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.

View More On Wikipedia.org
Recent contents Top users
  1. 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...
  2. M

    I Analyzing a 2nd Order Non-Linear ODE with Variable Substitution

    Can someone check my work here? Both ##f=f(x)## and ##y=y(x)##. $$f'y'+\frac{fy''}{1+y'^2}=0\implies\\ \frac{y''}{y'(1+y'^2)}=-\frac{f'}{f}\\ \frac{y''}{y'(1+y'^2)}=-\ln(f)$$ Now let ##v=y'##, which implies $$...
  3. G

    System simplification through state space

    Hi guys! I study Electrical Engineering and, don't-know-why, I don't know anything about state space. Now I'm working on a project where I have a non-linear system. The first task in the project is to separate the linear and the non-linear part of this system, and then obtain the state space...
  4. lep11

    Determine if the non-linear set of equations has unique solution

    Homework Statement Determine if the following set of equations has unique solution of the form ##g(z)=(x,y)## in the n-hood of the origin. $$\begin{cases} xyz+\sin(xyz)=0 \\ x+y+z=0 \end{cases}$$ Homework Equations I assume I am supposed to use the implicit function theorem...
  5. M

    A Help solving non-linear ODE analytically

    Hi PF! Anyone have any ideas for a solution to this $$0 = F F''+\left.2F'\right.^2+ xF' + F$$ where primes denote derivatives with respect to ##x##. So far I have tried this $$0=\left( F F'\right)'+\left({xF}\right)'+\left.F'\right.^2$$ Which obviously failed. I also thought of this...
  6. H

    Non-Linear optics vs The 2nd Law of Thermodynamics

    Hi, rank newbie here, with my first post. This one is something I figure every first year student comes up with at some point, but I don't know enough keywords to Search for an answer. (I'm not a student except in the category "of life": this isn't assigned homework) I figger, using a bit of...
  7. Mohamed_Wael

    Ansys non-linear stress strain curve

    If I am using a new material in ansys, let's say I assigned it as solid linear elastic and I assigned density, young's modulus and poisson's ration and then I applied large deformation for this material so that it should be experience non-linearity ...I wonder how Ansys can give the results...
  8. R

    Non-linear conductivity of tap water

    Hi everbody My plots seem to show that conductivity between two nails immersed a few mm in tap water falls off at low voltages instead of being a horizontal flat line as would be the case for a linear ohmic medium. And the deeper the immersion, the less linear the conductivity. Is this normal...
  9. srinaath

    Dynamic model of non-linear system

    i am finding it difficult to get insight about modelling of non linear system (electrical). my queries are, consider the system is DC DC converter... 1) I read that to study the dynamics of the system we linearise the non-linear system...why do we want to linearise it?... 2)can some one please...
  10. M

    MHB How Do I Solve this 2nd Order Non-Linear ODE and Find Its Equilibrium Points?

    Good morning everyone. First let me thank you for your help in advance! I have to solve this 2nd order non-linear ODE, and I'm stuck at the beginning. We have to find the equilibrium points, linearize the system, draw the phase portraits and classify the eq points, and solve it numerically. I...
  11. T

    Question about Non-Linear LED Loading

    I often create LED arrays, and have encountered a common issue where, as the LED load is increased, the total current requirement is not a linear factor as each branch (LED strip) is added to the circuit. It is common to have many branches of LED strips on a 12VDC 60W power supply. Power...
  12. kamhogo

    Light bulb & Diode: linear or non-linear elements.

    Homework Statement These questions are related to an experiment where we had to record the current for varying values of potential difference (10 readings), then repeat the process with the applied potential difference was reversed a) Why is the resistance of a light bulb not constant? b) Does...
  13. A

    Other Books for waves in Non-linear and dispersive media?

    What are the best book out there to learn about wave dynamic in Non-linear and dispersive media?
  14. Charles Link

    Non-linear properties in linear systems

    Common sense would suggest that all observed properties in a linear system would likely be linear. The response of a pendulum to an impulse can be computed, and for multiple impulses, the individual solutions can be superimposed to obtain the resultant solution. Qualitatively however...
  15. R

    Suggestion for an inexpensive non-linear load for 230/115?

