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Questions tagged [nonlinear-system]

In mathematics, a nonlinear system of equations 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.

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Degenerate perturbation theory to nonlinear equation

I want to use perturbation theory to find the steady-state solution to the following nonlinear equation: $$ x_i\left(\sum_{j=1}^Nx_j^2\right)-a x_i + \epsilon \sum_{j\neq i}^N J_{ij}x_j=0, $$ where $i=...
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Solving the system $\frac{xy}{ay+bx}=c$, $\frac{xz}{az+cx}=b$, $\frac{yz}{bz+cy}=a$ for $x$, $y$, $z$ [closed]

I came across a question regarding sytems of linear equations. I have tried elimination,substituition and simon's factoring trick etc but still not able to extract x,y,z. $$ \begin{cases} \dfrac{xy}{...
Aryan Malik's user avatar
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Time difference between peaks of Lotka-Volterra equations [closed]

In the Lotka-Volterra equations: \begin{align} \frac{dx}{dt} & = \phantom{-}\alpha x - \beta xy\\ \frac{dy}{dt} & = - \gamma y + \delta xy \end{align} The peak of the predator population ...
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How does extra equations affect Newton-Raphson method's performance on solving system of non-linear equations? [closed]

I'm working with model updating, in which the model's parameters are adjusted in order to reduce its ouput error in relation to a reference. For this, I would like to compare minimizing a single ...
Marcus Vinícius Medeiros's user avatar
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2 answers
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How to linearize first order ODE

I want to linearize the first order ODE, given as: \begin{equation} f' + \frac{f^2}{a} + \frac{f}{2a} - \frac{3}{2a} =0\,, \end{equation} The solution should be $f(a)=1$. But when I just drop the $f^...
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Is there a study of Quadratic Transformations?

In Linear algebras I remember learning about linear transformations, with examples being integrals and derivatives. The basic definition I remember is that $f(x)$ is a linear transformation if $f(a +b)...
Michael Diamond's user avatar
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Which nonlinear PDEs can be converted into linear PDEs?

In Section 4.4 of Partial Differential Equations by Evans, the author describes several techniques for converting certain nonlinear equations into linear equations. First, the author introduces the ...
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Help solving a simple nonlinear system of equations [closed]

I need some help making sure Im solving a nonlinear system of equations correctly. The system in question is the following: $ \begin{cases} x(y+1)(z+k) &= 0\\ \alpha(y+1)(z+k) &= 0\\ \alpha x (...
st30's user avatar
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Solving $\frac{M e^{-M}}{1-e^{-1}} - \epsilon = 0$ for $M \in \mathbb{R}$

I am trying to solve the following nonlinear equation analytically: $$ \frac{M e^{-M}}{1-e^{-1}} - \epsilon = 0 \, , $$ where $ M \in \mathbb{R} $ and $ 0 < \epsilon \ll 1 $. A solution can be ...
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The identity of the variational equation (ODE)

Consider the nonautonomous system: $\dot{x}=f(t,x)$, $s(t, t_0,x_0)$ is the solution trajectory starting from $s(t_0,t_0,x_0)=x_0$. I wonder why $\frac{\partial}{\partial t} \Vert s(\tau, t, x)\Vert ^...
Rui Tachibana's user avatar
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How to express the originals system of equation in terms of its Groebner bases?

As a network engineer I need to explain some mathematical stuff to my fellow coleagues. Particularly, I need to explain the fact the the Groebner Basis will create an equivalent system. One particular ...
Tuong Nguyen Minh's user avatar
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How to Handle a Singularity: Seeking Solutions for a Differential Equation System

I am trying to solve the following differential equation system: $\begin{aligned} & r^2 K^{\prime \prime}=K(K-1)(K-2)+\frac{1}{4} h^2 K \\ & r^2 h^{\prime \prime}=\frac{1}{2} h K^2+\left(\...
Hendriksdf5's user avatar
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How do nonlinear relationships affect casuality determination

Let's assume that I have only one independent variable and one dependent, and I have a great model with minimal error which deals well with predicting. Let's also assume that I do no know the true ...
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Solving underdetermined nonlinear equations

