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. N

    Attenuation numerical instability via numerical diss. nonlinear ODE

    Hi, I'm solving a problem numerically that takes the form Q_{ij} \ddot{y}_j +S_{ijk}\dot{y}_j\dot{y}_k +V_i=0, where (Q_{ij},S_{ijk},V_i) are all functions of the dependent variables y_i. The dependent variables are all functions of the variable t. The resolution of this spectral...
  2. K

    Calculating work done on nonlinear spring

    Can anyone help me solve this problem? It seemed straightforward at first, but I am not getting the correct answer of -12 Nm. Thank you! A nonlinear spring is modeled by a force law given by F(x) = -10x + 3x^2, where F is measured in Newtons and x in meters. How much work is done stretching the...
  3. G

    Statistical thermodynamics - mean energy of a nonlinear oscillator

    Homework Statement Consider a classical one-dimensional nonlinear oscillator whose energy is given by \epsilon=\frac{p^{2}}{2m}+ax^{4} where x,p, and m have their usual meanings; the paramater, a, is a constant a) If the oscillator is in equilibrium with a heat bath at temperature T...
  4. R

    MHB Nonlinear oscillator and simple harmonic motion

    The nonlinear oscillator $y'' + f(y)=0$ is equivalent to the Simple harmonic motion: $y'= -z $, $z'= f(y)$ the modified Symplectic Euler equation are $$y'=-z+\frac {1}{2} hf(y)$$ $$y'=f(y)+\frac {1}{2} hf_y z$$ and deduce that the coresponding approximate solution lie on the family of curves...
  5. R

    MHB Is the Nonlinear Oscillator Equivalent to a System of Equations?

    Show that the nonlinear oscillator $y" + f(y) =0$ is equivalent to the system $y'= -z $, $z'= f(y)$ and that the solutions of the system lie on the family of curves $2F(y)+ z^2 = constant $ where $F_y= f(y)$. verify that if $f(y)=y$ the curves are circle.=> nonlinear oscillator $y" + f(y)...
  6. D

    Nonlinear optics: second order polarization calculation

    This is a problem from Boyd Nonlinear Optics chptr 1 problem 2. Homework Statement Numerical estimate of nonlinear optical quantities. A laser beam of frequency ω carrying 1 W of power is focused to a spot size of 30μm diameter in a crystal having a refractive index of n =2 and a second order...
  7. E

    Solving Nonlinear ODE: magnetism with varying particle charge

    Hello. I have a set of ODE where 1) \frac{dv_x}{dt}=\frac{q(t)B}{m}v_y 2) \frac{dv_y}{dt}=\frac{q(t)B}{m}v_x 3) \frac{dv_z}{dt}=0 Following the strategy to solve a simple harmonic oscillator, I differentiate (1) to get 4) \frac{d^2v_x}{dt^2}=\frac{q(t)B}{m}\frac{dv_y}{dt}+q'(t)v_y...
  8. E

    Nonlinear optics: second harmonic generation

    Hello, I'm studying basic nonlinear optics and I would like to solve a couple doubts about (basic) photon interaction. Let a monocromatic (of frequency ω) electromagnetic field propagate through a nonlinear medium and let the third(and higher)-order terms in the relation between the...
  9. K

    MHB Solving Linear & Nonlinear System w/ Reparametrization Function

    I have a problem on which I am stuck and would like help on how to proceed. The problem and my work is fairly lengthy, so please bear with me. **Problem:** A model for transport of a solute (moles of salt) and solvent (volume of water) across a permeable membrane has the form...
  10. C

    Solving nonlinear equations for yaw/pitch/roll given rotor/quaternio

    I've been doing linear algebra for so long, that I've become quite a dunce at solving nonlinear systems of equations. tl;dr: Can anyone suggest a fruitful plan of attack for the following system of equations? \begin{align} \cos\phi_1 \cos\phi_2 \cos\phi_3 + \sigma \sin\phi_1...
  11. djh101

    Gouy Balance Nonlinear Slope (Magnetic Susceptibility Lab)

    I am currently analyzing my data for the physical chemistry magnetic susceptibility lab. All the data is giving nice, smooth graphs for force vs. current, but I am unable to linearize them. F = \frac{1}{2}\chi\mu_{0}AH^{2} I have plotted force vs. current and force vs. current squared...
  12. J

    Fundamental frequency of an object with nonlinear stiffness.

