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 PDE is $$ \frac{1}{a^2 x^2} (u_y)^2 - (u_x)^2 =1$$ I know the solution, its ## u=x senh(ay) ##, but I dont know how I can get it. I've tried variable separation and method of characteristics but they dont seem to work.
I think I managed to solve the entire problem, as I show below. My main doubt is about item (e), the incremental circuit.
Part (a)
Using the node method and KCL we reach
$$\frac{v_I-v_A}{2}=10(1-e^{-v_A/5})\tag{1}$$
Part (b)
We can simplify (1) to
$$v_A=5\ln{\left ( \frac{20}{v_A+20-v_I}...
I am quite stumped here.
This is what I did so far.
1) The first thing I did was to forget about the terminals to the left and the diode, and to write the circuit as a Thevenin equivalent circuit. Then I put in the diode. However, the terminals we are interested in are lost when we do this...
Let ##D## be a smooth, bounded domain in ##\mathbb{R}^n## and ##f : D \to (0, \infty)## a continuous function. Prove that there exists no ##C^2##-solution ##u## of the nonlinear elliptic problem ##\Delta u^2 = f## in ##D##, ##u = 0## on ##\partial D##.
Hello,
Regression analysis is about finding/estimating the coefficients for a particular function ##f## that would best fit the data. The function ##f## could be a straight line, an exponential, a power law, etc. The goal remains the same: finding the coefficients.
If the data does not show a...
I was trying to fit a set of data to the nonlinear equation
$$
y=\frac{kx}{5+cx}
$$
and find the parameters k,c that will result in a best fit, but (I was told without explanation) that the parameters change as we increase x, so regular fitting techniques such as nonlinear least square will not...
Hi all,
I want to know if a second solution exists for the following math equation:
Ce^{At} ρ_p+(CA)^{−1} (e^{At}−I)B=0
Where C, ρ_p, A and B are constant matrices, 't' is scalar variable. I know that atleast one solution i.e. 〖t=θ〗_1 exists, but I want a method to determine if there is...
Hello,
Models can be linear and nonlinear and I just learned that a "linear model" is not just a model where the dependent variable ##Y## is connected to independent variables ##X## raised to the 1st power... The model is called "linear" because the coefficients/parameters are not raised to...
hi, i am going through differential equations which are nonlinear and singular - like Lane-Emden equation etc.
my question is how to tackle singularity - while coding.
regards
hi, i am working on nonlinear differential equation- i know rules which decide the equation to be nonlinear - but i want an answer by which i can satisfy a lay man that why the word nonlinear is used.
it is easy to explain nonlinearity in case of simple equation i.e when output is not...
I have a machine I am designing that for all intensive descriptions, is a simple press designed to compress loose product into a puck like shape.
The press force comes from a roller bearing mounted to a piston shaft, the rod sliding through a rigid linear bearing and the piston on the end of...
Hello everyone!
I am analysing an 18 m per 1.2 m truss, simply supported, with 140x5 chords and 90x8 braces. I then loaded the superior nodes with 500 KN. The top nodes were also laterally constrained to prevent out-of-plane displacements.
After imputing the structure in Abaqus (FEA software), I...
Hi. I would like to use a CAS to solve systems of nonlinear equations symbolically. The JavaScript library Nerdamer can solve single nonlinear equations symbolically shown here, and I would like to use that function iteratively to solve systems. For example, if I have a system of three equations...
Given the hamiltonian:
\hat{H} = \hbar \omega_{0} \hat{a}^{+}\hat{a} + \chi (\hat{a}^{+}\hat{a})^2,
where ##\hat{a}^{+}##, ##\hat{a}## are creation and annihilation operators.
Find evolution of the state ##|\psi(t) \rangle##, knowing that initial state ##|\psi(0)\rangle = |\alpha\rangle##...
That's pretty much it. If there is a very basic strategy that I am forgetting from ODEs, please let me know, though I don't recall any strategies for nonlinear second order equations.
I've tried looking up "motion of a free falling object" with various specifications to try to get the solution...
What are the possible ways of solving the operating point of air gapped transformer with nonlinear B-H curve? For example let's consider 3C90 E34 sized core with 0.5 mm airgap. I know that the magnetomotive force over the ferrite part can be formulated as function of the reluctances...
Hello Everyone!
