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 ...

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15 views

Unique solution of a system of equations by continuation argument

Here is my conjecture (let's call it "Uniqueness Extension Theorem"): Conjecture: Let $x\in$$\mathbb{R}_{++}^{N}$, $\alpha\in[0,1]$, and $F(x,\alpha)$ be a continuous and differentiable function ...
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1answer
38 views
+100

simplify/solve nonlinear equations for constrained least squares problem

I am trying to find a simple, ideally closed form formula for the (not necessarily unique) unit vector $\vec{x}$ minimizing total squared cosine distance from a collection of unit vectors $\vec{v_i}$. ...
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2answers
25 views

Solving a system of polynomial equations

How can I solve a system of polynomial equations like this one Maybe I'm missing a very basic trick... Can anybody suggest me an approach?
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0answers
21 views

Significance of multiplying by weight

I have been reading optimization problems in communication area where it is a common practice to maximize rate of users as below objective function: $\hspace{28mm} \text{ Maximize } \sum_k w_k \log ...
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0answers
27 views

Is the given system of differential equations solvable?

I am trying to implement a system of differential equations (equations of motion of a roll axis vehicle, which are part of a vehicle model) in Matlab but it does not work for me. I have figured out ...
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1answer
19 views

Is it linear or nonlinear, time-invariant or time-varying?

The equation of motion can be expressed as $M(t)\ddot{q}(t) + D(t)\dot{q}(t) + K(t)q(t) = f(t)$ where $q(t)$ is the defection, $M(t)$, $D(t)$, and $K(t)$ are the mass, damping, and stiffness ...
2
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0answers
23 views

In the Glycolysys Sel'kov model, what are the meaning of “a” and “b” values?

In the Sel'kov model of glycolysis wich I put on next $u'=-u+av+u^2v\\ v'=b-av-u^2v$ wich have a limit cycle and have all sense because it is a glycolytic cicle. What are the ...
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1answer
22 views

Multiscale analysis with non-integer exponents

I am dealing with the following non-linear differential equation: $$\frac{d^2 x}{d t^2}=2\varepsilon\frac{d x}{d t}-\left(\frac{d x}{d t}\right)^3-x$$ I found that $x=0$ is the only one fixed point ...
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2answers
33 views

Multiplication of two variables, Linear or Nonlinear?

This should be very easy but I am confused as I am getting different answers in different sources. Say I have an equation: $z = x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4 + x_5 y_5 $ where, $0 < x_k ...
2
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0answers
29 views

“Dictionary” of linearizations for nonlinear dynamical system

I have recently jumped on a control project that involves predicting output of a nonlinear system given some input. The team has used $N$ training input/output relationships to build a 'dictionary' ...
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1answer
46 views

Finding value of (y) of logarithmic equation given (x)

I have an logarithmic equation $$\left[ r=a\,e^{b\,\theta} \right] $$ And I plot it to visualise it (see plot below). I can tell by the plot when (t=0), x1=0, y=1 (point AA) but how can I find out ...
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2answers
72 views

Linear or Nonlinear function?

I want to know if the below two function are Linear or not. I have been searching in Google but found some confusing results. The first one is an inequality and the second one is just a function and ...
1
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1answer
31 views

Find the nodes and coefficients of Gauss-Lobatto Quadrature with $n=4$

I am stucked at this problem: Gauss-Lobatto quadrature is defined as: $\int_{-1}^1 f(x)dx\approx w_1 f(-1)+w_n f(1) + \Sigma_{k=2}^{n-1}w_k f(x_k)$ ($2\leq n\in\Bbb{N}$) Where the nodes $x_k$ ...
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1answer
25 views

3d parametric spiral to 3D goldean mean spiral

I know I can create a 3d parametric spiral with the formula below but How can I do the same thing with goldean spiral? I looked at https://en.wikipedia.org/wiki/Golden_spiral but I don't see how to ...
3
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0answers
70 views

How to solve this first order diff. equation?

I'm looking for an analytic solution to a first order non-linear differential equation that I'm unable to solve : $$\frac{2}{1 + \Theta^2} \; \frac{d\Theta}{d r} = 1 - \frac{1}{r (\Theta - r)}$$ ...
4
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0answers
38 views

Solving non-linear equations in a chosen subspace

I'm trying to find the root $\mathbf{f(x)=0}$ to the following sets of equations $$ f_1(x,y,z) = x^\prime - \frac{x}{\sqrt{x^2+y^2+z^2}} = 0 \\ f_2(x,y,z) = y^\prime - \frac{y}{\sqrt{x^2+y^2+z^2}} = ...
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0answers
7 views

Non linear Optimization for resource allocation

I want to maximize the sum rate of a wireless system while maintaining fair allocation by using fairness constraint. $R_k$ is the rate for each user. I have set up my objective function as : Maximize ...
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0answers
40 views

How to find values of non linear equations / system and solve for given values

I'm trying to find the value for the variable phase in a equation / system if amp=0.5 and freq=2.5 (note: i'm looking for several different phase values given amp and freq but this is a small ...
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1answer
49 views

