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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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How can you solve a couple system of quadratic differential equations?

Suppose there is a differential equation in the form of $$\mathbf y'(t)=\begin{bmatrix} y_1 \\ y_2+y_1^2\\ \end{bmatrix}.$$ Not only is it non-linear, but it is coupled. Is there a general method of ...
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How can one solve a differential equation with polynomial vector components?

Suppose there is a differential equation in the form of $$\mathbf y'(t)=\begin{bmatrix} y_1+y_1^2 \\ y_2+y_2^2 \\ y_3+y_3^2 \end{bmatrix}.$$ What is the general method of finding the solution to ...
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Can system of nonlinear algebraic and transcendental equations contain more equations than variables and still be consistent?

I have system of nonlinear equations. Each equation involves algebraic or tanscendental functions (usually step/threshold functions, nothing fancy), but no euqation involves differentials or ...
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39 views

Solving a System of Quadratic Equations for Sound Triangulation

I am currently attempting to solve a system of quadratic (and linear) systems that I have run into while trying to triangulate sound. My hypothetical setup includes 3 sensors on a perfectly ...
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27 views

Find the critical points of the following system

I want to find the critical points of the following three-dimensional system: \begin{align} \dot{x_1} &= x_1 - x_1x_2 - x_2^3 + x_3(x_1^2 + x_2^2 - 1 - x_1 + x_1x_2 + x_2^3)\\ \dot{x_2} &= ...
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1answer
48 views

Squared-derivative PDEs

Is there a general theory for equations of the type $ f_y^2 = A(x,y) f_x$? where one first derivative is expressed as a multiple of the other one. Concretely, I'm interested in the equation $$ ( x+...
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Convergency of Adomian decomposition for solving on nonlinear algebraic systems of equations

As far as I've managed to understand this method, idea is the following. If we have equation like $x=1-a x^2$, then we assume that $a x^2$ is small and get zero approximation for the root $x_0 = 1$. ...
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A rather challenging non-linear system

In one of the solutions to an AIME question from a few years back (reference at the end), one comes across the following system: $$ P\sin\theta + Q\cos\theta = \cos\theta - \frac{1}{2}P$$ $$ P\cos\...
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1answer
15 views

Proof request: All values a for which a quadratic-linear system has exactly one point of intersection

Let $f$ and $g$ be the functions defined by $f(x)=x+2$ and $g(x)=(x^2)−a$, where $a$ is a positive constant. What are all values of $a$ for which the graphs of $f$ and $g$ have exactly one point of ...
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How to reduce number of nonlinear terms in nonlinear equation by linear substitution?

Consider system of nonlinear equations: $2a-2b-3=0$ $a-a^2-b-2ab-b^2=0$ If you replace variables this way: $x=a+b$, $y=a-b$, then this system can be simplified to $2y=3,y=x^2$. My question is ...
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How to do a sensitivity analysis on a non-linear equation?

In the company, it is very difficult to actually do quotations for our customers properly because we do not have perfect information regarding the factors that affect the cost and profit. So I created ...
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22 views

Relation eigenfunction of linearized PDE and solution of the original PDE

Consider $$ \frac{\partial^2}{\partial x^2}u+\mu \sin(u) = 0 \\ u(0) = 0 = u(1) $$ The linearized version is for small $u$ $$ \frac{\partial^2}{\partial x^2}u+\mu u = 0 $$ This gives for the general ...
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2answers
32 views

How many equations for a system of non-linear equations?

Let's suppose we have a system of n non-linear equations with m variables. How many equations does my system need to produce a single solution? Is there a theorem that states this?
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Calculate RGB color values given amounts of two types of melanin

I've been trying to create a function that can take values representing amounts of eumelanin and pheomelanin and return reasonably accurate numbers for red, green, and blue that I can use to tint a ...
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11 views

Non linear matrix system with average values

I am currently working on a system for the evolution of a vector $X=(x_1(r),\cdots,x_N(r))$ as a function of the variable $\displaystyle{t}$. I have: For $j=1$: $\dfrac{\mathrm{d}x_1(r)}{\mathrm{d}t}=...
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38 views

Showing $x_1x_3 + 2x_2 + x_1 = 4$ is not a linear equation [closed]

Prove that $x_1x_3 + 2x_2 + x_1 = 4$ is not a linear equation. I do understand that in order to be a linear equation, it must fall inside the graph linearly. This equation does show a linear equation ...
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equilibrium point in an inverted pendulum

The unstable upright position of an Inverted Pendulum on a cart does this corresponds to a Hyperbolic equilibrium point or a non- hyperbolic equilibrium point? please give me a lucid explanation on ...
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2answers
46 views

Singular points of non-linear ODE

EDIT: Sorry i messed up, I forgot a minus sign in front of the left hand side. I added it now. I am not sure how to proceed with this. Given this non-linear ODE$$\partial_{t}u(t,x)=-\cot(t)\left[\...
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1answer
21 views

Detecting independent parts in non-linear system of equations

When solving systems of non-linear equations using Newton's method, it is often observed that the system has an independent sub-system, e.g. : $$ f(x,y)=0 $$ $$ g(x,y)=0 $$ $$ h(x,y,z)=0 $$ If I am ...
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How would I solve this system of equations?