I read about the function of the ruby laser which made from ruby crystal and has three energy levels. There is radiationless transition to what so called metastable level and then the electrons there stimulated to transit to ground energy state. The question is: does this...
Hello. I am trying to fit some nonlinear equations.
I have some data ##\vec \lambda, \vec n, \vec \kappa ##. Now I would like to use them to fit two equations $$n=n(\lambda ; \alpha, \beta, \gamma)$$ $$\kappa=\kappa(\lambda ; \alpha, \beta, \gamma)$$ where ##\alpha, \beta, \gamma## are the...
This paper getting some press, with promises that NNs can crack Navier-Stokes solutions more efficiently than traditional numerical methods.
https://www.technologyreview.com/2020/10/30/1011435/ai-fourier-neural-network-cracks-navier-stokes-and-partial-differential-equations/...
Hi there can someone please help me with this differential equation, I'm having trouble solving it
\begin{cases}
y''(t)=-\frac{y(t)}{||y(t)||^3} \ , \forall t >0
\\
y(0)= \Big(\begin{matrix} 1\\0\end{matrix} \Big) \
\text{and}
\
y'(0)= \Big(\begin{matrix} 0\\1\end{matrix} \Big)\end{cases}
\\...
I am trying to solve a PDE (which I believe can be approximated as an ODE). I have tried to solve it using 4th Order Runge-Kutta in MATLAB, but have struggled with convergence, even at an extremely high number of steps (N=100,000,000). The PDE is:
\frac{\partial^2 E(z)}{\partial z^2} +...
hi guys
i saw this problem online about using the MATLAB ode45 to solve the nonlinear Hoock's law and its specifically stated that
the nonlinear hoock's law is given by
$$F = k\;u + \epsilon\;u^{3}$$ , but when expanding the potential function in a Taylor series where you obtain the force...
This equation, is non linear, non-separable, and weird. I would like to have a direction to start working on this.
I tried writing sin(2y) = 2sin(y)*cos(y).
See,
##xy' = x^3sin^2(y)-2sin(y)cos(y)##
Can't separate.
Writing in this way:
##(x^3sin^2y-sin2y)dx-xdy=0##
Also, I checked that it is...
Show that the nonlinear system
$\dot{X_1}=2\cos X_2, X_1(0)=a$
$\dot{X_2}=3\sin X_1, X_2(0)=b$
has a unique solution for the arbitrary constants $a$ and $b$.
how to solve this system? Thanks.
I am trying to derive the adjoint / tangent linear model matrix for this partial differential equation, but cannot follow the book's steps as I do not know the math. This technique will be used to solve another homework question. Rather than posting the homework question, I would like to...
We are given that ##i_D = 8\cdot 10^{-12} (e^{v_D/20m} - 1)##
Hence ##i_D' = e^{50 v_D}/2500000000## and ##i_D'' = e^{50 v_D}/50000000##
Then I have that ##\delta i_D \approx\frac{ e^{50 v_D}}{2500000000} \cdot \delta v_D = \delta v_D / 5## Cancelling ##\delta v_D## from boh sides and solving...
I have no problem getting the ##R_{TH}## since from the special element's POV, the resistors are in parallel, and that's the answer.
However, I don't really understand how to get ##V_{TH}##.
Ignoring the special element, it seems that I have the resistors in series this time. But I'm not too...
I don't really understand or see the correct way to approach this.
Letting the current in question be ##i_x## (as shown in Fig. 1), and the unknown (changing) resistance be $R_x$, I can write:
##-V_s + R_s i_s + i_x R_x = 0##, and ##R_p i_p = i_x R_x##. Hence we can also write ##-V_s + R_s i_s...
Homework Statement: first order non linear equation
Homework Equations: dT/dt=a-bT-Z[1/(1+vt)^2]-uT^4
a,b,z,v,u are constant
t0=0 , T=T0
Hi,
i need find an experession of T as function of t from this first order nonlinear equation:
dT/dt=a-bT-Z[1/(1+vt)^2]-uT^4
a,b,z,v,u are constant...
I am trying to understand attracting, Liapunov stable, asymptotically stable for given coupled system. I don't have any Liapunov function. Just two coupled systems such as
##\dot{x} = y##, ##\dot{y} = -4x##
or sometimes normal systems
##\dot{x} = -x##, ##\dot{y} = -5y##
How can I approach...
