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 am having a little trouble solving this equation:
\frac{d^2y}{dx^2} + \frac{A}{y}(\frac{dy}{dx})^2 + \frac{B}{(y+C)^2} = D - Ex
where A, B, C, D, and E are constants and, sadly, not related.
So far, I've found this
http://eqworld.ipmnet.ru/en/solutions/ode/ode0344.pdf...
Homework Statement
Find the fixed points and classify them using linear analysis. Then sketch the nullclines, the vector field, and a plausible phase portrait.
dx/dt = x(x-y), dy/dt = y(2x-y)
Homework Equations
The Attempt at a Solution
f1(x,y) = x(x-y)
x-nullcline: x(x-y) = 0 \Rightarrow...
Hello,
Does anyone know of a nonlinear 2nd order DE which must be solved numerically?
I've got a new idea about how tackle it analytically...
So I need one with a known solution to check my results.thanks!
Hello team PF! I have been out of touch from calculus for quite a while and have been trying to solve a differential equation which I believe is nonlinear and non-homogenous. Haven't found any thread much relevant here, so I need this new one. The problem is as follows:
-(d2 u)/(dx2 ) + γ*u...
hi,I want to solve this problem g=(2*k-1)*y^(k-1)-((1-y^(k-1))/(1-y));
with different k and compute real and positive y so I wrote this problem in MATLAB :
clear
clc
syms y k
g=(2*k-1)*y^(k-1)-((1-y^(k-1))/(1-y));
s=input('k=');
r=subs(g,k,s);
d=solve(r,y)
for...
Say one had three intense laser pulses and one weak reference pulse coming together on the surface of a thin nonlinear medium as if coming from the corners of a square. The three intense laser pulses mix, and there will be the third order mixing k1+k2-k3 or k1-k2+k3 or whatever coming out in the...
Homework Statement
m*dv/dt = - mg - kv^2, where m,g and k are constants and v(t) is what I should solve
Homework Equations
The Attempt at a Solution
At first I solved homogeneous equation m*dvh/dt = -kvh^2 and got vh = m/(kt - cm). Where c is also constant.
Then I treid to...
Homework Statement
A certain op amp has a maximum output voltage range of ±9 V. The maximum output current magnitude is 20 mA. The slew-rate limit is SR = 300 kV/s. The op amp is used in the amplifier circuit, shown in the diagram below. For a frequency of 10 kHz and RL = 2.2 kΩ, what peak...
Homework Statement
Find the orthogonal trajectories of the given families of curves.
x^2 + y^2+2Cy=1
Homework Equations
The book has covered homogeneous and separable methods.The Attempt at a Solution
To find the orthogonal trajectories, we simply find the curves whose tangents are...
Please teach me this:
I hear that linear sigma model has application in pion,nuclear physics.But do not understand what is the application of nonlinear sigma model in QFT?
Thank you very much in advance.
I was trying to solve a calculus problem (about the equation of the curve traced by a dog chasing a postman) and I came across the following equation. I would like to know how to solve it.
\frac{w}{v}(1-\frac{y(x)y''(x)}{{y'(x)}^{2}})=\sqrt{1+{y(x)}^{2}}-\sqrt{1+{C}^{2}}
Thanks.
I need to derive the solution for the differential equation analytically:
y'' + g(t,y(t)) = 0
y'(0) = z_o
y(0) = y_o
I know the solution is:
y(t) = y_o + z_ot - single integral from 0 to t of (t-s)g(s,y(s))ds
I believe I need to assume something about the solution being a function...
Hi there,
I've having problems solving a particular nonlinear ODE. Any help/suggestions will be highly appreciated.
The nonlinear ODE is:
v[t]*v'[t] + (4*v[t])/(t^2 - 1) = t/(t^2 - 1)
Thank you.
Please teach me this:
In QTF theory book of Schoeder say:
In nonlinear sigma model L=\intdx\frac{1}{2g^{2}}(\delta
_{\mu}n^{\rightarrow})(consider2<d<4 where d is the dimension number of spacetime).If we consider the Lagrangian as the Boltzman weight of a partition function,then the...
I'm doing my economics thesis on game theory, and need to solve a system of nonlinear equations (these will be the reaction functions). Unfortunately, I have no idea how to solve it explicitly--indeed, I am a bit worried its impossible with the parameters. It is rather symmetric though.
The...
Hi All,
I have been trying to solve following nonlinear differential equation, but I couldn't solve it.
0 = a*[f(t)]^{z/(z-1)} + (-t+C)*f(t) + b*[df(t)/dt]
where a, b and C are constants and 0< z<1.
Could you please help me how to solve this nonlinear differential equation? I would...
