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.
I am currently an undergrad studying physics and am doing research on PPLN and nonlinear optics. I have a basic understanding of the math involved with my research, but would like to know more on nonlinear optics and why these materials behave the way they do. I am currently reading...
Is there an approach to the following 2nd order nonlinear ODE?
xy'' + 2 y' = y^2 - k^2
I am interested in learning how to analyze for asymptotic behavior, proof of existence, etc.
Hi everyone!
I have perhaps a basic question, but I can't dealt with it.
I have a rectangular sample 50x150mm of let's say wood. The sample is compressed from the top over the width of 4mm. I know the shortening of the sample at 70 mm from the bottom (from experimental testing) and I know the...
Homework Statement
Solve the following equation.
Homework Equations
( 3x2y4 + 2xy ) dx + ( 2x3y3 - x2 ) dy = 0
The Attempt at a Solution
M = ( 3xy4 + 2xy )
N = ( 2x3y3 - x2 )
∂M/∂y = 12x2y3 + 2x
∂N/∂x = 6x2y3 - 2x
Then this equation looks like that the integrating factor is (xM-yN).
IF =...
Homework Statement
$$y''+6y^{2/3}=0$$
Homework Equations
Nothing comes to mind
The Attempt at a Solution
I don't really know where to start. Any tricks or tips are appreciated. This isn't a homework question, but I posted here since I didn't know where else to post.
Thanks for your time
One of my friends needs to numerically solve this two dimensional boundary value problem but has now idea where to begin. Could anybody help?
## [(K H )(f g_x-gf_x)]_x+[(K H )(f g_y-gf_y)]_y=0 #### K H G^2 (f^2+g^2)+\frac 1 2 [KH (f^2+g^2)_x]_x+\frac 1 2 [K H (f^2+g^2)_y]_y-K...
1. Show that T isn't a linear transformation and provide a suitable counterexample.
##T \begin{bmatrix}x\\y \end{bmatrix} = \begin{bmatrix}x - 1 \\ y + 1 \end{bmatrix}##
2. The attempt at a solution
##\text{let}\, \vec{v} = \begin{bmatrix}0\\0 \end{bmatrix}. \text{Then,}##
##T(\vec{v}) =...
Hi all,
I have a nonlinear equation of the form:
\frac{TP_x}{TP_R} = c_0 + c_1 U_R^n + c_2 \frac{T_R^2}{\sqrt{U_R}}
This equation describes the relationship between tidal parameters and river discharge (velocity) in tidal rivers derived from the 1-D St. Venant equations. TPx is some tidal...
Homework Statement
In this problem, you will calculate the perihelion shift of Mercury simply by dimensional analysis.
(a) The interactions in gravity have
##\mathcal{L}=M^{2}_{Pl}\Big(-\frac{1}{2}h_{\mu\nu}\Box...
What would be the best book for me if I want to learn nonlinear dynamics ? I have my basics clear in linear differential equations, linear system theory, integral transforms and random process if they suffice as prerequisites.
According to my textbook the nonlinear Schrödinger equation:
$$\frac{\partial A(z,T)}{\partial z} = -i \frac{\beta_2}{2} \frac{\partial^2A}{\partial T^2} + i \gamma |A|^2 A \ \ (1)$$
can be cast in the form
$$\frac{\partial U(z,\tau)}{\partial z} = -i \frac{sign \beta_2}{2} \frac{1}{L_D}...
This is actually a straightforward question, but I'm struggling to find answers because I don't know very much about lasers. I want to use a nonlinear crystal for frequency doubling in an infrared laser (1480 nm) so that the output is half @ 740 nm. I know that nonlinear crystals like KTP are...
I'm trying to find a closed form (an algebraic solution) for the following system:
x² - y² = 5
x + y = xy
It's a bit tricky but I manage to end up with the quartic equation:
x^4 - 2x^3 + 5x^2 -10x + 5 =0
And this is where I get stuck looking for a closed form root.
Any suggestion would be...
Here is the table of contents of Nonlinear Dynamics and Chaos (by Strogatz)
Overview
Flows on the Line
Bifurcations
Flows on the Circle
Linear Systems
Phase Plane
Limit Cycles
Bifurcations Revisited
Lorenz Equations
One-Dimensional Maps
Fractals
Strange Attractors
Last quarter, there was a...