    What can you recommend as a non-linear load? Preferably one that can be bought or constructed inexpensively. Thank you!
  16. M

    Numerical solution of non-linear diffusion equation

    Hello. I have some questions regarding the equation: k\frac{\partial}{\partial x}\left( u\frac{\partial (u-r(x,t))}{\partial x} \right) = \kappa \frac{\partial u}{\partial t}. u is positive. r(x,t) is given as an input. I have implemented this non-linear diffusion equation using backward Euler...
  17. L

    Python Non-linear elliptic differential equation - python

    Hey guys, I'm going to be honest and say I'm so stuck on this assignment - I really need help! I've took on a third year computational physics course last year - turn your weaknesses into strengths someone told me. Well, I failed and I'm back doing it again this year! So, I just have to pass...
  18. C

    Non-linear non-constant coefficient second order ODE

    I would like to solve the steady-state one dimensional heat equation for a two piece material system. The thermal conductivity in each segment is a linear function of temperature, where ##\kappa_1=a_1T+b_1## for material 1 and ##\kappa_2=a_2T+b_2## for material 2. ##a_1, a_2, b_1, and \;b_2##...
  19. D

    Periodic motion -- Potential as a function of a non-linear Force(x)

    Homework Statement Please see the attached.I don't know how to do (ai). potential function is the potential energy defined by f = -dV/dx e is the total energy of the system where e = KE + PE = (dx/dt)^2 /2 + V Note:m=1 because the particle has a unit mass If you integrate f,you get V(the...
  20. D

    Proving Non-linear Wave Equation for Riemann Tensor

    Hello, I am working through Hughston and Tod "An introduction to General Relativity" and have gotten stuck on their exercise [7.7] which asks to prove the following non- linear wave equation for the Riemann tensor in an empty space: ∇e∇eRabcd = 2Raedf Rbecf − 2Raecf Rbedf − Rabef Rcdef I have...
  21. C

    System of Implicit Non-Linear First Order ODEs

    I have an extremely messy system of differential equations. Can anyone offer any ideas for a general solution? p(t) is a function of t, and A is a constant.
  22. C

    Non-linear extrapolation graph

    Hi, Please note: this is not a homework question! It is a real world problem I am trying to solve. I have some values in mA and tonnes which I need to extrapolate but they are not linear. I know it the mA curve drops off the higher the tonne values go. I have plotted the values in excel...
  23. M

    Exploring Non-Linear Operators: An Intro to Derivatives

    Hello every one . If the derivative is a linear operator ( linear map ) Then what is the example of non-linear operator Thanks .
  24. M

    Solving Non-Linear ODE: Tips and Guidance from PF Community

    Hi PF! Can any of you help me reduce this ODE to find a solution? $$y y''+2y'^2+xy'+\frac{1}{2}y = 0 \implies \\ (y y')'+y'^2+xy'+\frac{1}{2}y = 0 \implies\\ (yy')'+(xy)'+y'^2-\frac{1}{2}y=0$$ but here I am stopped. Am I even going the correct route? I know I can re-write this equation as...
  25. R

    Distribution between non-linear branches

    Hello My question could be about bars and litres/sec but I'll express it as volts and ampères, mathematically it's the same puzzle for me. How do you calculate the currents in circuits when there are devices whose resistances are not linear? In the following network there are 3 low value linear...
  26. evinda

    MHB Non-linear differential equation

    Hello! (Wave) I want to find the solution $\psi$ of the non-linear differential equation $y'=1+y^2$ that satisfies the condition $\psi(0)=0$. (Notice that the solution $\psi$ exists only for $- \frac{\pi}{2}< x < \frac{\pi}{2}$) We notice that: $(tan^{-1})'(x)=\frac{1}{1+x^2} (\star) \left(...
  27. M

    Variational solutions to non-linear ODE

    Hi PF! I have a system of nonlinear ODE's, wherein the only constant ##C## in the ODE takes on several values depending on the geometry; thus once a geometry is defined for the ODE, ##C## is uniquely determined. Let's say I want to guess a quadratic solution to the ODE, call it ##\phi(x)##...
  28. M

    How do I numerically solve a non-linear ODE in Mathematica?

