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. K

    Nonlinear gain in a SHG crystal

    I'm not a native English speaker so I need some clarification about the term "nonlinear gain" concerning a laser experiment. In detail I'm working on a lab report where I'm supposed to estimate the nonlinear gain of a KTP crystal in a Nd:YAG LASER using the intracavity second harmonic...
  2. F

    Any idea for this nonlinear equation?

    Hi I have a nonlinear equation for diffusion of multiphase fluids in porous media, and it is like 1/2(Laplacian(P^2)+d(p)/dy=d(p)/dt I couldn't find any analytical or semianalytical solution for this equation, do you have any idea?
  3. D

    Solve Linear vs Nonlinear Homework Statement

    Homework Statement I've started differential equations and I'm trying to understand the how to figure out if an equation is linear or not. The relevant equation I don't really understand either.Homework Equations http://img138.imageshack.us/img138/4158/8ac6f972e84a7e33c291f42.png The Attempt...
  4. K

    Nonlinear ODE with Hint: Solving for x and y

    Hello everyone and thanks for looking at my thread, I had some trouble solving this ODE which was in a textbook by Henry J. Ricardo: x(e^y - y') = 2. This problem is from a section dealing with linear equations, but there is a hint beside the problem which reads, "Hint: Think of y as the...
  5. B

    Problem with first-order nonlinear ordinary differential equation

    i have problem to find the solution for : (3x3y+2xy+y3)+(x2+y2)dy/dx=0 i have tried the exact equation method : (3x3y+2xy+y3)dx+(x2+y2)dy=0 thus M(x,y)=(3x3y+2xy+y3) and N(x,y)= (x2+y2) then deltaM/deltay=3x3+2x+3y2 and deltaN/deltax=2x Since deltaM/deltay does not equal to...
  6. K

    How Do You Solve Nonlinear Oscillations Using Poincaré Mapping?

    Homework Statement http://img51.imageshack.us/img51/853/39983853.jpg 2. The attempt at a solution Q3.1 I get the general solution as x(t) = Ae^{3t}+Be^{-t} + cost - 2sint . Q3.2 Letting y=\dot{x} and using the general solution, we get...
  7. E

    How to Solve a First-Order Nonlinear PDE using the Method of Characteristics?

    i have to solve this equation : du/dx * du/dy = x*y u(x,y) = x for y =0 with putting this equation in the form : F(x,y,u,du/dx,du/dy) = 0 . it can be solved. But mine book does not explain how to do this, there are no examples. Can someone help me ? or any links of examples on the...
  8. V

    Finding Equilibrium Points of Nonlinear Systems

    Hi, So I keep making mistakes trying to find all of the equilibrium points of different simple nonlinear systems. These problems aren't difficult, it's just that I keep taking different approaches to finding the equilibrium points. Is there a methodological way to know that I have found...
  9. E

    Separable D.E., Nonlinear in terms of y(x) after integration

    So, this is where I am stuck: ln\left(y\right)+y^{2} = \sin{x}+c_{0} I am confrused... :blushing:
  10. T

    Phase plane analysis for nonlinear and linear systems near (6,2)

    i need help on this part, does anyone have any idead about maple lab? I should get the cure and trajectories in the red rectangular. But i try to fix the points and range, i still didn't get it http://img193.imageshack.us/img193/387/deqp.jpg trange1 := -3..3: window1 := x=1..3,y=-3..-1...
  11. S

    Nonlinear State-Space Modeling for UAV Dynamics

    Anyone have any good online sources for nonlinear state-space modeling? I need to place nonlinear UAV dynamics into state-space form for modelling and seriously running out of good reference material for information.
  12. W

    Nonlinear vs Chaotic: Is There a Difference?