Let $u,v,b \in \mathbb{R}^3$ $k_d$ is a scalar constant. Let $u^2$ denote element wise square of $u$ $H$ is a $3\times 3$ non-invertible matrix. $G$ is a $3 \times 3$ invertible matrix. I have an ...
V Adarsh's user avatar
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Solving the system $x^4+y^4+z^4=a$, $xy+xz+yz=b$, $xyz=c$

I am trying to solve the following system of equations: $$ \begin{cases} x^4+y^4+z^4=a\\[4pt] xy+xz+yz=b\\[4pt] xyz=c\end{cases} $$ where $a$, $b$ and $c$ are constants and $x$, $y$ and $z$ are the ...
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Real part of complex solution to PDE still solution?

EDIT: I updated my question in order to make it more clear. Here is my concrete problem: I am trying to solve some PDE of the form: $$\partial_t u = L(x,u) \quad , \qquad u \equiv u(t,x) \tag{1}$$ ...
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Solving a System of Equations Involving Complex Variables and Their Magnitudes

I’m trying solving a system of equations with two complex variables, x and y . The equations are given by: $$ x + y = c_1 \\ |x| + |y| = r $$ where $c_1$ is a given complex number and r is a ...
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Question about the proof that uniform asymptotic stability can be characterized by KL function. (Lemma 4.5 in Nonlinear Systems (3rd) by Khalil)

Lemma 4.5 in Nonlinear Systems (3rd): Consider the nonautonomous system \begin{equation} \dot{x} = f(t,x) , \end{equation} where $f : [0,\infty) \times D \to \mathbb{R}^n$ is piecewise ...
Lau's user avatar
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Liapunov’s Second Method proof for $V(x) = ax^2 + bxy + cy^2$

For this theorem, Theorem 9.6.4: The function $V(x,y)=ax^2+bxy+cy^2$ is positive definite if, and only if, $a>0 \text{ and } 4ac-b^2>0$. I'm trying to prove why $a > 0$ and $4ac - b^2 > ...
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Finding where a function is positive definite

For this problem, Is the following function sign definite, sign semidefinite, or sign indefinite? $$V(x,y)=1-cos(x)+y^4$$ Solution: V is positive definite on $D=${$(x,y)\in \mathbb{R}^2:x^2+y^2 < ...
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A function of specific form passing through two given points

Let $$s(t; a_0)=a_{0}t^{2}\left(\frac{1}{2}-\frac{t}{3T(a_0)}\right)$$ with $T(a_0)=\sqrt{\frac{6d}{a_{0}}}$ (where $d$ is some positive real constant). Then, let $$ s^*(t; a_0, t_w) = s\left(\frac{t-...
Airat Valiullin's user avatar
2 votes
1 answer
31 views

Does the presence of neural bias change the hypothesis space of a NN?

This question has to do with neural networks, but it is a purely mathematical question, so I think it belongs here. Consider $f: \mathbb{R}^N \to \mathbb{R}^M$ such as to be implementable with a ...
Noumeno's user avatar
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Polynomial formula for orthogonal vector in odd dimensions [duplicate]

I have been thinking about this problem recently. In 2 dimensions there is an easy formula for a nonzero vector orthogonal to a given vector $(x, y)$, namely $(-y, x)$. By taking pairs of coordinates, ...
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normalizable solution of a nonlinear equation

How to find a normalizable solution of the nonlinear differential equation below? $$ R'' + \frac{R'}{r} - R + R^3 =0 . $$ The domain is $[0,\infty ]$ and we want the norm of the solution to be ...
poisson's user avatar
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System of non-linear differential equations, $\dot{\vec{\theta}} = K^{-1} \hat{J}^{T} \vec{h}$

Suppose I have following system $\hat{J} \dot{\vec{\theta}} = \vec{h}$ with $\hat{J}$ is a function of $\theta$. I want to solve $\vec{\theta}$. Naively, one starts with the following construction. ...
phy_math's user avatar
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invertibility of overcomplete system of non linear equations

I am currently working on a research problem involving 8 nonlinear equations in 5 variables. While these variables are all real, the equations themselves are complex in general. A colleague has ...
Kobamschitzo's user avatar
1 vote
2 answers
156 views