    Hi all. It's been a few years since I've posted here, but it's remained a great go-to resource for me. Any time I have dealt with mechanical vibrations, the fundamental frequency was based on a constant stiffness. However, I have never encountered the subject of finding the fundamental...
  13. N

    Nonlinear Least Squares Fitting

    Hello, So I was hoping to get some help implementing a nonlinear least squares fitting algorithm. Technically this is an extension of my previous thread, however the problem I am having now is correctly computing the algorithm So the problem definition is this: Given two sets of n 3D points...
  14. R

    NUMERICAL approach to NONLINEAR PDE

    Hi guys, I need to simulate wave propagation for a nonlinear dispersive wave PDE and since I can't find proper resources for handling nonlinear PDEs numerically, I would appreciate any help and clues. the PDE is in the form of utt-(au+bu2+cu3+duxx)xx=0 Romik Ps: BC: Clamped at both ends IC...
  15. B

    How to solve this nonlinear differential equation

    dy/dx=2x+y^2 By the way, methods of solving linear differential equation are useless, such as integrating factor and Bernoulli method.
  16. E

    Nonlinear dynamic modelling of A380 aircraft

    Hi there, I am currently a student at Heriot-Watt University and have been given a project of deriving a nonlinear equation of motion for an a380 wing engine. The engine is to be considered as a lumped mass attached to the cantilever beam and the main fuselage is considered having translational...
  17. P

    How to solve a system of nonlinear equations in python

    I have the following system of 3 nonlinear equations that I need to solve in python: 7 = -10zt + 4yzt - 5yt + 4tz^2 3 = 2yzt + 5yt 1 = - 10t + 2yt + 4zt Therefore I need to solve for y,z, and t. Attempt to solve the problem: def equations(p): y,z,t = p f1 = -10*z*t + 4*y*z*t - 5*y*t...
  18. F

    Nonlinear Optics Books: Recommendations for Robert Boyd Style

    Hi, I'm an electrical engineering student starting research in nonlinear optics, and I'd like some good books to do with nonlinear optics. I'm looking for book similar in style to Nonlinear Optics by Robert Boyd as I really quite like that book. Other books I've gone through include Optical...
  19. R

    Comp Sci Fortran programming for nonlinear ode

    Adapt the fortran programming using second order adams bashforth method to generate a numerical solution of the Lorenz system: dx/dt =-10x+10y dy/dy=28x-y-x*z dz/dt= x*y- (8/3)*z with initial condition x(0)=y(0)=0, z(0)=2 slightly perturbed. Plot x and z against t runs from 0 to 15, and also z...
  20. N

    Analytical solution of nonlinear ordinary differential equation

    Dear All, I have following first order nonlinear ordinary differential and i was wondering if you can suggest some method by which either i can get an exact solution or approaximate and converging perturbative solution. \frac{dx}{dt} = 2Wx + 2xy - 4x^{3}\frac{dy}{dt} = \gamma \, (x^{2} -...
  21. E

    Nonlinear Optics: Split Method & Results

    Hello I want to ask you about the split method used to solve the nonlinear schrodinger equation numerically I just want to know what are the results that I am expecting to get how many graphs ??
  22. D

    MHB Project topics in Nonlinear PDEs

    I have to work on a project for my nonlinear PDEs class. What are some topics that are suitable for a project in nonlinear PDEs? Thanks.
  23. H

    How to Solve Nonlinear System of DEs in Physics Homework?

    Homework Statement Solve the System of DEs: \sqrt{1+y'^{2}+z'^{2}}-\frac{y'^{2}}{\sqrt{1+y'^{2}+z'^{2}}}=C_{1} \sqrt{1+y'^{2}+z'^{2}}-\frac{z'^{2}}{\sqrt{1+y'^{2}+z'^{2}}}=C_{2} Homework Equations The two equations above are quite relevant. The Attempt at a Solution I...
  24. T

    Shooting Arrows - Nonlinear Spring Problem

    Hello awesome physics people! Someone asked me for help on their first year physics homework, and I couldn't really solve it. This kept bugging me, because I should know how this works by now :P Homework Statement See attachment for the full problem statement. Basically, a bow is strung with...
  25. C