    Are the two words interchangeable?
  13. A

    Nonlinear Lifting Line method to estimate wing lift distribution

    Hi all, I have implemented a nonlinear lifting line method to estimate lift/induced drag of a typical wing. This method utilizes 2D lift coefficient data, and uses a numerical technique to solve the nonlinear equations. The program I have written in MATLAB is working, though I would like to...
  14. L

    Second-order nonlinear ordinary differential equation

    Homework Statement Given the Second-order nonlinear ordinary differential equation x''(t)=1/(x(t)^2) Find x(t).Homework Equations I tried use Laplace transforms, and solving it using linear methods but that is not useful.The Attempt at a Solution I tried to find t(x) and got to...
  15. V

    Nonlinear Dynamical Systems: Book Suggestions

    Hi all, I'm in a position where I have to teach myself an entire course in Dynamical Systems (mostly nonlinear, though there is a linear component) I've got several books, including Strogatz (Nonlinear Dynamics and Chaos) and Bauer/Nohel's classic book (A Qualitative Theory of Ordinary...
  16. P

    Phase portrait of nonlinear system of differential equations

    Homework Statement Describe the phase portrait of the nonlinear system x' = x^2, y' = y^2 Also, find the equilibrium points and describe the behaviour of the associated linearized system. The Attempt at a Solution We have an equilibrium point at (0,0). The associated linearized...
  17. O

    A Nonlinear Second Order Differential Equation Problem

    A Nonlinear Second Order Differential Equation Problem: very frustrating please help! Hello, I am a first year engineering undergraduate student, and this is my question. Homework Statement A dust particle of negligible mass starts to fall, t=0, under the influence of gravitational force...
  18. S

    Solving Nonlinear Integral Equation with Newton Method

    Homework Statement If I have a non linear integral equation of the form: y(s)+\int^x_0{K(x,s,y(s)}ds=f(x) and i want to find a way to solve this numerically using the Newton method Homework Equations The Attempt at a Solution after discretizing, and using the quadrature...
  19. Y

    Difference between linear and nonlinear transformation

    right now, my concept for their difference is that linear transformations are 1 to 1, where as nonlinear transformations are not. However, P^n to P^(-1) is a linear transformation, but it's not 1 to 1. the textbook def of linear transformation is that it must be closed under addition and...
  20. T

    Nonlinear System of Equations Newton-Raphson and SOR Sucessive Over Relaxatiom

    Homework Statement 3X1 - COS(X2*X3) – 0.5 = 0 X1^2 - 81*(X2 + 0.1)^2 + SIN(X3) + 1.06 = 0 EXP(-X1*X2) + 20*X3 + (10pi – 3)/3 = 0 Homework Equations Newton Raphsom and Gauss Sceidal with relaxation term The Attempt at a Solution I've already solved the above system using Newton...
  21. M

    Simple 2nd order nonlinear ODE

    I'm trying to solve the equation y'' = x^2 * y. This looks like it should be simple, but I don't have mathematica and the only reference I've found calls it a special case of the Emden-Fowler equation and refers to a solution in a book I don't own. Does anyone know the solution to this...
  22. Pythagorean

    Nonlinear vacuum permittivity?

    I've been told recently that the vacuum permittivity, given a sufficiently strong electric field, is not a constant, as it can cause positron-electron pair to split out of the vacuum. 1) is this true? 2) if so, where do such pairs as the positron-electron come from in a vacuum? I did try to...
  23. U

    Unable to show proposed transformation is nonlinear

    Homework Statement Show whether or not the following transformation T is a linear transformation, given the description of T: T maps each point in R2 w/ polar coords. (r, θ) to R2 w/ polar coords. (r, 2θ). T maps zero-element to itself (T(0) = 0)Homework Equations I suppose (x, y) = (rcos(θ)...
  24. Z

    Can a distribution or delta function solve a NONlinear ODE or PDE

    the question is , can a delta function /distribution \delta (x-a) solve a NOnlinear problem of the form F(y,y',y'',x) the question is that in many cases you can NOT multiply a distribution by itself so you could not deal with Nonlinear terms such as (y)^{3} or yy'
  25. D

    Problem of nonlinear curve fitting

    helo, I am dealing with a problem of nonlinear curve fitting. I have estimted samples of a continuous Fourier transform function, and my goal is to reach from Fourier to laplace transform using the known equality: F(S=jw)=laplace{h(t)}. i want to represnet the laplace trnsform in the...
  26. D

    Discrepancy in the solution of a nonlinear dynamic system

    Hi, I am solving the following nonlinear dynamical system using Energy Balance Method (EBM*). My intention is to arrive at an approximate analytical expression for the frequency of oscillation and the excitation force. u''+u=A(1+2*u) with u(0)=u'(0)=0, where A is a constant (Physically it...
  27. W