    Hi PF! I am wondering if any of you have experience numerically solving second order ODE's? Basically, I'm trying to solve one and am trying to do it numerically in mathematica. Can anyone help? For those curious, the equation is ##y y'' + 2y'^2 +xy' = 0## where ##y## is a function of ##x##...
  29. M

    Solving Non-linear First Order ODEs with Variable Coefficients?

    Homework Statement $$y' y + \frac{y}{x} = 1 - 2x$$ Homework Equations nothing comes to mind The Attempt at a Solution i've guessed a quadratic but that didn't work. now I'm stuck. any ideas? also, this is not homework, but a problem I am working on. Thanks!
  30. A

    System of linear and non-linear equations

    Homework Statement find the vector with magnitude 5 and perpendicular to 3i-2j+4k and 4i-3j-kThe Attempt at a Solution lets name the vector component x for the i y for the j and z for the k i got three equation 1- x^2+y^2+z^2=25 ( magnitude) 2-3x-2y+4z=0 3-4x-3y-z=0 (2, 3 from dot product by...
  31. U

    Pendulum - Stability and fixed points

    Homework Statement (a): Show the lagrangian derivative in phase space (b)i: Show how the phase space evolves over time and how they converge (b)ii: Find the fixed points and stability and sketch phase diagram (c)i: Find fixed points and stability (c)ii: Show stable limit cycles exist for T>ga...
  32. Puma

    Non-linear inertia from rest seems common?

    It seems to me that there are many instances of 'initial stickiness' when you move an object. Pushing a book over a table for example might be initially hard due to adhesive forces to the table which have formed over time - a plastic coating forming some kind of seal with a varnish is the...
  33. PhiowPhi

    Net Lorentz force, of a non-linear wire.

    From the diagram above, there are two types of wires a straight wire and a non-straight wire(zig-zag shaped wire), that are both placed inside a magnetic field(B), and have current flowing in both of them as well. The straight wire will experience a force and will move towards the left as...
  34. topsquark

    MHB Approximate solution for non-linear ODE

    I've decided to finish off this stage of my GR problem by finding an interval over which the acceleration of the object is "roughly" constant. I don't need help with the Math per se, but I would like your opinion on the method I am proposing. The Math is sufficiently ugly that I'd like some...
  35. topsquark

    MHB Coupled non-linear system of ODEs

    If you really want to know where this comes from I am solving the GR equations for a rectilinearly isotropic metric. In other words, I can express the metric as d \tau ^2 = -T(x) dt^2 + X(x) dx^2 + dy^2 + dz^2. (It may be simpler to use a cylindrical coordinate system, but the equations come...
  36. P

    Controllability of non-linear systems via Lie Brackets

    In http://www.me.berkeley.edu/ME237/6_cont_obs.pdf , page 65, the controllability matrix is defined as: $$C=[g_1, g_m,\dots,[g_i,g_j],[ad_{g_i}^k,g_j],\dots,[f,g_i],\dots,[ad_f^k,g_i],\dots]$$ where the systems is in general given by $$\dot{x}=f(x)+\sum_i^m{g_i(x)\mu_i}$$ Lets say you have a...
  37. NATURE.M

    Nonlinear System Homework: Potential Energy & Stability of Fixed Points

    Homework Statement In this question we will consider a damped mass-on-a-spring system whose spring exhibits cubic deviations from Hooke’s law. We will consider a damped spring with a restoring force F such that F/m=−βx−αx^3 where β is the “Hookian” part and α is a new nonlinear term. Unlike...
  38. datafiend

    MHB Systems of Non-Linear equations

    Hi all. I'm really at a loss on how to find the "other" points of intersection for this system of equations: x^2-y=4 x^2+y^2=4 Obviously we have a parabola and circle. I have solved for x and got "2", and plugged that back into get "0". However the answer has 4...
  39. A

    How to do Linearization for Non-linear least squares?