    MHB First Order Nonlinear Ordinary Differential Equation

    Hello, can you guys help me please with this differential equation from Demidovitch book, is to find the solution transforming to polar coordinates :
  26. twoski

    Program to solve system of nonlinear equations

    Homework Statement Solve the aforementioned system of nonlinear equations using Newton's method. write a program to carry out the calculations (it must use gauss elimination). Use the values 0-3 for x_{1}^{(0)}, x_{2}^{(0)} (ie. 16 data sets total). The Attempt at a Solution...
  27. D

    Calculating the nonlinear speed of Erosion

    Hi there, I'm trying to find a calculation to work out the speed of erosion and required rotation of water to cause erosion on a selection of 10 rocks. This is theoretical rather than an actual conducted experiment. Each rock has a density of 100 to 1000, ie rock one is 100, rock 2 is...
  28. L

    Nonlinear coupled differential equation system - kinetics

    Dear All, Recently, I have measured a series of nonlinear vibrational spectra from which I would like to extract some useful information about kinetics of the exchange process occurring in the studied system. I need to fit my experimental data to kinetic model that is a solution of coupled...
  29. P

    Linear approximation of a nonlinear component.

    Hello, I am trying to find the effective resistance of the NLR in the attachment (to the first order). It is given that IL = gVL2 + I0. I understand that this is normally achieved via ∂g/∂V at V=V0, but when I do so I get that R should be 1/(2gV0), and not 1/2g as shown in the solution. Could...
  30. S

    First-order nonlinear ordinary differential equation

    hii, how to solve this differential equation: A*(dT(x)/dx)(1873.382+2.2111T(x))=90457.5-2.149*10^-10* (T(x))^4 where A is a constant Thank you
  31. F

    Physics nonlinear ODE example for numerical methods project?

    I am doing a little research project into numerical methods of solving ODEs where I do 1 half of learning about the basics of numerical methods and then look at a particular method (Linear multistep) and then the second half is looking at a particular example, applying what I've learned and...
  32. K

    Need resources for solving nonlinear matrix equations

    In my work I've encountered equations of the type: (Ax).*(Bx) + Cx = d Where A,B and C are non-unitary square matrices, x and d column vectors and .* denote component-wise multiplication. I have a few books which discuss nonlinear matrix equations, but not of this kind. Any suggestions?
  33. N

    Archived Nonlinear Dynamics and Chaos, Strogatz: 2.1.5

    Homework Statement a) Find a mechanical system that is approximately governed by \dot{x}=sin(x) b) Using your physical intuition, explain why it now becomes obvious that x*=0 is an unstable fixed point and x*=\pi is stable. Homework Equations \dot{x}=sin(x) (?) The Attempt at a Solution...
  34. N

    Where do I fit? Nonlinear Dynamics and Chaos, and High Energy Physics

    Hello all, My question is really simple. I really like working on problems that involve Nonlinear Dynamics and Chaos, and I also really enjoy fields of Theoretical Physics that probe the nature of reality (quantum mechanics, high energy & elementary particle physics, string theory, etc.) I...
  35. M

    Numerical method to use on a system of second order nonlinear ODE's

    I'm trying to create a java application that models the path of a double pendulum. To do so I have been attempting to use Lagrangian Mechanics to find the equation's of motion for the system. The problem is that I have never seen a set of equations like the one yielded by this method and need...
  36. I

    Nonlinear second order differential equation

    Hello! It is the first time that i write on this forum. I'm doing a PhD but i can't solve this equation: it's a non-linear second order differential equation. ay''+b|y|y'+cy+dx=0 Some ideas?
  37. Z

    Solving a pair of nonlinear coupled DEs

    Hi everyone, new member zeroseven here. First, I want to say that it's great to have a forum like this! Looking forward to participating in the discussion. Anyway, I need to solve a pair of differential equations for an initial value problem, but am not sure if an analytical solution exists. I...
  38. L

    Can an additive function be nonlinear without AC?

    A function is additive if f(x+y) = f(x) + f(y). Intuitively, you might think that an additive function on R is necessarily linear, specifically of the form f(x) = kx. But assuming the axiom of choice, that is wrong, and the proof is rather simple: you just take a Hamel basis of R as a vector...
  39. M

    Nonlinear programming problem, mathematical programming, Summation Function.

