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.
The 3 equations are:
3xy-2xz=-1,
-xy-xz=-1,
-2xy+3xz=2
I've never learned how to solve these nonlinear equations. Is there anyway to solve this?
I tried wolframalpha, but this was the result:
http://www.wolframalpha.com/input/?i=3xy-2xz%3D-1%2C+-xy-xz%3D-1%2C+-2xy%2B3xz%3D2
Preface: just want to start by saying that I'm 99% sure I'm having a stability issue here in the way I'm implementing the time step since if I set \Delta t \ge 1 then for any stopping time > 1, the algorithm works as it should. For time steps smaller than 1, as the time step gets smaller and...
[Wasn't sure if each problem needed a separate post. Please feel free to edit if needed.]
Also \~ and \^ are tilde and hat respectively.
1a. Homework Statement
Use perturbation theory to derive the 3rd order nonlinear susceptibility \chi^{(3)}(3w;w,w,w) (problem gives potential energy, etc...
Homework Statement
(y^2 + xy)dx - x^2dy = 0
The Attempt at a Solution
Put it into derivative form.
y^2 + xy - x^2 \frac{dy}{dx} = 0
\frac{dy}{dx} - \frac{y^2}{x^2} - \frac{xy}{x^2} = 0
\frac{dy}{dx} + \frac{-1}{x}y = \frac{1}{x^2}y^2
I recognized this as a Bernoulli equation...
Homework Statement
Verify whether or not the operator
L(u) = u_x + u_y + 1
is linear.
Homework Equations
An operator L is linear if for any functions u, v and any constants c, the property
L(c_1 u + c_2 v) = c_1 L(u) + c_2 L(v)
holds true.
The Attempt at a Solution
I feel...
Hi,
I'm trying to find a toy (i.e. analytic) example of a nonlinear system that has very different behavior for two different types of forcing:
1) \frac{\partial u(x,t)}{\partial t}+ N(u(x,t)) = F(x)
where u(x,t) is the dependent variable, N represents some nonlinear operator with only...
hi,
I have some confusion for performing nonlinear analysis in ANSYS with NLGEOM... I was following the tutorial, the APDL is given below
/prep7 ! start preprocessor
/title,NonLinear Analysis of Cantilever Beam
k,1,0,0,0 ! define keypoints
k,2,5,0,0...
Hi everyone,
I've got an optimisation/computing question. I have a system of nonlinear equalities and inequalities, which I've written below for reference. It's the conditions for a minimiser of a Karush-Kuhn-Tucker problem. Would anyone be kind enough to explain how I could use software to...
Take for example a system
\frac{dx_i}{dt}=(x_i,t,a,b,...) i-number of state equations.
What would be the maximum number of parameters permitted for this system of non-linear differential equations?
Is it finally determined by the solution space?Is there a criteria for number of...
Homework Statement
http://www.freeimagehosting.net/t/9369y.jpg
Homework Equations
The Attempt at a Solution
a) is as follows: http://www.freeimagehosting.net/t/4oqft.jpg
Then for b), I have the equilibria as (0,0,0) and (r-1,\frac{r-1}{r},\frac{(r-1)^2}{r})
To...
I need to solve the following ODE:
http://www.sosmath.com/CBB/latexrender/pictures/041ee1419e05bc0776451b294c1dcc0e.png
but i can't figure out a way to. Please help!
I need to solve 2 ODEs:
1. http://www.sosmath.com/CBB/latexrender/pictures/7b213e6c9e4d5fd9d92877694610ac22.png
2. http://www.sosmath.com/CBB/latexrender/pictures/528f96046147932945da54b7a47f97a9.pngbut i can't figure out a way to. Please help!
Hello, my question is that almost all textbooks say that a linear system will give the output to a weighted sum of impulses which equals the superposition of scaled responses to each of the shifted impulses. But if we apply the same input which is a weighted sum of impulses to a non linear time...
Strogatz's Nonlinear and Dynamics book states that
$$
\langle\sin^{2n}\rangle = \frac{1\cdot 3\cdot 5\cdots (2n-1)}{2\cdot 4\cdot 6\cdots 2n}
$$
for $n\geq 1$.
However, $\langle\sin^6\rangle = \frac{5}{16}\neq\frac{15}{48}$.
What is the deal here?
Hello everybody!