I need to solve the well known momentum equation in 3D cylindrical coordinates:
ρ(∂v/∂t +(v.∇)v)=A
where A and the velocity v are both local vector variables.
I am actually looking for the stationary solution to the equation, i.e. no ∂/∂t term)
I have tried evolving the velocity and tried...
Homework Statement
The state space model of a nonlinear system is
x'_1(t) = 2x^2_2(t) - 50
x'_2(t) = -x_1(t) - 3x_2(t) + u(t)
Where x_1(t) and x_2(t) are the states, and u(t) is the input. The output of the system is x_2(t).
Find the state space model of this system linearized at the...
I am learning nonlinear optics and recently got my hand on Nonlinear Optics by Robert W Boyd.
Any other suggestions?
Also is there a solution manual available for the above textbook?
http://www.sciencedirect.com/science/book/9780123694706
When I point a 5 milliwatts red laser at a pile of barium borate crystals, all I get is red speckles of scattered light.
When I point a 100 milliwats blue laser at the same pile of barium borate crystals, I get blue speckles.
When I point a 2000 milliwats green laser at the pile of barium borate...
Hi there,
I'm having a little trouble understanding the "distinguishability" of frequencies in the nonlinear electric susceptibility tensor. As far as I understand, if we have a SHG process with two collinear beams of the same polarization and frequency ω, there is only one susceptibility...
Homework Statement
Evening all, I'm trying to solve the 2-D diffusion equation in a region bounded by y = m x + b, and y = -m x -b. The boundary condition makes it complicated to work with numerically, and I recall a trick that involves a coordinate transformation so that y = m x + b, and y =...
Evening all,
I'm trying to solve the 2-D diffusion equation in a region bounded by y = m x + b, and y = -m x -b. The boundary condition makes it complicated to work with numerically, and I recall a trick that involves a coordinate transformation so that y = m x + b, and y = -m x -b are mapped...
Recently I was a witness and a minor contributor to this thread, which more or less derailed, in spite of the efforts by @Samy_A. This is a pity and it angered me a bit, because the topic touches upon some interesting questions in elementary functional analysis. Here I would like to briefly...
5000
In each of Problems 21 through 24,determine the order of the given partial differential equation;also state whether the equation is linear or nonlinear. Partial derivatives are denoted by subscripts.
21. $u_{xx} + u_{yy} + u_{zz}= 0$
23. $u_{xxxx} + 2u_{xxyy} + u_{yyyy} = 0$
22. $u_{xx} +...
I have a nonlinear least squares problem with a set of parameters \bf{g}, where I need to minimize the function:
\chi^2 = \sum_i \left( y_i - M(t_i ; {\bf g}) \right)^2
The t_i are some independent parameters associated with the observations y_i and the model function has the form
M(t_i ...
Homework Statement
I've been asked to graphically verify that the system of equations F (that I've uploaded) has exactly 4 roots. And so I did, using the ContourPlot function in Mathematica and also calculated them using FindRoot. Now, I've to approximate the zeros of F using the fixed point...
Hi,
In quantum physics, solution of Shrodinger equation live in a Hilbert space which is a vector space. The state superposition is obtained by mixing solution of the équation which is LINEAR so a linear combination of solution is a solution.
Now i have a non-linear equation of a scalar field...
I have an equation of motion given by
$$f(z(t),t) = \frac{d^2z}{dt^2} + A\frac{dz}{dt} + B $$
where
$$f(z(t),t) = [(\frac{C}{z^2+C^2})^2-(\frac{D}{z^2+D^2})^4]^2(1+cos(wt))$$
and ##A,B,C,D,## and ##w## are constants
Is it possible to solve this for ##z(t)##? I have been solving it numerically...
I have read a book that demonstrate the origin of electrical susceptibility of high order in harmonic generation: (in Robert Boyd's book : "Nonlinear optics").
For example, he show clearly for the case of second harmonic generation, how \chi^{(2)} depends on matrix element of electric dipole...
