Non-linear Definition and 340 Threads

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. R

    MATLAB Newton-Raphson Method for Non-linear System of 3 variables in Matlab

    I am trying to solve 3 non-linear system of 3 variables using the Newton-raphson method in matlab. Here are the three equations: \begin{equation} c[\alpha I+ k_f+k_d+k_ns+k_p(1-q)]-I \alpha =0 \end{equation} \begin{equation} s[\lambda_b c P_C +\lambda_r (1-q)]- \lambda_b c P_C =0 \end{equation}...
  2. J

    Solution of system of non-linear equations

    1. Is there a general condition for the existence and uniqueness of solution of a system of simultaneous non-linear equations similar to the determinant test for a system of linear equations. 2. What are the solution methods (theoretical and numerical) for solving a system of simultaneous...
  3. R

    MHB How to Solve Non-Linear Equations of 3 Variables Using Newton-Raphson Method?

    The three non-linear equations are given by \begin{equation} c[(6.7 * 10^8) + (1.2 * 10^8)s+(1-q)(2.6*10^8)]-0.00114532=0 \end{equation} \begin{equation} s[2.001 *c + 835(1-q)]-2.001*c =0 \end{equation} \begin{equation} q[2.73 + (5.98*10^{10})c]-(5.98 *10^{10})c =0 \end{equation} Using the...
  4. M

    How to Solve a System of Non-Linear ODEs in Physics?

    Hello everybody. Solving a problem in Physics I run into a system of equations that I do not know how to solve, I would appreciate some help. Here is the system: \ddot{x}+4\dot{x}^2=C_1e^{y} \dot{y}^2=C_2\ddot{x} The dependent variables are x,y. C_1 and C_2 are some constants. I try...
  5. MarkFL

    MHB David's Math: Solving a Non-Linear System in 2 Variables

    Here is the question: I have posted a link there so the OP can view my work.
  6. ETBunce

    MHB Non-linear function with two unknown constants and one variable

    Am am presented with the problem: h(t) = c - (d - 4t)^2 At time t = 0, a ball was thrown upward from an initial height of 6 feet. Until the ball hit the ground, its height, in feet, after t seconds was given by the function h above, in which c and d are positive constants. If the ball reached...
  7. J

    Condition for a vector field be non-linear

    If a vector field ##\vec{v}## is non-divergent, so the identity is satisfied: ##\vec{\nabla}\cdot\vec{v}=0##; if is non-rotational: ##\vec{\nabla}\times\vec{v}=\vec{0}##; but if is "non-linear" Which differential equation the vector ##\vec{v}## satisfies? EDIT: this isn't an arbritrary...
  8. J

    Non-linear acceleration in swim cap for data consistency(help)

    Dear Experts! I am currently try to measure the drag of an buoyant object. I did 10 replicates, in general terms 10 tests all with the same set-up and same release mechanisms. To start off with I will explain my project: Drag reduction in swim cap First of all I got one accelerometer...
  9. V

    Uncertanty in a non-linear regression with least squares method

    Homework Statement Ok, so I'm trying to fit a set of data (21000 points to be exact) to a sine function. Homework Equations Y = A*sin(ωt) The Attempt at a Solution I used NumPy to get the parameters A and ω with the least squares method. So far, so good. However, i appear to...
  10. D

    Ohm's Law & Non-Linear Circuits

    Ohm's law only applies to linear circuits. If most loads in life are non-linear, what use is ohm's law?
  11. C

    First order non-linear differential equation

    Homework Statement Hello, I was given an extension problem in a Dynamics lecture today and am struggling to solve it. It is a simple scenario: a particle of mass m is accelerating due to Galilean gravity, but is subject to a resistive force that is non-linear in the velocity of the particle...
  12. 5

    Second order non-linear differential equation involving log

    EDIT: my problem is solved, thank you to those who helped Homework Statement Solve: x y^{\prime \prime} = y^{\prime} \log (\frac{y^{\prime}}{x}) Note: This is the first part of an undergraduate applications course in differential equations. We were taught to solve second order...
  13. P

    A system of 2 non-linear equations with 2 unknowns

    Homework Statement i'm trying to find a solution for these two equations, p & q are variables and c is known constant (it's given randomly) : $$ \begin{align}...
  14. I

    How can I solve a non-linear functional problem numerically?

