with an equation that would model the motion of a non-linear pendulum with air resistance. [/B]Homework Equations Fc=mgsintheta Fdrag=(1/2)p(v^2)CA The Attempt at a Solution I started with mgsintheta-(1/2)p(v^2)CA=ma After substituting v=r*omega and...
41. ### A Does Commutativity Affect Linearity?

Hi, I have in a previous thread discussed the case where: $$TT' = T'T$$ 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...
42. D

### B Do Parabolas and Other Non-Linear Graphs Have Slope?

So the slope is of course a ratio of the change in y-coordinates to the change in x-coordinates. This is easy to see with a linear equation. I just came across a cool math simulator ( https://phet.colorado.edu/sims/equation-grapher/equation-grapher_en.html), and I left the first value (ax^2)...
43. ### A Commutation and Non-Linear Operators

Suppose ##A## is a linear operator ##V\to V## and ##\mathbf{x} \in V##. We define a non-linear operator ##\langle A \rangle## as $$\langle A \rangle\mathbf{x} := <\mathbf{x}, A\mathbf{x}>\mathbf{x}$$ Can we say ## \langle A \rangle A = A\langle A \rangle ##? What about ## \langle A \rangle B =...
44. L

### Linear and Non-linear Equations (QM)

1. Problem Recall that we defined linear equations as those whose solutions can be superposed to find more solutions. Which of the following differential/integral equations are linear equations for the function u(x,t)? Below, a and b are constants, c is the speed of light, and f(x,t) is an...
45. ### A Equilibrium in system of non-linear difference equations

I have a little question. I want to know if there is a process in which I can find equilibrium solutions to some system of difference equations. For example, if I have something crazy like \begin{cases} x[n+1]=(x[n])^2y[n]+z[n]e^{-ax[n]} \\ y[n+1]= z[n]x[n]+x[n+1]y[n+1]\\ z[n+1]=...
46. ### Filtering an LED light's power supply

Hi, Out of interest, today I was mucking around with a 160W LED, 240V, 50hz, outdoor light. I looked at the Power Factor (it was 0.96) and the THD was 14.something %. I then wondered what would happen if I put a capacitor in parallel with the light. It was a three phase 90uF cap, so since they...
47. ### Understanding Drag Force against a non-linear Force

Homework Statement I'd like to understand Drag Force better; but school always ignores it. Thus, I'm asking this purely out of obssession. I'm picturing a scenario where a non-constant force is pushing an object horizontally, ignoring friction. But, I'd like to understand how Drag Force...
48. ### I Second order, non-linear, non-homogeneous differential eq.

I have a physics project at university, we designed an experiment to measure the effectiveness of Poiseuilles law in a 'quasi non-steady state'. Poiseuilles law, simply being the measurement of the flow rate of a fluid in a pipe, holding only under steady state though. So by quasi steady state I...
49. ### I Matrix Mechanics and non-linear least squares analogy?

I have some experience with non-linear least squares curve fitting. For instance, if I want to fit a Gaussian curve to a set of data, I would use a non-linear least squares technique. A "model" matrix is implemented and combined with the observed data. The solution is found by applying well...
50. ### 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...