Please teach me this:
In linear and nonlinear sigma model, the spontaneous symmetry broken is inevitable happened or it may be happened and it may be not happened?Because the expectation value of field may be zero or may be not zero,then it seem that sometimes the hidden symmetry broken not...
Is there any good arguments or reasons for why it seems the general consensus is that the linear Schroedinger equations is fundamental?
I know that if it turns out either of these are nonlinear it would falsify certain readings of QM so it would be interesting if there is indeed no real good...
Hello everybody,
Background and problem description
I have derived an analytical expression for an implicit frequency response function. To verify it, I would like to check with a numerical solution. For very weak nonlinearities, congruence is obtained. For weak nonlinearities, the...
Hello All,
I'm working with a microcontroller and want to take analog measurements using its built-in ADC. However, I want to be able to measure voltages much higher than just 0-5V. My *seemingly* simple solution to this was to grab a 500k 25-turn potentiometer which I could connect up as a...
Hi,
what are some good textbooks dedicated to this subject? I have a short time to learn everything about this subject (upto a certain level). I'm trying to fit a non-analytic curve (whose form I know) to some data.
edit- sorry, I don't only mean non-linear, I mean non-analytic. The curve is...
Hi,
I have some equations I have derived though energy balances and have been using the Integration Factors method to solve equations in the form:
dy/dx + Py = Q
However, my original equations only work if I assume some variables to be constants. Removing this assumption leaves me with...
Hello,
I'm trying to (numerically) solve the equation y''*y=-0.5*y'^2 in Mathematica.
I know there's an analytic solution (and I know how to calculate it), but I want to modify this equation and thus need to verify that the numerical solution for the original equation is exact.
I'm using...
Hi everyone,
I am studying a 2-D nonlinear dynamics system, with two key parameters. But I have trouble when I want to locate where the homoclinic bifurcation occurs in the parameter space. Can anyone give me some ideas or reference readings? Thx
y^2=y' \Rightarrow y=\frac{y'}{y} \Rightarrow \int y dx = ln \left( y \right) \Rightarrow y=e^{\int y dx}=e^{\int e^{\int y dx} dx}=e^{\int e^{\int e^{\int y dx} dx} dx}=\cdots
Is that correct?
Homework Statement
2.1.5(A mechanical Analog)
a) Find a mechanical system approximately governed by dx/dt=sinx
b)Using your physical intuition explain why it becomes obvious that x*=0 is an unstable fixed point and x*=pi is a stable fixed point.
(note*this is exactly how it appears in...
While solving a problem involving equilibrium positions of charges on a line, I came up with a recurrence relation which is nonlinear, and moreover implicitly defined. Here it is: x_{0}=0 and \sum^{n-1}_{i=0} \frac{1}{(x_{n}-x_{i})^{2}} = 1. I should also mention that 0 \leq x_{n}< x_{n+1} for...
Hi all,
I have more or less convinced myself through trial and error that the following three-dimensional non-linear simultaneous equation cannot be solved. However, it would be great if someone could provide me with a proper mathematical reason as to why this is not solvable, rather than me...
I propose 27 scalar functions {fn : n = 1,2,…,27}, 9 delay-coupled 3D unit vectors {eij : i,j = 1,2,3} of general periodic nature, 7 ortho-normal bases {Zk : k = 0,1,2,…,6} each having three 3D unit vectors, 4 tensors {Y3,Y4,Y5,Y6} of order 2 and a tensor E of order 3. Z3-Z6 and E are defined...
Hello, everyone
There is a control theory problem that i would like to ask for help here.
Suppose a very simple nonlinear system as follows
x' = u
y = (2-cosx)*(2-cosx)
how to find a control law u that makes the output y can track a constant asymptotically,
of course the x should...
For a general dynamic system: dXi/dt = Fi(X1, X2,...,Xn), i=1,...,n,
Q.1
do you have some ideas of the existence conditions of following PDE:
a) (grad U, grad U + F) = 0 in n-dimension domain, (,) is inner product;
b) U >=0
Does it need a first type or second type of boundary...
Hello, I've started learning how to numerically integrate ODE's and I've run aground when i posed a 2nd order system for mathematica.
Here is my initial code:
NDSolve = [{x'[t] == -y[t] - x[t]^2, y'[t]==x[t], x[0]==1, y[0]==0.5},{x,y},{t,0,30}]
and i get an error message saying the...
Homework Statement
dv/dt = (k+v2)/h
Homework Equations
k is a constant, and v and h are variables where h is independent of v but v is dependent of h (v is a function of h and t).
The Attempt at a Solution
dv/(k+v^2) = dt/h.
The problem I have is with dealing with the right side...
