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.
Hello,
I have a system of trigonometric equations from which I should find theta1,..., theta5. Is it possible you can give me a hint on how to proceed. Thanks.
theta, phi, psi, Px, Py, Pz, l1, l2, l3, l4, l5, d1, d2, d3, d4, d5 are all constants.
Cos[t1+t2] Cos[t3+t4] Cos[t5]+Sin[t1+t2]...
I need some help with the problem that follows. Any help is highly appreciated.
Problem:
"Consider the initial value problem y^{\prime}=y^{1/3}, \mbox{ }y(0)=0 from Example 3 in the text.
(a) Is there a solution that passes through the point (1,1)? If so, find it.
(b) Is there a solution...
I need to solve the following second order nonlinear differential equation:
z''(b) * [6(1 - f)z(b) + (1+f)b z'(b)] = (15 - 9 f)[z'(b)]^2 + [2(1 - f) z(b) z'(b)] / b + [4 f z'(b)^(5/2)] / b^(1/2)
where f is a constant between [0,1].
initial conditions are z(0)=0 and z'(0)=0
I...
The problem is on competing species. For the problem, I am supposed to find the critical points. For each critical point, I need to find the eigenvalues and eigenvectors and classify the type of critical point and its stability.
dx/dt = x(1.5 - x - 0.5y)
dy/dt = y(2 - y -0.75x)
I got...
what is a non linear finite element finite element analysis. When does a structure behave non linearly and how do we calculate non linearity
of an element by the nodal displacements.
I recently came across Nonlinear Dynamics and Chaos by Strogatz and I'm recommending it to all my Physics/Applied Math friends. This is a great introductory book on the subject and you don't need any more Math than is taught in a basic Diff. Eq. course.
I love this book 'cause
a) Strogatz...
I'm posting this under analysis because of the method I'm thinking about using to solve it.
Here is the equation:
\varepsilon \left( x\right) =e^{\varepsilon \left( x-1\right) }
and the initial condition is
\varepsilon \left( 0\right) =1.
My main goal is to consider x as a number in...
1.
a line in my code is something like
int(phi*x^2,0,2)
where i want phi to be a function which the user
has choosen. But how can i let the user choose a function? For numbers you can use input but how to do
it with a function?
2.
i want to solve a nonlinear system of equations...
Relevant to the NLS is the differential equation,
\left( -\sum^N_{i=1} \frac{\partial^2}{\partial x^2_i} +c \sum_{i\neq j} \delta(x_i-x_j)\right)f_N = E_Nf_N(2.87)
How does one show that
\left(\prod_{i<j}(\theta(x_i - x_j) +...
I have recently borrowed a new book from the local library:
Calculus of Linear and Nonlinear Functions
by Francis J. Flanigan and Jerry L. Kazdan.
I have already finished first-year Calculus and I was wondering if this type of book would prepare me well for second-year Calculus in...
Is there anyone here studying chaos in graduate school? I have read a little on the subject and find it interesting. I cannot however, find many graduate schools offering studies in this field. (I know of only two, University of Maryland and Cornell) Also, what kind of real world applications...
Hello
I study animation, and I wanted to make a script where I can automate pendulums in my scene with some kind of realistic physics, rather than animate them manually.
So 'right' I think, 'I need to go study some physics and maths' - but it wasnt as easy as I hoped, I've hit a wall, and...
this is the problem: xy'' -x(y')^2 = y'
my book says that i need to substitute u=y' and du/dx=y''...
so i get:
x(du/dx)-xu^2 = u
so next the book says i need to separate the x's/dx' to one side and u's/du's to the other. however, i cannot do it
am i using the correct technique...
hi!
i want to know how to deal with the nonlinear coupled dfifferential equations? can anybody help me in that regard? if some one can paste MATLAB links/programs to solve the equations, it will be highly appreciated.
regards,
chap
problem: xy'' -x(y')^2 = y'
what i have so far:
u=y' and du/dx=y''
du/dx - u^2 = (1/x)u
int[(1/u)-u]du = int[1/x]dx
ln u - (1/2)u^2 = ln x +c
ok, now is what I've done so far correct? what do i do next?
ps: i'd like to say hi to everyon :) I am new here
Denying the unification of general relativity and quantum mechanics seem to be the simultaneous requirements of relativism and nonlinearity for the wavefunction. Wavefunction linearity and relativity are seamlessly incorporated under the physics of Fermi. Has there yet been a successful theory...
Here's a problem I ran into in complex analysis. Given z = x + iy and w = u + iv, I need to find all w such that w² = z. It reduces to solving this system:
x = u² - v²
y = 2uv
My professor mentioned that we should try to deal with the problem in at least two cases: y = 0, and y does not...
I am thinking of solving a nonlinear equation, namely, Gross Pitaevskii equation that applied in Bose Einstein Condensates. Can I solve the equation with Mathematica? Could someone give me some clues?
Hi, everybody.
I have an ODE:
y'''+2y''y-3y'^2=0
I know that it has an analytic solution, but I cannot get it (yet).
Can anybody help me?
(I don't need a full explanation how to solve it, just some hints or just the solution with 3 arbitrary constants).
Thanks beforehand,
Max.
Hi all,
Is there a way to have "hotspots" in a powerpoint/openoffice presentation document? I'm making a presentation and there are several areas where people might ask questions about a specific point and I would like to start a new line of slides, and then go back to where it branched off...
Hello there. I have just finished a biological experiment on "effect of trypsin concentration on rate of casein hydrolysis"
I have already obtained a graph, and I used a program called "Graphpad Prism" to analyse the data usin nonlinear regression (3rd degree polynomial). I have got all the...
Having some troubles getting further in this system I'm trying to solve. The notebook's at http://www.gabrielwyner.com/fluids5.nb
PDF form:
http://www.gabrielwyner.com/fluids5.pdf
I guess the most current problem involves mathematica complaining about DSolve[h'[t] = f[h[t],h[0]]] (with...