While solving some physical problem I got stuck with some system of integral equations.
The problem is formulated in .pdf file below.
I will be over-satisfied with the following
1) to know whether and why this system has/doesn't have a solution
2) how it could be...
Homework Statement
How do you derive the time-dependent velocity equation for motion along a curve, such as a skateboarder on a half pipe?
For the sake of abstraction, I ask myself the following:
A uniform sphere of mass m and radius r is set free from the top edge of a semicircle half pipe...
Show that all vector fields on the line are gradient systems.
This is exercise 7.2.4 in the book "Nonlinear Dynamics and Chaos" by Steven H.Strogatz
Thanks very much!
Homework Statement
We have the equation:
y'(x)^2+2 (x+1) \left(y'(x)+x\right)+2 y(x)+2 x=0
2. The attempt at a solution
None. I don't even know how to proceed with this problem, except for, of course, expansion.
I tried the factorization method, but no luck here. I have a feeling I...
Homework Statement
Show that the following nonlinear system has 18 solutions if:
0 ≤ α ≤ 2∏
0 ≤ β ≤ 2∏
0 ≤ γ ≤ 2∏
sin(α) + 2cos(β) + 3tan(γ) = 0
2sin(α) + 5cos(β) + 3tan(γ) = 0
-sin(α) -5cos(β) + 5tan(γ) = 0
using the substitutions x = sin(α) y = cos(β) z = tan(γ)
The Attempt at a...
Has anyone seen this form of a nonlinear equation with respect to X, but linear with respect to Y & Z? I provided a contour plot within the region for all 3 variables between -2 & 2. The plot is actually Z*conjugate(Z) so that the magnitude is above ZERO. If I am correct I may have seen this...
Homework Statement
given: dt=-\frac{1}{2}\sqrt{\frac{l}{g}}\frac{d\theta}{\sqrt{sin^2(\alpha/2)-sin^2(\theta/2)}}
make the change of variables sin(\theta/2)=sin(\alpha/2)sin(\phi)
to show that: dt=-\sqrt{\frac{l}{g}}\frac{d\phi}{\sqrt{1-k^2sin^2(\phi)}}
where k=sin(\alpha/2)
Homework...
The basal metabolic rate (in kcal/day) for large anteaters is given by:
y=f(x)= 19.7x0.753
where x is the anteater's weight in kilograms
a) find the basal metabolic rae for anteaters with the following weights
i. 5kg
ii. 25kg
My answer:
i= 66.19kg
ii= 222.39 kg
Hopefully I got a right...
I am having a problem finding the solution for this eq:
y''(x)+(2/x)y'(x)+(w^2)y(x)=0
I couldn't find examples in the textbook that goes on a similar line, and have been searching the internet as well, but no use. I am thinking of using substitution v=y' but not sure how to do that in the...
Homework Statement
i need to solve this diff equation.
y' = x2 + y2
y = 1 when x = 0
i can assume that the answer is a power series on the form Ʃanxn
andi only need the 4 first non zero terms of the power-series answer
Homework Equations
Ʃanxn
The Attempt at a Solution...
Not sure if this topic belongs here, but here goes.
Homework Statement
From the AP physics C 1995 test there is a problem that gives the potential energy curve U(x). With F=-\frac{dU}{dx} in one variable,
F(x)=-\frac{a}{b}+\frac{ba}{x^{2}}
Where a and b are constants. Now I need to get...
Homework Statement
Hello everyone.
I'm trying to solve a non linear 11x11 system. (for eliminate harmonics in a power inversor)
I used Excel's Solver but it didn't work. (Solver couldn't solve the system). Then I found fsolve (a scilab function) but again it didn't work
I will attach...
Homework Statement
\frac{dy}{dx}=\frac{2y - x + 7}{4x - 3y -18}
Homework Equations
The Attempt at a Solution
I tried using v = y/x and got nothing. Same goes for trying to find an integrating factor to make the equation exact. I am given a hint, Find h and k so that the...
Hello,
I am confused as to how to transform nonlinear ODEs to linear ones by change of variables. Usually its pretty straight forward and I can do it, but this particular problem has me stumped and I don't know where to begin.
Homework Equations
Thank you guys!
I'm having trouble understanding the exponential map for nonlinear vector fields.