I have been deriving a formula from physics. Anyway, I came across a equation that goes by the form y = exp(1/(xy)) *(b - 1/x) solve for y ; b is a constant there are other constants but i removed them for simplicity. or x = exp(1/x) solve for x , it is a function
Could some one at least...
Homework Statement
The following equation turned up while I was trying to make an integral
stationary in a 'calculus of variations' problem.
y^{\prime}(x)^2 + 1 = y^{\prime\prime}(x) y(x)
How would one go about solving this nonlinear equation?
So if I have a general reaction diffusion equation and perform a liner stability analysis on it I would only get the domains in which Turing Instability occurs but not which pattern I would get (stripes , spots etc.). So I need to use a nonlinear analysis. One technique people use is a...
Hey guys, I'm having a conceptual problem implementing the Crank-Nicolson scheme to a PDE with nonlinear boundary conditions.
The problem is the following:
u_t + u_{xxxx} = 0,
u(0,t) = 1,\quad u_x(0,t) = 0, \quad u_{xx}(1,t) = 0,
u_t(1,t) - u_{xxx}(1,t) = f\bigl(u(1,t)\bigr).
Taking m...
I am trying to solve the following system of 2n variables:
w1 + w2 + ... + wn = b0
w1x1 + w2x2 + ... + wnxn = b1
w1x12 + w2x22 + ... + wnxn2 = b2
...
w1x12n-1 + w2x22n-1 + ... + wnxn2n-1 = b2n-1
for w1, w2 ... wn and x1, x2 ... xn.
The problem is the using the Solve command returns...
\frac{d\mu }{dt}=-\left( kx\right) \left( \frac{\mu _{m}^{3}-\mu ^{2}\mu
_{m}}{\mu ^{2}+\mu _{m}^{2}-2\mu \mu _{m}+\mu ^{2}K_{s}}\right)
\frac{dx}{dt}=\mu x
Any method for me to solve the pair of nonlinear equations or numerical graph of the differential equation.
*\mu_{m} and K_{s} are...
Homework Statement
I'm given two equations
first
(d^2)*r/dt^2 - r((d*theta/dt)^2)= (-A)/r^2 --- this is a non linear second order differential equation
second
(r^2)*((d*theta)/dt)=B
B and A are...
How should I go about solving this problem? This is only to get a better understanding of how NLLS works.
F(x;a) = (1+a1*x)/(a2+a3*x) (so n = 3)
I am choosing a1,a2,a3 to be 2,3,5 respectively. I am also picking 6 data points (so m = 6):
(0, 0), (-1/4, 1/4), (-1/2, 1/10), (1/4, 1/4)...
First off, though I've been reading through these forums for a while now, this is my first post here, so let me briefly introduce myself.
I'm finishing up my third year as an undergraduate in mathematics. Next year, I want to apply to grad school in math, specifically, I'd like to study...
According to this article, researchers have obtained measurements with an accuracy larger than the upper bound set by HUP.
http://physicsworld.com/cws/article/news/45535
My first thought is that this is not a violation, since we are talking about a finite number of measurements, but then...
We did an experiment using the physical pendulum to measure gravitational acceleration g. A graph is shown in this link:
http://i593.photobucket.com/albums/tt20/omicgavp/measuringggraph.jpg"
A nonlinear (possibly chaotic) trend can be seen in our graph even though we assumed small-amplitude...
Nonlinear ODE, Howto attack...
Hi,
I've got a general nonlinear ODE equation that I have been solving in various situations, and I needed to make an approximate correction to -- but after the correction, an analytical solution to the new form evades me...
So I am studying it, but my math...
I am participating in a reading course on nonlinear optics, which is a little difficult since I haven't had any formal education in quantum mechanics other than the standard introductory solving of Schroedinger's eqn. in 1D. Happily this course takes the semiclassical approach, in which the...
Can anyone please suggest whether I can use MATLAB ode45 for the numerical solution of the following equations?
mx ̈+ c_x x ̇ + k_x x= F_x0+ μ(v_r ) (K 〖VB〗^2 y ̇/v) sgn(v_r )
my ̈+ c_y y ̇+ k_y y= F_y0+ (K 〖VB〗^2 (y/v) ̇ )
Where,
m, c_x, k_x, c_y, k_y, F_x0, F_y0, K, v are known...
Homework Statement
Let P2 be the vector space of real polynomials of degree less or equal than 2. Define the (nonlinear) function E : P2 to R as
E(p)=integral from 0 to 1 of ((2/pi)*cos((pi*x)/2)-p(x))^2 dx
where p=p(x) is a polynomial in P2. Find the point of minimun for E, i.e. find...