    I was wondering if you might have some insight into a problem, where we consider an optimization problem: max ∑ from j=1 to n of fj(xj) such that ∑ to n of xj <=B xj>=0, integers where B is a positive integer and fj is real to real I am trying to formulate a solution using dynamic...
  40. S

    Perturbation Techniques and Theory for Nonlinear Systems

    Homework Statement Given the equation \ddot{\theta}=\Omega^2\sin{\theta}\cos{\theta}-\frac{g}{R}\sin{\theta} Determine a first-order uniform expansion for small but finite theta. Homework Equations Other than the equation above, none so far as I am aware. The Attempt at a...
  41. M

    Nonlinear programming problem, mathematical programming

    Hello, I need to minimize {- f (x) | a <=x <= b} where f ( x) is a concave and twice differentiable function. In addition, a and b are positive constants such that a <b. Assume that -f (x) exists in the given interval [a, b] . Show that if the optimal solution is at x*= a , then delta f...
  42. A

    Stability of Nonlinear System: Can the Zero Solution be Nominally Stable?

    i need show that at the following system the zero solution is nominally stable, using some change of variable that transforme in a linear system \frac{dx}{dt}=-x + \beta (x^2+ y^2) \frac{dy}{dt}=-2y + \gamma x y i tried with the eigenvalues of the Jacobian matrix at (0,0), but one of...
  43. N

    Testing symmetry properties of nonlinear governing equations

    Hi, I'm a bit uncertain about the validity of my argument/approach to the following: I'm trying to prove that the solution to a partial differential equation \frac{\partial u(x,t)}{\partial t} + N[u(x,t)] = 0, where N is some nonlinear operator, CAN BE (not necessarily is) asymmetric...
  44. D

    MHB Can Matlab or Python find a solution to this complex nonlinear equation?

    I let Mathematica run for over an hour but it couldn't solve this equation. Can someone run this in Matlab or Python and see if they can get a solution? $\alpha = 2\arcsin\left(\sqrt{\frac{10014.6}{2*a}}\right)$ $\beta = 2\arcsin\left(\sqrt{\frac{10014.6 - 6339.06}{2*a}}\right)$ $$...
  45. P

    Numerical methods for nonlinear PDEs in large domains

    Hi all, first post :) I have a system of z-propagated nonlinear PDEs that I solve numerically via a pseudo-spectral method which incorporates adaptive step size control using a Runge-Kutta-Fehlberg technique. This approach is fine over short propagation lengths but computation times don't...
  46. S

    Direction of nonlinear polarization

    Homework Statement A linearly-polarized electromagnetic wave with a frequency w and with an intensity of 1MW/cm2 is propagating in x-direction in a nonlinear crystal with a refractive index n=1.5. Assume that a second-order nonlinear optical susceptibility tensor for second-harmonic...
  47. Z

    Nonlinear Operators on Finitely Generated Vector Spaces: Group Property Example

    Can someone give an example of a nonlinear operator on a finitely generated vector space(preferably ℝn)? I'd be particularly interested to see an example of such that has the group property as well.
  48. J

    A second order nonlinear ode in an electrostatics problem

    I encountered the following second order nonlinear ODE while solving a problem in electrostatics. The ODE is: \frac{d^{2}V}{dx^{2}} = CV^{-1/2} How can I solve this? Regards. Homework Statement Homework Equations The Attempt at a Solution
  49. H

    Nonlinear ODE - dy/dt=F(y(t),x(t))

    I'm having issues approaching this problem. I need to solve for Homework Statement Given the following equation, I need to find the max change in x(t) as y(t) changes, given bounds y_{max} and y_{min}. \frac{dy}{dt} + a \sqrt(y(t)) = b x(t) Homework Equations All ODE methods, MATLAB, or...
  50. schmiggy

    Nonlinear system phase portraits

    Homework Statement See attached image Homework Equations Classification of critical points chart (unless you remember it) The Attempt at a Solution See attached. Now, I'm not entirely sure what exactly I'm doing. With linear systems, the goal is to find the eigenvectors...
Back
Top