Suppose we have the following system of non-linear equations $t_1 = \alpha_1^{d_1}+\alpha_2^{d_1} \\ t_2 = \alpha_1^{d_2}+\alpha_2^{d_2}$ where each $t$ is a known complex number, each $d$ is a ...
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33 views

Stability of the RC Circuit with Two Capacitors

I want to find poles and zeros of the circuit shown below. So I can determine stability of the system. $$ V_{C1} + V_{C2} + V_R = 0 $$ where $I$ is current of the loop, $$ V_{C1} + V_{C2} + RI = 0 $$...
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How to show uniqueness of the non-degenerate solution of $g(x)=0$?

Let $g(x): \mathbb{R}^n \rightarrow \mathbb{R}^n$. We call a root $x_*$ of $g(x)=0$ non-degenerate if $Jg(x_*)$ is invertible, where $Jg(x_*)$ is the Jacobian at $x_*$. How can we show if $x_*$ is ...
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31 views

Solve simple nonlinear equations in the form [A]x=b

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 (...
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How would I solve this non linear systems of equation with trig. [closed]

How would I solve this. I'm kinda stuck... $$\cos(y) + \cos(z) = -1$$ $$\sin(y) + \sin(z) = 0$$
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How to show that a non-linear differential equation is convex.

This question is relevant to my investigation, and looks like a question asked by some other user(Convexity of the solution of an o.d.e.) but I tried to apply what they were given as a solution and I ...
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1answer
25 views

Logarithmic non-linear system of equations

As part of an heat exchanger problem, we are given two unknowns $T_{cs}$ and $T_{fs}$ solving the following equations: $\forall (Q,U,F, m_c,m_f, C_c,C_f, \Sigma , T_{ce},T_{fe}) \in \mathbb{R_+^*}^{10}...
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1answer
20 views

Solution of System of Bivariate Cubic Polynomials

Apart from numerical solutions, is there a method to find the real roots of $X$ and $Y$ for this system of nonlinear equations ? $X^3 -3 X Y^2 + b_1 X - b_2 Y + c_1=0$ $Y^3 -3 X^2 Y - b_1 Y - b_2 ...
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Sketch phase portrait for autonomous systems

I have thus far mostly (only) been dealing with phase portraits where you compute eigenvalues and eigenvectors and from there are able to sketch the portrait. Now I came across a different type of ...
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3answers
29 views

Given a planar system, show that $H$ is constant along solutions

Let $\mathbf{x}=[x_1\quad x_2]^T$ and consider the system $$\begin{bmatrix}\dot x_1 \\\dot x_2\end{bmatrix}=\begin{bmatrix}f_1(x_1,x_2)\\f_2(x_1,x_2)\end{bmatrix}.$$ Let also a function $H:\mathbb{R}^...
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1answer
46 views

Finding the equation of a rational function

I have a formula for a rational fuction in the form $$\frac{1}{(a(x-b)^c)}=y$$ and I am given the points $(0,30.8493),(0.75,1.1392)$, and $(1,0.2838)$. When simplifying I always seem to reach a dead ...
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1answer
47 views

Visualise the bifurcation diagram

Can anyone help me visualise the bifurcation diagram that would be produced by $\dot x = (x−μ)(1+μ−x^2)$ and $\dot x = (μ^2−1)(μ−2)−x$ I know for 2. there is only one equilibrium point so ...
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Attempting to simplify a quadruple system

I am attempting to simplify a horribly complicated equation that can be expressed in the form $\frac{ay^2}{(b(x-c))^d)}=z$. By plugging in x and y values I am able to find points to set up a system ...
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1answer
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find all the vectors such that the pairwise differences of whose entries are all different.