    Looking for a nonlinear equation with the following property

    Hi, I am looking for a nonlinear equation capable of approximating a sigmoid curve that can be multiplied by another equation of the same type with different parameters and this product can be made linear in the parameters. I am an ecologist, so I hope I am using the right terms. I...
  28. J

    Nonlinear ODE by an infinite series expansion

    I have to solve the nonlinear DE y'=x²-y² by using an infinite series expansion y=\sum_{n=0}^{\infty} a_n x^n, but I've tried in vain. Maybe a change of variables would make it easier, but I don't know which one. Thanks
  29. A

    Systems of Nonlinear Differential Equations

    Hi, I trying to solve a system of Nonlinear Differential Equations. I'm using Runge-Kutta on the Differential equations and Newton Method for the system. I have some doubts in how to create the JAcobian to the differential equations. Could somebody help me, please? Thank you, Aline
  30. B

    Nonlinear ODE Help: Strategies to Solve a Challenging 1st Order PDE

    Haiya :P In the process of trying to find the solution of a 1st order PDE, I've reached a point where I have to solve the following ode: \frac{dy}{dx} = \frac{x^2y-4(y-x)^3}{xy^2+4(y-x)^3} and I am stuck here :( It's not separable, homogeneous, or exact and I...
  31. S

    A second order nonlinear ODE? Whoa

    Homework Statement I'm not sure if this is actually solvable, or a typo on my homework... but here's the problem in question: Solve the ODE: \frac{d^{2}y}{dt^{2}} + t^{2} \frac{dy}{dt} + y^{2} = 0, y(0)=0, y'(0)=0Attempt at solution I've been stumped on where to even start with this one, but...
  32. J

    Nonlinear diff. eq. involving Besselfunction of first kind

    Hi there, I hope my post is not against forum rules (not sure if this section is only intended for general questions or if it is ok to ask about specific problems .. it is definitely not Homework). Anyway, I was hoping that there were some guru's out there that can help me with the following...
  33. S

    Solving Nonlinear System: 3x2+2y2=35, 4x2-3y2=24

    Homework Statement solve the system of 3x^{2}+2y^{2}=35 and 4x^{2}-3y^{2}=24 Homework Equations The Attempt at a Solution I re arranged for y^2 and got 1\frac{1}{3}x^{2}-16=y^{2} I keep getting x to equal \pm 2.473 this is clearly wrong, the answers...
  34. M

    How do I solve a 2nd order nonlinear ODE with specific boundary conditions?

    Hi, I need some help, I must solve the following nonlinear differential equation, -k1*(c'') = -k2*(c^0.5) - u*(c') subject to the bc, u*(c - 0.5) = k1*(c') where k1, k2, and u are constants, thanks
  35. I

    Coupled nonlinear partial differential equations or simple matrices?

    Why is it impossible to find ALL of einstein's equations in one place? well I suppose its irrelevant, I'd just like to know what math I have to do to define the energy-momentum tensor for a particle if I know say... its energy and momentum, or is that illegal? I'm struggling to grasp general...
  36. P

    System of nonlinear algebraic equations

    Hello, I came across a somewhat special system of nonlinear algebraic equations which I think must have been the subject of consideration in some book or article. I failed however to find such a resource, so I hope you can help out and point me somewhere. The system consists of n equations and...
  37. L

    Nonlinear spring energy problem

    Homework Statement The stretch of a nonlinear spring by an amount x requires a force F given by: F=40x-6x^2 where F is in Newtons and x is in meters. What is the change in potential energy U when the spring is stretched 2m from its equilibrium position? Homework Equations U=.5kx^2...
  38. M

    How can I solve this nonlinear ODE homework on Apollo reentry?