Intercept another object while matching velocity in 2D

There are dozens of solutions online for finding how to intercept an accelerating object, etc. But I have a more stringent requirement for interception. Suppose object I can accelerate and is trying ...
HiddenBabel's user avatar
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How to obtain the result of Ott-Antonsen ansatz of the classic Kuramoto model

This is my derivation process: $\frac{\partial f}{\partial t} +\frac{\partial (f*\frac{\partial \theta }{\partial t})} {\partial \theta } = 0.$ Then $\frac{\partial}{\partial \theta }\left(f \cdot \...
Putin's user avatar
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3 votes
1 answer
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A question on the qualitative analysis of solution of a system of ODEs [closed]

Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be a non-zero smooth vector field satisfying $\text{div} f \ne 0.$ Which of the following are necessarily true for the ODE: $\dot{\mathbf{x}}=f(\mathbf{x})$? (a) ...
MathRookie2204's user avatar
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2 answers
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simultaneous non linear equations [closed]

$x^3y^3(x^3+y^3) = 905$ and $x^4y^4(x+y) = 810$. Find values for x and y. Divide the first equation by the second equation, and factor to give $(x+y)^2 = \frac{25}{6}$. If I had stopped at this point, ...
Bob's user avatar
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Solving Systems of Linear PDE's in R3 w/ Singular Matrix Coefficients

been curious whether first-order linear systems of PDE's of the form $A\vec{u}_{x}+B\vec{u}_{y}+C\vec{u}_{z}=\vec{0}$ can be solved in the case where all matrices "A, B and C" are singular, ...
Reuben Miller's user avatar
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Does the accuracy of the iterations of the Newton method transfer to parts of the underlying non-linear equation system?

I'm just wondering one thing. Suppose I have a non-linear system of equations $F(z) = z - d(z) = 0$ for $F: \mathbb{R}^{n} \longrightarrow \mathbb{R}^{n}$. If I apply a Newton method with respect to $...
Donnie's user avatar
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Help in Solving a non-linear elliptic PDE

I am trying to solve the following PDE numerically, but I am unsure of what algorithm I should use to do so. The following snippet shows what I am working on: There is also a periodic BC in $\theta$ ...
scruby's user avatar
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Convergence analysis for $x_{k+1}=A\lvert x_k\rvert+c$

I have the following iteration $$x_{k+1}=A\lvert x_k\rvert+c $$ where $x_k \in \mathbb R^n$ and $A \in \mathbb R^{n \times n}$ is a square matrix. The absolute value if taking over the elements. I ...
Fathi's user avatar
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What are the eigen values of Fischer's equation?

What are the eigen values of Fischer's equation $u_t=u_{xx}+u(1-u)$? I am asking in the sense just like for heat equations we have eigen values $\lambda=n\cdot \pi/l$, so what will be the eigen values ...
Simran bedi's user avatar
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How many Boundary and initial conditions are needed for nonlinear coupled PDEs? Is there a theorem?

I would like to solve the following coupled PDEs, and I wonder how many initial and boundary conditions I need. $$ \partial_t z(x,t) + \partial_x (z(x,t) b(x,t))=0 $$ $$ \partial_x (z(x,t) |b_x|^{m-1}...
questionerno8's user avatar
2 votes
1 answer
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Clarifications on the solution of a double integral: $\iint_X\frac{x^2y}{x^2+y^2}dxdy$

Calculate the following double integral: $$\iint_X\frac{x^2y}{x^2+y^2}dxdy$$ where $X=\{(x,y)\in \Bbb R^2\colon 1\leq x^2+y^2\leq2x\}.$ Here my confusion arises. Looking at the integrand the polar ...
Sebastiano's user avatar
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How to find the general form of a vector acted upon by linear and non linear operations

I stumbled upon this problem, originally I considered it easy to solve, I tried, and failed, and now I am beginning to believe it is quite a formidable adversary; so I am posting here, in hope of ...
Noumeno's user avatar
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0 answers
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Find the probability distribution density of a random process at the output of a nonlinear circuit

I know: Probability distribution density at the input of the circuit $$ W(x) = \frac{\mu}{2\left(1+\mu^2\left(x-m\right)^2\right)^{3/2}} $$ Non-linear function $$ f(x) = a_0 + a_1 \cdot x - x $$ ...
Антон's user avatar
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Solution to a system of nonlinear equations with certain conditions