Homework Statement
I have a particle moving with uniform velocity in a frame ##S##, with coordinates $$ x^\mu , \mu=0,1,2,3. $$
I need to show that the particle also has uniform velocity in a frame ## S' ##, given by
$$x'^\mu=\dfrac{A_\nu^\mu x^\nu + b^\mu}{c_\nu x^\nu + d}, $$
with ##...
Good evening I have these coupled equations and was wondering if there is any chance solving them analytically. If not, how would you approach it numerically? (shown in attachment)
Thank you very much
Hello. I was wondering if anyone here had come across an equation similar to this one:
\alpha(uu_x)_x= u_t
Any info regarding this equation or tips on how to solve this would be appreciated :)
I came across these solutions: http://eqworld.ipmnet.ru/en/solutions/npde/npde1201.pdf, but how do...
Hey! I'm currently solving the heat equation using finite differences. I have a conductivity k(u) that varies greatly with temperature. It even drops to zero at u=0.
I have discretized the equations the following way:
\frac{\partial}{\partial x}\left( k(u) \frac{\partial u}{\partial x}\right) =...
Hello,
I'm a student in mechanical engineering and right now, we're studying non Linear Calculation of Structures. We have a project which consists on evaluating one of the solutions to stop a tourism aeroplane CESSNA 172, like a cable to stop it when it lands. So we started our study with a...
For the linear differential equation, it says that a_n and b_n are constants or functions of x. This implies that a function of x = f(x) = constant. How can a number be a constant if it is a function? I think I'm misunderstanding something.
Hi guys! I am trying to fit a function whose x data depends nonlinearly on the parameter of the fit and I am having hard time doing that!
I will explain better: from my experiment I was able to measure my ydata e my x0 array and I know that my xdata are:
x=x0+a/(1+4x^2), with a being a...
Homework Statement
Consider a consumer with wealth ##w## who consumes two goods, which we shall call goods ##1## and ##2.## Let the amount of good ##\mathcal{l}## that the consumer consumes be ##x_{\mathcal{l}}## and the price of good ##\mathcal{l}## be ##p_{\mathcal{l}}##. Suppose that the...
Hi guys,
I was browsing in regards to differential equations, the non-linear de and came up with this site in facebook:
https://www.facebook.com/nonlinearDE
Are these people for real? Can just solve any DE like that, come up with a series? Not an expert in this area, so I do not know what if...
Hi,
I have a small two dimensional nonlinear objective that has a very well defined minimum and maximum. Here is the function:
$$f(x,y)=2(1-x)^{2}e^{-x^{2}-(y+2)^{2}}-9(\frac{x}{5}-x^{3}-y^{5})e^{-x^{2}-y^{2}}-\frac{1}{5}e^{-(x-1)^{2}-y^{2}}$$
Attached is it's plot and contour.
Notice this...
Hi. I'm a bit confused on determining whether a certain PDE is linear or non-linear.
For example, for the wave equation, we have: u_{xx} + u_{yy} = 0, where a subscript denotes a partial derivative.
So, my textbook says to write:
$L = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial...
Homework Statement
The problem and its solution are attached in the TheProblemAndSolution.jpeg file.
Homework Equations
V = RI
G = 1/R
The Attempt at a Solution
I tried to understand what the solution said, but I'm still very lost.
Here's what I'm stuck on.:
When the solution says "Since...
how do you solve this equation?
y´´ + k/(y^2) = 0 ? I got it from applying Newton's 2nd law of motion to an object falling from space to Earth only affected by gravitational force. Thank you!
I have this system of equation: A = \frac{\alpha + \beta + \gamma}{3} B = \sqrt[2]{\frac{\beta \gamma + \gamma \alpha + \alpha \beta}{3}} C = \sqrt[3]{\alpha \beta \gamma} And I want to solve this system for α, β and γ. In other words, I want to express α, β and γ in terms of A, B and C...
This isn't a precise homework question, but this seemed like the most reasonable place to post. If not, please feel free to move it.
I have a large set of data points that should fit to a known equation (the Drude-Smith model for conductivity)
The equation the data should fit to is: σ(ω) =...
Hi guys,
after hours of searching internet I couldn't find much real-life examples of second order nonlinear dynamic systems (only tons of tons of equation and system theory... got totally frustrated). They will serve as a base process for modeling controllers.
So far I found propeller pendulum...