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 need to find a trapping region for the next nonlinear ODE system
$u'=-u+v*u^2$
$v'=b-v*u^2$
for $b>0$.
What theory i need to use or which code in Mathematica o Matlab could help me to find the optimal trapping region.
Perhaps the title says it all, but I should expand it more, I guess.
So I am trying to explore more about constrained optimization. I noticed that there are very little to no formal (with examples) discussions on algorithms on nonlinear constrained optimization in the internet. They would...
I'm trying to find a solution to a system in which a clamped free Euler-Bernoulli Beam system rests on top of a mass-spring-damper system. The MSD system has nonlinearities in both the spring and the damper and is of the form:
I have extended the nonlinear restoring force to its 3rd term and...
Hello
I have a system of differntial equations:
dx/ds = sin(p)
dy/ds=cos(p)
dp/ds = k
dk/ds = -1/EI(s)*(k*dEI/ds+f*sin(p))
x(0)=y(0)=p(0)=p(L)-pl = 0
These are nonlinear differential equations. I should use some sort of nonlinear finite difference. But I do struggle to setting up the finite...
I am trying to apply a force to the tube structure via the plate. I am struggling to get this to solve.
Current set up is as follows:
Frictionless contact between the two compenents
displacement constraing the plate in the x direction
force applied to the plate
Could anyone offer some...
Nonlinear sigma models are particular field theories in which the fields take values in some nontrivial manifold. In the simplest cases this is equivalent to saying that the fields appearing in the lagrangian are subject to a number of constraints. Since the lagrangian fields are not independent...
What textbooks would you recommend for self studying Nonlinear Dynamics? I am a undergraduate junior who will be doing research on nonlinearity of spiking neurons. I have taken courses on ODE, vector calculus, probability, statistics, and linear algebra.
i want to solve a nonlinear PDE with finite difference method ,but using just discretization like in linear PDE , it will lead to nowhere , what's the right way to use FDM to solve nonlinear PDE or could someone provide me with book's titles or articles that can help me solving a nonlinear pdf...
Homework Statement
Okay, here's the deal:I have been given a second order nonlinear differential equation, and I have also been given the general solution with constants A and B. I am supposed to find the constants A and B. The solution represents a fermion at rest, since the solution does not...
Homework Statement
Solve from the differential equation below numerically for the function \phi(x) for x \in [0,L]
\phi '' (x) + D(x) sin(\phi (x) ) + E sin(\phi (x) ) cos( \phi (x) ) = 0
with D(x) a polynomial.
Homework Equations
Matlab.
The Attempt at a Solution
I can rewrite it in a...
Hi everybody,
In Robert W. Boyd's book "Nonlinear Optics", the quantum treatment of the nonlinear optical susceptibility lead to the next expression, for the second order case:
\chi^{(2)}_{ijk}(\omega_{\sigma},\omega_q,\omega_p)=\frac{N}{\hbar^2} P_F\sum_{mn}...
Homework Statement
y'=(x^2 +xy-y)/((x^2(y)) -2x^2)[/B]Homework EquationsThe Attempt at a Solution
I know that really the only way to solve this one is to use an integrating factor, and make it into an exact equation. My DE teacher said that to make it into a exact equation you need to take...
Hello,
I'm a doctoral student in civil engineering. In my research I came across a differential equation for the net force acting on an object as it impacts a granular medium at low velocities.
z'' + a[ z' ]^2 + b[ z ] = c
Where a, b, and c are all constants
I believe that this equation will...
I am solving a nonlinear ODE in the form of Newton's Second Law. In the equation, there is a Heaviside Theta Function of the function which I am solving (##\theta (x(t)##). Since it is quite troublesome to have both the left side of the ODE and the imput of the ODE to contain function of unknown...
Homework Statement
Find whether this system is stable or unstable at the steady state (x1,x2)=(0,0)
dx1/dt = -x1+2sin(x1)+x2
dx2/dt=2sin(x2)
Homework Equations
The Attempt at a Solution
z1=x1-0
z2=x2-0
dz1/dt=-z1+z2+2z1
dz2/dt=2z2
Jacobian =
[ 1 1 ]
[ 0 2 ]
so the system is unstable.
This...
Dear forum colleagues,
I'm looking for universities which have a good research lab in the field of nonlinear optics, located in Canada.
Can you please give me some hints or contacts?
PS: If anyone need information about Brazil and Portugal, please don't hesitate and just ask! ;)
Luis
In my introductory ODE class we have focused mostly on linear differential equations. I know that nonlinear differential equations are much harder to solve, and I am wondering what exactly the "state of the art" methods are for dealing with them, or also what recent developments have been made...
Hi guys! I have a question on applying constraint on Linkage systems. Assumed that there is a two dimensional one-bar linkage, one end can only rotate and one end is free (Such as the figure above, please neglect the damper-spring system if you want).
This link can rotate only 180 degrees, not...
I have some experimental circuit element that exhibits both resistance and capacitance, and results in nonlinear I-V curves.
can i extract capacitance and resistance of this element just from the I-V curve? or do I need time-axis data?
a suggestion on what equation to use to fit this data? thanks.
So I'm reading the Example on page 161 of Differential Equations, Dynamical Systems and an Introduction to Chaos by Hirsh, Smale, and Devaney.
I'm not understanding everything.
So given the system
x' = x + y^2
y' = -y
we see this is non-linear. I get it that near the origin, y^2 tends to...
I'm trying to solve question 4.12 from Cross and Greenside "pattern formation and dynamics in nonequilibrium systems".
the question is about the equation
\partial_t u = r u - (\partial_x ^2 +1)^2 u - g_2 u - u^3
Part A: with the ansatz u=\sum_{n=0}^\infty a_n cos(nx) show that the...