    Hello, I'm not really sure where does this question fit and what title should it bear, but here is my problem: \psi(x) \exp (a\psi(x)^2) = C f(x) given a positive definite f(x), find ψ(x) and the constant C, subject to the condition \int \psi(x)\, dx = 1 I want to solve this numerically...
  15. ajayguhan

    Understanding Normal Modes of Objects - Linear vs Non-Linear Particles

    What does one mean by normal mode of an object? Why is it 3n-5 for linear particles, 3n-6 for non linear particle where n is the number of particle.
  16. D

    MHB First Order Non-Linear Ordinary D.E.

    Hello people, I couldn't solve the given D.E by using exact d.e & substitution method :( Thanks in advance. (x*y*sqrt(x^2-y^2) + x)*y' = (y - x^2*(sqrt(x^2-y^2) ) gif file of d.e can be found in the attachments part.
  17. M

    MATLAB Polymath or Matlab non-linear algebraic solver

    Hello, I am having difficulty inputting a non linear algebraic equation into polymath to solve for reference the equation is x*(100-.5*x)^0.5/(15-x)/(20-.5 * x)^0.5 - 87.824 == 0 and I want to solve for x, but haven't gotten anything. Also I don't know how to program it into matlab...
  18. H

    [Wolfram Mathematica] Using Newton's method to solve non-linear system

    Hi. This is not actually not part of the homework; but it's something I'd like to do. I have to solve the following system using Newton-Raphson's method: $$\begin{matrix} \frac{X}{\mu }+Y=1 \\ X=\left( \lambda -\left( K-1 \right)X \right)Y \\ \end{matrix}$$ Surfing the...
  19. M

    Non-linear multivariable functions

    I wanted to know if there is any way of classifying the set of all non-linear multivariable functions. I wish to analyse something over all possible non linear functions with 4 variables. In fact these variables are binary variables. for example f(x,y,u,v)= x.y - u\oplusv
  20. B

    Non-linear convolution and power series

    Homework Statement Hi, suppose we have the summation \sum_{i=0}^{n-1} \sum_{j=0}^{n-1} a_j b_{i-j}^{j} x^i, where the subscripts are taken modulo n, and a_i^n = a_i, b_i^n = b_i for each i. Write the above power series as a product of two power series modulo x^n - x.Homework Equations I'm...
  21. L

    Coupled non-linear differential equations

    Homework Statement x'= E - sin x + K sin (y-x) y'= E + sin y + K sin (x-y) E and K >0 Find fixed points for this system of equations Homework Equations This system is the form of coupled oscillators described in Strogatz. θ1'= ω1 + K sin (θ2-θ1) θ2'= ω2 + K sin (θ1-θ2)...
  22. D

    Programs Best path to Non-Linear Dynamics/Chaos theory for non-science major?

    Hi, I think some background is necessary: I'm a Psychology undergraduate student. I also really love math. I've incorporated some pure math into my studies - next semester I'll have room for a course on "Foundational Mathematics" (I'm not sure if that's exactly what they call it outside my...
  23. J

    Non-linear second order from calculus of variation I can't solve

    Hi, I derived the equation: 1+(y')^2-y y''-2y\left(1+(y')^2\right)^{3/2}=0 Letting y'=p and y''=p\frac{dp}{dy}, I obtain: \frac{dp}{dy}=\frac{1+p^2-2y(1+p^2)^{3/2}}{yp} I believe it's tractable in p because Mathematica gives a relatively simple answer: p=\begin{cases}\frac{i...
  24. L

    Non-linear difference equation transformation

    Homework Statement The problem is tough to type out correctly. Pasting problem statement image http://postimg.org/image/a0r92a0wl/ http://postimg.org/image/a0r92a0wl/ The Attempt at a Solution I just need to know how to proceed with the problem. Not the answer. This is the scan...
  25. dexterdev

    How to find the input output relation of an unknown non-linear system?

    Hi friends, I have a system (with unknown properties) which takes an input vector of length 10 and outputs a vector with length 6. I select Inputs vector 'I' which is a 1000x10 matrix, : 1000 samples of 10 elements. And outputs vector 'O' is a 1000x6 matrix,: 1000 samples of 6...
  26. Z

    MHB What are the fixed points and stability of a non-linear system of ODEs?