I am working in a model and I found a problem relating a nonlinear system of equations. Let $\mathbf{D}(\mathbf{Q})\in \mathbb{R}^N$ for $\mathbf{Q}\in \mathbb{R}^N$ be a continously differentiable ...
Tan1278's user avatar
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1 answer
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Solving system of power sum symmetric polynomials

I'm interested in solving the following system of $n$ equations in the unknowns $x_i, i=1, ..., n$ $$ \sum_{i=1}^n x^k_i = \alpha_k$$ where $k=1, ..., n$. The LHS is the power sum symmetric polynomial ...
blundered_bishop's user avatar
1 vote
0 answers
31 views

Does topological conjugacy hold in the hyperbolic subspace of linearizations?

Suppose you have a nonlinear vector field $\mathbf{f}(\mathbf{x})$ and a fixed point $\mathbf{x}'$ s.t. $\mathbf{f}(\mathbf{x}') = \mathbf{0}$. Let $\mathbf{f}'(\mathbf{x}) = \mathbf{A}(\mathbf{x} - \...
cisprague's user avatar
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3 votes
0 answers
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Non-autonomous system of two nonlinear ordinary differential equations with conditions

Consider the ODE system: $$ \frac{df}{dx}= -\sqrt{g},\tag{1} $$ $$ \frac{dg}{dx}= -\sqrt{x}f,\tag{2} $$ where $f=f\left(x\right)$ and $g=g\left(x\right)$ are the functions on the interval $x\in\left[0,...
Khristo Mikhail's user avatar
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0 answers
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Method for solving polynomial system without multilinear form?

I am an engineer who is currently working with some network optimization problem during my post graduate study. During my study time, I see that sometimes I need to look for solution of polynomial ...
Tuong Nguyen Minh's user avatar
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0 answers
48 views

Help with finding stationary solutions of a system of differential equations

I need help/guidance to solve the following question with the following system of equations: $$\theta_{1}^{\prime}(t)=\omega_1+\gamma\sin[2\pi(\theta_1-\theta_2)]\\\theta_{2}^{\prime}(t)=\omega_{2}+\...
nutshell_A's user avatar
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Scaled Relative Graph Analysis

The drug concentration levels $c_1$ and $c_2$ in two compartments are modeled by the following differential equations: $V_1\dot c_1=-\rho_1(c_1)-\alpha_{12}c_1+w_1$ $V_2\dot c_2=-\rho_2(c_2)-\alpha_{...
BeNavon's user avatar
  • 13
7 votes
2 answers
184 views

Finding an analytical solution to a particular class of second order ODEs

Let $$\ddot{y}+f(x)y=0$$ Be a differential equation such that $\{y,f\}\subseteq\mathcal{C}^\infty$. We make the substitution $y=e^{u(x)}$ to get: $$\frac{d^2}{dx^2}e^{u(x)}+f(x)e^{u(x)}=0$$ Or $$(\...
Simón Flavio Ibañez's user avatar
3 votes
1 answer
72 views

how to show global asymptotic stability with $V(x)=f(x)^{T}Pf(x)$ as a lyapunov function.

consider the system $f(x)=\dot{x}$ with $f(0)=0$, $f(x)$ is continuously differentiable. $f(x)$ can be written as $f(x)=\int_{0}^{1}\frac{\partial f}{\partial x}(x\sigma)x\partial\sigma$ (The first ...
TiredMechanicalEng's user avatar
0 votes
1 answer
33 views

Example 1.1 Nonlinear Control Khalil

$f( x) =\begin{bmatrix} -x_{1} \ +x_{1} x_{2}\\ x_{2} \ -x_{1} x_{2} \end{bmatrix}$ is continuously differentiable on $R^2$. Hence, it is locally Lipschitz on $R^2$. It is not globally Lipschitz since ...
SS1's user avatar
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4 votes
1 answer
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On the solution of these equations

Do not forget to see the Good News at the end of the problem. This problem is linked to the previous one, up to a changes of coordinates. However, that question is actually about only the first three ...
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