Given a DE in the general form of either y'' = y^2 or y'' = (y-1)^2, is there a general method to solve these?
I separated the equations to get y''(y^-2) = 1 and then integrated, which left me with (-y^-1)dy = (t + c)dt, and then integrated once more.
Is this correct so far? I have essentially...
Homework Statement
This is not the exact problem that I want to solve but I will use this as a guidance tool:
##y'' - (y')^2 + y^3 = 0##
where y is the function of x
2. The attempt at a solution
I tried doing a substitution ##u(x) = y'(x)## which leads to
##u' - u^2 + y^3 = 0## where both u...
While I am studying the wave propagation in fluids, the amplitude modulation seems to be governed by the Nonlinear Schrodinger (NLS) equation. In some of the journal papers the nonlinearity parameter, N seems to be of high value (N≈O(104)) and so on. I understand that weak nonlinearity...
Homework Statement
By truncating the differential equation below at n=12, derive the form of the solution, obtaining expressions for all the ancoefficients in terms of the parameter \lambda .Homework Equations
The ODE is:
\frac{\mathrm{d^2}\phi }{\mathrm{d} x^2} = \frac{\phi^{3/2}}{x^{1/2}}...
Hi, everyone. I am having a hard time finding explicit values of non-linear susceptibility tensor values for any sort of crystals. Specifically, I'm looking for values of a BBO crystal, but I would like to know where to find others for my future research.
I should say that I am looking for the...
Homework Statement
I've determined the dispersion relation for a particular traveling wave and have found that it contains both a real and an imaginary part. So, I let k=\alpha+i\beta and solved for \alpha and \beta
I found that there are \pm signs in the solutions for both \alpha and \beta...
This is part of a personal project... I've recently become addicted to modeling various physical systems from scratch, such that I find explicit solutions of position as a function of time, and graph em.
But I've hit a glass ceiling trying to find an analytic solution to the 1-dimensional...
Is it possible to solve a nonlinear differential equation of the form below such that the dependent variable y can be expressed as a function of time t?
(second time derivative of y) = y + y squared + y cube
Hi, I need help solving this ode, when I try to solve it I end up with a big crazy answer and I believe it should be simpler.
(dy/dx)^2=((ay^4)/2)-(a+1)y^2+1
y(0)=0, y'(0)=1 and a is within [0,1]
Solve the 2nd order nonlinear differential equation, with initial conditions y(0)=0 and y'(0)=1
y''=2ay^3-(a+1)y with a within [0,1]
I am pretty much lost on how to go about solving this. It would be greatly appreciated if someone could point me in the right direction on this. Thanks!
Homework Statement
Solve the 2nd order nonlinear differential equation, with initial conditions y(0)=0 and y'(0)=1
y''=2ay^3-(a+1)y with a within [0,1]
It would be greatly appreciated if someone could point me in the right direction on this. Thanks!
Homework Equations
The Attempt at a...
Homework Statement
A nonlinear spring has a temperature dependent force law,
F = -\frac{K}{T}(L-L_o)^3
At a temperature T = T_o and length L = L_o the specific heat at a constant length is C_L = C_o. What is the specific heat at T = T_o when the spring is stretched to length 2L_o...
Hi everyone,
I need some help to solve this differential equation.
The question states "Use the perturbation or multiple scale method to find the third-order approximate solution for the following system:
diff(x(t), t, t)+w^2*x(t)*(1+epsilon*x(t)^2) = 0 "
Currently, I am still...
Hi everyone
I am trying to solve non linear heat conduction where thermal conductivity is function of temperature, I am solving it by finite difference method. this is my equation
∂2t/∂x2+∂2t/∂y2 *k(t)= -q (x,y)
i have solved the equation taking k(t)= a-b*t,and when i further solved the...
Hi everyone,
i am trying to solve and program a non linear differential equation in MATLAB where thermal conductivity depends on temparature.I am trying it to solve by explicit finite difference method.
the given equation is ∂2t/∂x2+∂2t/∂y2 *k(t)= -q (x,y)
i have solved the equation taking...
i just signed up here so i hope this is the right place.
i need to solve a set of 2 non-linear ordinary differencial equations.
i tryed using NDSolve but it doesn't really work so I am not sure what's wrong with my code.
here is my code (copy paste):
c = 0.1;
Subscript[sys,
B]...
Definition/Summary
Nonlinear optical processes that occur due to the presence of a second-order nonlinear susceptibility are termed 2nd order processes, or three-wave mixing processes. There are four second order processes, second harmonic generation, sum and difference frequency generation...
Definition/Summary
Light propagating through a vacuum will obey the principle of superposition, however this is not generally true for light propagating through gaseous or condensed media. As light propagates through transparent media, it induces a dipole moment on any atoms present in the...
Homework Statement
Find the stable/unstable manifold for the nonlinear system dx/dt=y^2-(x+1)^2; dy/dt=-x
Homework Equations
The Attempt at a Solution
I'm trying to solve the below nonlinear system using Matlab, but got the following warning message. Any idea...
Hey there,
I have modeled a propagating wave in a 1D dispersive media, in which square and cubic nonlinear terms are present.
u′′=au3+bu2+cu
the propagating pulse starts to steepen with time which is the effect of nonlinearity, but there is an effect which I can't understand.
so...
A new variational principle is presented in this paper: http://arxiv.org/ftp/arxiv/papers/1112/1112.2286.pdf
When trying to derive something like the equation of motion of a Duffing oscillator, I take the following approach:
Set up the functional as such:
$$...
Hello everyone
I really need to ask if someone could show me a paper for the maximum gain peak of SBS in HNLF or any parameters having to do with SBS for highly non linear fiber ?? I really tried to look at many papers but nothing concerning Stimulated Brillouin Scattering(SBS)
Thank you