    I need help with the following so please help me -- Consider the following non-linear system X’ = x² - ay Y’ = y² - y(a) Find the fixed points of this system. (depending on a, there may be different fixed points!) (b) Study stability of each fixed point via linearization. In the case the...
  27. B

    Scaling Invariant, Non-Linear PDE

    Homework Statement Consider the nonlinnear diffusion problem u_t - (u_x)^2 + uu_{xx} = 0, x \in \mathbb{R} , t >0 with the constraint and boundary conditions \int_{\mathbb{R}} u(x,t)=1, u(\pm \inf, t)=0 Investigate the existence of scaling invariant solutions for the equation...
  28. S

    Shooting method for non-linear equation

    shooting method for non-linear equation(urgent) Homework Statement for shooting method , in non-linear equation, we're find $$t_{k}=t_{k-1}-\frac{[y(b,t_{k-1})-β](t_{k-1}-t_{k-2})}{y(b,t_{k-1})-y(b,t_{k-2})}$$ but how can we find the $$y(b,t_{k})$$ ? I am suppose to use Euler method for...
  29. F

    How can I solve a first-order non-linear ODE?

    Hi! I'm having a lot of trouble solving the following ODE: dx/dt = A - B*sin(x) where A and B are constants. My ODE skills are a bit rusty, and I wasn't able to find anything on the Internet that could help me, so could someone please show me how to solve for x in terms of t? I've...
  30. J

    Solving Non-linear Problem for x between 0 and L

    Assume we have a straight piece of wire with two end points A and B and with length L where x_{A}=0 and x_{B}=L. The wire has non-ohmic resistance and hence the current is not proportional to the potential difference, i.e. \left(V_{A}-V_{B}\right). In fact the current is a function of the...
  31. anemone

    MHB Solving a Non-Linear System: Approaches and Techniques

    Hi members of the forum, I am given to solve the following non-linear system: Solve (1+4^{2x-y})(5^{1-2x+y})=1+2^{2x-y+1} and y^3+4x+\ln(y^2+2x)+1=0 I'm interested to know how you would approach this problem because I don't see a way to do so. Thanks!
  32. P

    Can we apply non-linear smoothing to a linear looking like data ?

    My doubt is that whether we can apply non-linear smoothing to a almost linear data ( without one or 2 discontinuity) I have attached the pic in which the red data is the smoothed one. Blue is the original one. I multiplied each point with an increasing like 1, 1.1, 1.2, 1.3, 1.4...so on...
  33. Ackbach

    MHB A Method for Proving Some Non-Linear Limits

    I received permission from my father to post this from his (unpublished) Calculus text. Note that this method will, I believe, work for proving existence of a limit for a nonlinear function at any point that is not a local extremum. My father thought it would be good to give you this proviso...
  34. E

    Is this PDE linear or non-linear?

    hello, guys Below is the equation I am concerned with: Is the above equation non-linear because of (delta P/delta x)^2 term assuming other variables are constant and don't change with pressure , P?
  35. C

    How to find error between non-linear plot and data points?

    Is there a formal way to measure the error between some arbitrary points and a non-linear curve in order to minimize it?
  36. S

    Thoughts on a non-linear second order problem

    h''(t)=-\frac{1}{h(t)^2}, h(0) = h_0, h'(0)=v_0 The first step is to, I think, reduce this to a fist-order problem: h'(t)h''(t)=-h'(t)\frac{1}{h(t)^2} --- Multiply both sides by h'(t) h'(t)^2=\frac{1}{h(t)}+c_1 --- Integrate both sides 1/h'(t) = \sqrt{\frac{h(t)}{c_1 h(t)+1}} ---...
  37. E

    Can Fourier Series Simplify Solving Nonlinear ODEs with Oscillatory Inputs?