Continuing the summary, the author argues that if ##g## is nearly 1, i.e ##g(u)\approx 1+\epsilon(u)##, one obtains the solution:
##f(u)\approx 1-\epsilon(a)##.
The derivative in the summary, i.e the dots represent derivatives with respect to ##u##.
Then how to deduce the solution for ##f##?
If...
Unlike in electromagnetics, the nonlinearity of mechanical structures is not only due to the material property. It could be due to large deformation and contact as well. Even though I will be implementing the existing and popular methods, I am still scared and I feel it is not a job that can be...
Hello, I have been working for hours on this single problem, to no avail. I am supposed to find the Thevenin equivalent for Voltage and Resistance, which I found to be 1ohm for resistance, and for question 2, Vth is -3V, and for 3 and 4, Vth is 1V. Any help is appreciated, as I have no clue...
I developed a program to solve the nonlinear system below through the method of successive approximations and was only able to find one solution, namely ##x_1 = 0.93377## and ##x_2 =0.88417##, even though I tried many different starting points. I was wondering if there's a way to determine if...
Hi
I would like to find out please what it would mean to transform a vector based on some property that it has and if you do that to more than one vector would both operations be isomorphic in some respect.
Is there a set of vector transformations of this time that could be used to process non...
Hi! I have a simple set of nonlinear equations
1) 3x = 30
2) x+2y = 20
3) x + y*z = 15
Clearly the solution to this is (10,5,1) but I want to find a robust way to solve this type of problem [A]x=b (where [A] is a simple function of x) which doesn't involve numerically solving using Newtons...
Dear all,
I would like to perform numerical simulations of the heat transfer/temperature field in a static bath of superfluid helium. The heat conduction in superfluid helium can be modeled in two regimes depending on the heat flux. For low heat fluxes ##\dot{q}##, the temperature gradient...
I'm trying to solve the following nonlinear second order ODE where ##a## and ##b## are constants: $$\frac{d^2y}{dx^2}+\frac{1}{x}\frac{dy}{dx}-\frac{y}{ay+b}=0$$ It looks somewhat like the modified Bessel equation, except the third term on the left makes it nonlinear. I've been trying to...
Homework Statement
Hi, I'm trying to calculate the formula for the position vs. time of a rocket landing from an altitude of 100km. I'm neglecting a lot of forces for simplification but basically, I want to solve ##F_{net} = Drag - mg##.
Homework Equations
Drag Force: D = ## \frac {C_dAρv^2}...
Homework Statement
equations above are descriptive of a system with two configuration variables, q1 and q2. inputs are tau1 and tau2. d and c values are given.
the question is about conversion of above equations to a state-space equation where the state-variables are x1 = q1_dot, x2 = q1_2dot...
Hello to you all,
Can someone please explain non-linear evaporation to me. I have not studied physics and it was a very long time ago that I learned chemistry and maths at school. So please explain it to someone with no prior knowledge of physics.
A small glass tube is filled with the liquid...
I am looking for a comprehensive nonlinear transformer model, which should be available free, readily available for simulation in LTSpice and able to take account of various factors of nonlinear transformer including hysteresis characteristic, skin and proximity effect, leakage inductance...
I'm having trouble finding textbook material on nonlinear functions on vectors. Just as I could define a function ##f## such that:
$$f(x) = cos(x)$$
I'd like to write something like:
$$f(\vec{x}) = \begin{pmatrix}
f_1(x_1) \\
f_2(x_2) \\
... \\
f_n(x_n)
\end{pmatrix} $$
where ##f_i## is...
As is known, when calculating localized states in a crystal
it is roughly considered that the point charge is “immersed” on medium with
static dielectric constant ε.
However, a simple estimate, for example, shows that an impurity atom
with charge modulo equal to the electron charge creates at...
(a'[t]/a[t])^2 == K*(A + B*a[t]^-6)^1/2} is the equation to be solved for getting the solution of a(t) in terms of time(t). Any ideas on how to solve this problem? Use of Matlab or Mathematica is accepted.
Hi good day. I am trying to find the general Inverted Pendulum on a cart nonlinear state space equations with two degrees of freedom with x, x_dot, theta, theta_dot which represents displacement, velocity, pendulum angle from vertical, angular velocity. However from research, I am seeing...