    Hy I want to know how to make linearization for some function,...what should by in Non-linear least squares problems. In my book I have only this example how to do: http://i.imgur.com/MUFiHkr.pngSomeone could me help how to do, some receipt of method what I need to do? Non-linear least...
  40. Heros

    Uncovering the Mystery of Non-Linear Torque in Muscle-Powered Energy Conversion

    Hi, i didnt know where to put this question cos its kind of multitopic, here is why... I am working on a better way to take mechanical energy from muscles, i made a teste bench for pedalling in very different ways, including traditional(circular) pedalling, the thing is. I am measuring the power...
  41. Z

    Efficiency of non-linear optical device

    It is said that a non-linear optical device may convert light into its higher order components. So what is the highest efficiency of the conversion? Can a non-linear device convert all input light into its higher order frequency? Thanks.
  42. S

    Transformations for the non-linear sigma action

    For the non-linear sigma action, S_G=\frac{1}{4\pi\alpha^\prime}\int d^2\sigma\sqrt{-\gamma(\sigma)}\gamma^{\mu\nu}(\sigma)G_{ij}(X)\partial_\mu(\sigma) X^i\partial_\nu X^j(\sigma), Let us consider an infinitesimal target space transformation X^\mu\to X^{\prime\mu}(X)=X+\epsilon\xi^\mu(X). The...
  43. S

    CAM Design with Non-Linear Force Profile

    I need to design a CAM composite with a spring for an exoskeleton to aid elderly mobility. However, the problem I face right now is that the CAM I designed provides a linear force profile, i.e., the force varies proportionally with the extension of the leg. How can I delay this force applied so...
  44. L

    Differential Equations Help, non-linear first order with substitution

    (r^2) (dT/dr)+B*r*T=T^2, with initial condition dT/dr |r=0 =0 where B is a constant I've gotten it to this: dT/dr = -BT/r + T2 / r2 by dividing everything by r2, then I substitute using λ= T/r which gives: r * dλ/dr + lambda = -B * (λ) + λ^2 I don't know how to separate...
  45. S

    Renormalization of the non-linear [itex]\sigma[/itex] model

    I have some questions about this paper:http://users.phys.psu.edu/~radu/extra_strings/freedman_sigma_model.pdf In section 3, they renormalize the bosonic non-linear \sigma model at one loop level. The action is given by I_B[\phi]=\frac{1}{2}\int...
  46. D

    Unique solution to an arbitrary monotonic non-linear system

    Quick version: I have a vector field f:\mathbb{R}^n\oplus\mathbb{R}^m \to \mathbb{R}^n of two arguments x \in \mathbb{R}^n, y \in \mathbb{R}^m, which has the following properties: The jacobian matrix of f wrt to the first argument \frac{\partial f}{\partial x}: \mathbb{R}^n\oplus\mathbb{R}^m...
  47. S

    Convergence of perturbative solutions to a non-linear diff eq

    Hi. First off, sorry for the not so descriptive title. If one of you finds a better tilte I will edit it. We have the equation \begin{equation} \partial_{xx}\phi = -\phi + \phi^{3} + \epsilon \left(1- \phi^{2}\right) \end{equation} We will look for solutions satisfying...
  48. W

    Dissertation guidance -- Non-linear Diff. Eqs and Photosynthesis

    Hi everyone. I am doing research about what happens to light energy after it has been absorbed by a leaf but before it has been used in the photosysnthetic reactions. For that I need to use system of 3 nonlinear ordinary differential equations. My dissertation title is : A hybrid numerical...
  49. W

    Solving Non-linear System of 3 diff eqns using ode23s in matlab

    I am trying to solve 3 differentail equations(Lorenz equations) using ode solver: ode23s in Matlab. Here are the 3 lorenz equations: dc/dt= alpha*I*(1-c) + c*(- k_f - k_d - k_n * s - k_p*(1-q)) ds/dt = lambda_b * c* P_C *(1-s)- lambda_r *(1-q)*s dq/dt = (1-q)* k_p * c *(P_C /...
  50. W

    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 3 non-linear 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...
Back
Top