    Hi all, I have a nonlinear ODE in the following form: a x'' + b |x'|x' + c x' + d x = y where x and y are functions of time and a,b,c and d are constants. As far as I can tell the only way to solve this is numerically, something I've managed to do successfully using a Rung-Kutta scheme...
  38. D

    Sigma Plot, Non-linear regression, fitting a line to a set of points

    I model arterial baroreflex data that I have collected in humans using the Kent equation which is: y=p1/(1+exp((x-p3)*p2))+p4; where Y=heart rate, X= estimated carotid sinus pressure, p1=range of Y, p2=slope coeff, p3=centerpoint on X, p4 = minimum Y. I use Sigma Plot to do a best fit line...
  39. G

    Maths behind non-linear dynamics, driven damped oscillator more specifically.

    I am investigating the mathematics behind driven damped oscillators, I will then simulate it in MATLAB and observe the unpredictable long term behavior of the system. In order to create non-linearity in a oscillating spring I can no longer use hookes law but a form of it by introducing a...
  40. G

    MATLAB Is fortran 90 or matlab better for simulating non-linear dynamics (Chaos)?

    I am going to be simulating damped driven oscillators for my project and I was wondering if someone could give me a definitive answer on the matter. I know MATLAB is more of a 'tool' than a language so I'm thinking the maths behind damped driven oscillators might be easier to implement into...
  41. W

    MHB Four Non-Linear Simultaneous Equations

    4 equations, 4 unknowns: \[\frac{u(r^2 - u^2)}{r^2 + u^2}=156~~~~~~~~~~(1)\] \[\frac{v(r^2 - v^2)}{r^2 + v^2} = 96~~~~~~~~~(2)\] \[\frac{w(r^2 - w^2)}{r^2 + w^2} = 63~~~~~~~~~~(3)\] \[\frac{315uvw + 24336vw + 9216uw + 3969uv}{2r} = 943488~~~~~~~~~(4)\] Who can solve that mess?
  42. D

    How to Model Droplet Vaporization Using Coupled Non-linear Equations?

    Hi,I'd really appreciate any help on this as I've spent many many hours trying to get my head around it. I am simulating the vaporization of a droplet in hot air and I have two equations to use: m_lc_l\frac{dT_l}{dt} = \widetilde{h}A_0(T_\infty - T_l) + \dot{m}_l(h_v-h_l) \dot{m_l} = -\frac{\pi...
  43. M

    Linear or Non-Linear Differential Equations

    (d4x)/(dt4) + (1/(1+t))*(d2)/(dt2) = x(t) Is this differential equation linear or non-linear? I don't understand the difference.
  44. M

    System of non-linear partial differential eqs from electrostatics

    I have an electrostatics problem (shown here: https://www.physicsforums.com/showthread.php?t=654877) which leads to the following system of differential equations: \frac{\partial E_z}{\partial z}=\frac{\rho}{\epsilon_0} (1) Z_i E_r \frac{\partial \rho}{\partial r}+(u_g+ Z_i E_z)...
  45. B

    Substitution to turn a non-linear least squares problem into a linear one

    Hi, I want to solve an overdetermined non-linear equation with the method of least squares. Assume it's f(x) = 1 + ax + a^2 + b, and I want to estimate a and b. This is non-linear, as I said, so the derivatives of the squared residuals involve a^3 terms and are difficult to solve. Now I thought...
  46. A

    Cubic non-linear inequality (HELP)

    I just can't figure this out one question from my review test. I don't know hot to express it graphically or algebraically.
  47. F

    How to transform non-linear frequency vibration to constant frequency?

    How to transform non-linear frequency vibration to constant frequency of vibration of 2 Hz? Such as transform the different frequency of waves to constant frequency and then maintain it..
  48. mnb96

    How the inner product changes under non-linear transformation

    Hi, if we suppose x and y are two elements of some vector space V (say ℝn), and if we consider a linear function f:V→V', we know that the inner product of the transformed vectors is given by: \left\langle f\mathbf{x} , f\mathbf{y} \right\rangle = \left\langle \mathbf{x} ...
  49. G

    Solving Non-Linear Differential Equation with Fourier Transforms

    Hiya. I have to solve this bad boy under the assumptions that f, f' and f'' tend to 0 as |x| tends to infinity: 1/2(f')^2 = f^3 + (c/2)f^2 + af + b where a,b,c are constants. My thoughts are use Fourier Transforms to use the assumptions given, but not sure how to do them on these terms...
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