    Homework Statement http://www.math.udel.edu/~moulton/Apollo%20EC.pdf This is the full problem that I am working on for my ODE class. Homework Equations I would figure acceleration equals the second derivative so a=d^2s/dt^2 and V=ds/dt like the hint says. The Attempt at a Solution I...
  39. Y

    How to Solve a Nonlinear ODE using Variable Changes

    Homework Statement Solve the nonlinear ODE du/dx=(u+x√(x^2+u^2 ))/(x-u√(x^2+u^2 )) by changing variables to x=rcos(theta), u=rsin(theta) and converting the equation to one for d(theta)/dr. The Attempt at a Solution Not sure if I'm going in the right direction. du/dx =...
  40. S

    Unable to find the nonlinear least squares

    Homework Statement We have the following x, y values x ||| y 1.0 -0.15 1.5 0.24 2.0 0.68 2.5 1.04 3.0 1.21 3.5 1.15 4.0 0.86 4.5 0.41 5.0 -0.08 How can you find the equation y(x) = ax^2 + bx + c by least squares? The Attempt at a Solution I know how to...
  41. F

    Solving a Nonlinear ODE for Parachute Area in Free Fall

    Homework Statement We recently discussed a problem in class involving free fall and parachutes.One of the examples was to solve for the area of a parachute in drag,gravity,air density,mass and the speed at which the object deployed the parachute and the speed it hit the ground out. I'm pretty...
  42. P

    2nd order nonlinear non-seperable equation

    I've recently been trying to solve the following equation: d2x/dt2 + (x2 - a) dx/dt + (x2 - b)x = 0 I've reduced it to a first order equation by a simple substitution of y = dx/dt to obtain: dy/dx = (a-x2) + [(b-x2)x]/y = 0 However I cannot figure out how to solve this equation. Is it...
  43. F

    Nonlinear differential equation issue

    Homework Statement This is just a nonlinear differential equation. All I have to do it solve it, though it is an initial value problem as well. 2*y*y' + y^{2} = t Initial value: y(0) = -1. The Attempt at a Solution This should be easy, but it doesn't seem easily...
  44. M

    A nonlinear system of algebraic equations

    how to solve a nonlinear system of algebraic equations such as : \sum^{m}_{i=1}x^{n}_{i}=k_{n} n=0,1,2... m<\infty \sum^{m}_{i=1}x^{n}_{i} is a power sum polynomial
  45. H

    First order nonlinear differential equation

    Hi, I've come across this equation in my research, and I am so far unable to solve it. The equation is: \frac{dV}{dt} + \frac{1}{RC}V - \frac{I}{C}e^{(-V/n)} = \frac{V_f}{RC} - \frac{I}{C} both terms on the right-hand-side are constants; throughout the equation R, C, I, n, and Vf are...
  46. R

    How Does Changing Boundary Conditions Affect Nonlinear Shooting Methods in BVPs?

    Given the boundary value problem (primes denote differentiation w.r.t x): \begin{array}{l} y'' = f(x,y,y') \\ y(a) = \alpha \\ y(b) = \beta \\ \end{array} the nonlinear shooting method may be implemented to solve the problem. A bisection algorithm may be used or, with a little...
  47. V

    Solving First-Order Differential Equation in Nonlinear Optics

    Hi. Could someone help me? In Boyd's book nonlinear optics equation 6.2.24 How to solve it? Basically, it is a 4 variables first order partial differential equation. How to solve it analytically? Thanks
  48. S

    Solving nonlinear equations using matrix

    [b]1. To solve for six unknown variables using six nonlinear equations using matrix form and to prooce it as singular. Tp1a_d,Tp2a_d,Tp1b_d,Tp2c_d, Tp3b_d,Tp3c_d are known variables Tp1,Tp2,Tp3, a,b,c are unknown variables. [b]2. Tp1a_d =Tp1+a*Tp1 Tp1b_d=Tp1+b*Tp1...
  49. M

    Newton for a 4x4 system of nonlinear eqns

    Hi All. Can somebody give me a sample MATLAB code for Newton-Raphson Method for Nonlinear 4x4 System of Equations. I'm trying to set a very ugly one up and haven't seen NR before the beginning of this week. I've figured the NR for a single eqn but this is driving me nuts.
  50. M

    Analyzing Nonlinear PDE Systems with Polar Coordinates

    Homework Statement Hi, i have the following system of equation. In the task is that system have periodic solution and have to be used polar coordinates. Homework Equations x'=1+y-x^2-y^2 y'=1-x-x^2-y^2 The Attempt at a Solution After transfer to polar system i tried to use the method...
Back
Top