If dσ/dt=X(σ)
for vector field X, then how does one interpret the solution:
σ(t)=exp[tX]σ(0) ?
If X is nonlinear, then X is not a matrix, so this expression wouldn't make sense.
If X is a...
I saw this post at stackexchange:
I ran across this post when trying to solve a homework problem. But I have no idea how he got that solution for that. When I use the Euler-Lagrange, I get this diff eq below.
Here is the simplest form I have managed to get it in...
Hi all,
I've got a nonlinear differential equation of the general form
y' + f(x)y + g(x) = h(x)(y^n)
to solve.
For g(x) = 0 this is your standard Bernoulli equation. I've been trying to think of a way to solve it but haven't managed so far.
Any ideas would be appreciated.
Many...
Homework Statement
Is this differential equation linear or nonlinear? Assume that y' means dy/dx.
Homework Equations
1. Homework Statement [/b]
Is this differential equation linear or nonlinear? Assume that y' means dy/dx.
Homework Equations
\sqrt{xy'+2x2}=5
The Attempt...
Homework Statement
I want to find the general solution for y(x) if dy/dx = x + y^2 with initial cond't y(1) = 2
Homework Equations
The Attempt at a Solution
I can't figure out how to make it linear. (Obviously I don't think it's seperable)
Any suggestions/solutions...
Hello!
I am taking a self study diff e course, and I have run into a problem with no one to ask for help.
Here is the problem:
d/dt [ h^3(t) + 3h(t)^2 + 3h(t) ] = q(t)
h(t) is output.
q(t) is input.
is this Nonlinear First Order Differential Equation.
But I could not expand to...
I was wondering what the common methods for solving such a system are:
2 m \ddot{x} - m l \ddot{θ} θ + k x = 0
m l^{2} \ddot{θ} - m l \ddot{x} θ + m g l θ = 0
Lately I have been reading about nonlinear systems and chaos. It's fascinating and I would like to know more about how I could prepare myself to possibly study this in grad school. I would be interested in looking at biological systems such as neural networks or even animal populations. I'm a...
Like linear algebra goes in depth about linear systems, what should I look for to learn about the extension of linear algebra to nonlinear systems? Is there a name of the field of study? If I go into a book store to buy books about it, what should I be looking for?
Abstract Algebra? Complex...
Hi,
My understanding that one of the postulates of quantum mechanics is that the vector describing the quantum mechanical state of a system evolves in a linear fashion. My question is how this can be reconciled with systems where the system evolves in a non-linear fashion for example systems...
I am not too familiar with differential equations but am familiar with basic calculus, I came across this equation trying to describe a particular function:
dy/dx =((sqrt((y-x)^2+y^2)-abs(y))/(y-x))*abs(y)/y
Anyway I tried to separate the variables unsuccessfully and using v(x)=y(x)/x with...
Hi
I like to fit in Mathematica using NonlinearModelFit. When I look at the fitted parameters, there is an entry called "P-value". Here is what it means: "The p-value is the probability of observing a t-statistic at least as far from 0 as the one obtained.". I'm not quite sure what this...
Hi,
Part of my research, I nondimensionalized an ODE to eventually arrive at this form:
sin(τλ) = q^((n+2)/(n+1)) + κq' + q''
where q' = dq/dτ
The problem is of course the nonlinear q^n. n is an integer greater than 0.
Is there a Laplace transform for this?
Or what solutions are there for...
hi all,
I've been trying to work this problem out,
\frac{dv}{dt}-A(B-v)^{1.6}=G
A, B and G are constants
and Matlab can't give me a solution either. I'm wondering if there is even a solution?
A problem from a heat transfer book with conduction and radiation led me to a differential equation like this:
T'(t) = a - b*T(t) - c*T(t)^4
Although my professor said that there wouldn't be an analytical solution for this one and to get the answer by an iterative method I got curious and...
Homework Statement
\ddot{y} = - \dot{y} - y -sin(y)
Homework Equations
The Attempt at a Solution
to reduce the order I need to find a solution y1. it seems to me the only obvious solution is y1 = 0 but i can't use this to do a reduction can i
I've been reading through my mechanics of materials textbook recently, notably in regard to the section on the deflection of beams. The well regarded Euler-Bernoulli beam theory relates the radius of curvature for the beam to the internal bending moment and flexural rigidity. However the theory...