Given a natural number $n$, I want to find all vectors in $\mathbb{R}^n$ such that (1) The sum of the entries is $0$ (2) Pairwise differences of the entries are all different. Let $\mathbf{a}=(a_1,...
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1answer
78 views

Solving an ODE system which depends on a periodic function

I have the following ODE system and want to show that all solutions $\gamma(t)=(x(t),y(t),z(z))$ exist for all times (or can be extended on all of $\mathbb{R}$). The only problem is that the system ...
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Derivation of 2D Korteweg-de-Vries equation

Coming from engineering rather than mathematics, I am recently dealing with non-linear partial differential equations e.g. like the well known Korteweg-de-Vries equation: $$u_{t} + uu_x + u_{xxx} = 0$$...
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Factoring a general biquartic into two quartics

Let $W_0$, $W_1$, $W_2$, and $W_3$ be known real numbers. I have to solve a biquartic equation: \begin{equation} z^8+W_3z^6+W_2z^4+W_1z^2+W_0=0 \notag \end{equation} Of course I could solve the ...
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1answer
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What does the lack of singular points of non liniar system mean in phase portraits?

I'm going to plot the phase portrait of this system: $\dfrac{dx}{dt}=-x^2 + 4 y^2$ $\dfrac{dy}{dt}=-8 - 4 y + 2 x y$ The singular point $(x,y)$ can be found from the system: $-x^2 + 4 y^2=0$ $-8 -...
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Cauchy's theorem for existence of solution to nonlinear ODE

I was reading about Cauchy's theorem for existence of solution to nonlinear ODE and the book (Nhan T. Nguyen - Model-Reference Adaptive Control. A Primer) stated "continuity of f (x, t) in a closed ...
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37 views

show that the system has a limit cycle using a theorem

I have the following system, and I'm trying to use a theorem to prove that it has a limit cycle. I proceeded finding the fixed points(Strogatz), $(0,0)$ in this case and then calculating the ...
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1answer
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Solution of a system of linear equations when matrix is a function of a vector variable

My problem is actually a nonlinear equation: $A(x)x=0$, where x is a vector and matrix A is a function of vector variable x. Are there any works related to this kind of problem??
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36 views

Variation of the Diffusion Equation

Let $\mathcal{L}=D\dfrac{\partial^{2}}{\partial x^{2}}-v\dfrac{\partial}{\partial x}+\beta$ be a differential operator describing diffusion ($D$) with drift ($v$) and a source ($\beta$). As part of a ...
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Solution to a set of matrix equations involving an inverse

Let $\{x_1,\ldots,x_n\}$ be a set $d$-dimension Euclidean vectors whose span is $R^d$. With $c_1,\ldots,c_n$ positive reals, how can we find real-valued $w_1,\ldots,w_n$ satisfying the set of ...
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87 views

A Golden Ratio Functional Equation Sequence

I was looking at the equation $f^{-1}(x)=\int f(x)dx$ recently. One can note that it has an easy real-valued solution $f(x)=\phi^{\frac{\phi-1}{\phi}}x^{\phi-1}$ (by guessing for a solution of the ...
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1answer
75 views

Analytical Solution to Nonlinear Second Order ODE

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 ...
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30 views

Phase portrait of nonlinear system with repeated, complex eigenvalues

I am asked to describe the phase portrait of the following system: $$x' = |y|\\ y' = -x$$ My approach so far is to write the system in the form $$X' = \begin{bmatrix}0&|1|\\-1&0\end{bmatrix}X$...
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Gramian Observability of Nonlinear System

I understand that for a linear system, the observability Gramian is given by $W_{o}=\sum_{\tau=0}^{\text{inf}}(A^{T})^{\tau}C^{T}CA^{T}$. However, I am wondering, how would be the calculation of $W_{...
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44 views

Nonlinear ODE - possible transformations or analytical techniques?

I work with a problem that gives me a nonlinear 3rd order ODE. The equation in question is: $f'''(x) - \alpha f(x) f''(x) + \alpha f'(x)^2 + k f'(x)=0$ The coefficient $k$ always turns out to be an ...
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20 views

convergence criteria for system of non-linear equations

I have these two non-linear equations which I am able to solve using any good software like MATLAB/Mathematica but I need to find the conditions of convergence. How do I proceed on that ?? 1) $(x-k_0)...
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2answers
35 views

tracking error state space, non-linear control example

I am trying to understand an example from [1]. In detail I do not understand how the equation for the dynamic of the tracking error is chosen. I am not a mathematician so please forgive me if I may ...
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1answer
55 views

Solving nonlinear 1D advection pde with MoC

I would like to solve the 1D nonlinear advection equation with the method of characteristics. Here is my notation: \begin{equation} \begin{cases} \rho_t + (1+\rho)\rho_x = 0\\ \rho = \rho(x,t); \quad ...