Relation between Power Laws and Fractals

Are all power laws (i.e., of the general form $y=cx^{\alpha}$) fractal (exhibiting some form of self-similarity)? Does the scalability of power laws also mean by definition that they are also ...
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22 views

asymptotic matched expansion with transiently blowing inner solution

I have been trying to solve following set of equations with method of matched asymptotic expansion, $\frac{dy(t)}{dt}=k z(t) - 3 \alpha y(t) - y(t)^2 + \mu (M-z(t))^2$ $\epsilon ...
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0answers
22 views

Solving for 1 varible while missing two variables in non linear sin wave equation:

I can solve for y for the equation below but how do I go about solving for other freq(x) values? Example equation below: $$\left[ y={\it amp_1}\,\cos \left(2\,{\it freq_1}\,\pi\,t+ {\it ...
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0answers
22 views

Simple example of approximating a nonlinear system with Volterra series

I'm trying to understand Volterra series as a means of modelling/approximating nonlinear input-output relations. I'm having trouble to understand the abstract definitions of kernels/functionals and ...
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1answer
65 views

Locus of solution of an ODE system

I have an ODE system $$ \ddot p = \frac{ p \left( {2p - 4} \right) }{{p - 4}}{{\dot q }^2 } \\ \ddot q = \frac{{3p - 8}}{{p - 4}}\dot q \dot p $$ Short of finding closed-form expressions for ...
4
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1answer
41 views

3 unknown, 3 nonlinear equations of form $xy - z =$ constant

How can I solve: $$xy - z = a $$ $$xz - y = b $$ $$ yz - x = c $$ for $x, y, z$ (where $a,b,c$ are constants)? Let all variables and ...
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0answers
27 views

How to find a stable periodic solution of a nonlinear non-autonomous second-order dynamical system?

The system I am working on is in the following form: $$ \dot{x}=f(t,x,a)=f(t+T,x,a), $$ where $f(t,x,a) \in R^2$ is nonlinear and periodic in $t$, $T$ is a known constant, and $a$ is a vector of ...
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0answers
22 views

Binary solutions of multivariate polynomial system in special (factored) form.

In my personal research I've run into a system of multivariate polynomials (with coefficients in a field). I am aware that there is no polynomial time algorithm (in the number of indeterminates) for ...
2
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1answer
63 views

Is there a numerical solution for a system of three 1st order nonlinear ODE?

How would I go about solving the following system of non-linear ODEs for $x(t), y(t), z(t)$ $$x' = y $$ $$y'=\sin(x)+z$$ $$z'=y-z$$ I have the following initial conditions; $$x(0) = 0$$ ...
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1answer
24 views

How to solve $ \frac{1}{1+x}-\frac{c}{x}-2\log \left( \frac{1+x}{x}\right)+A=0$

How to find a solution to the following equation \begin{align*} \frac{1}{1+x}-\frac{c}{x}-2\log \left( \frac{1+x}{x}\right)+A=0 \end{align*} where $c$ and $A$ are some constants such that $c\ge 1$ ...
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1answer
32 views

Automatic way to have a good initial guess for the iterative methods ( newton method) and for high dimensional nonlinear problems

I am solving a variety class of nonlinear systems where I need to reduce in optimal manner the number of iterations of either the newton or modified newton method. For this end I am trying to figure ...
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0answers
22 views

Non-linear Systems, Impulse Responses, and Convolution

In linear system theory, it is easy to find the particular solution to differential equation by means of convolving the system's impulse response with the forcing function. My question is why can we ...
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2answers
22 views

How to represent non-linear operators computationally?

I have a finite dimensional vector space V, and want to compute a non-linear operator $R: V \rightarrow V$. I want to have a "general" form of this operator R. I think of the following series ...
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0answers
17 views

dominant balance for coupled differential equations

I have been trying to solve following set of nonlinear differential equations: $\frac{dy(t)}{dt}=k z(t) - 3 k y(t) - y(t)^2 + \epsilon_1 (M-z(t))^2$ $\epsilon_2 \frac{dz(t)}{dt}=Mz(t) - z(t) y(t) - ...
2
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2answers
64 views

Non-linear system of equations

Solve following system of equations over real numbers: $$ x-y+z-u=2\\ x^2-y^2+z^2-u^2=6\\ x^3-y^3+z^3-u^3=20\\ x^4-y^4+z^4-u^4=66 $$ This does not seem as hard problem. I have tried what is obvious ...
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1answer
45 views

exact solution to lotka-volterra equations

I am looking for exact or perturbative solution realistic lotka-volterra (the one with logistic term in one of the equations) equations in population dynamics. Any reference where they have done it ...
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0answers
29 views

Well-posedness of nonlinear PDE system

The surface is parametrized by two variables $\sigma_1$ and $\sigma_2$. Moreover, this surface evolves in time. As a result, coordinates of the surface are: $\vec{F} =[x(\sigma_1,\sigma_2 , t), ...
1
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1answer
59 views

Averaging for nonlinear systems

I am trying to figure out how the following result has been obtained. Consider a function $J:\mathbb{R} \longrightarrow \mathbb{R}$ and a dynamical system: $$ \dot{ \hat{x} }(t) = k a \sin ( \omega ...
2
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4answers
40 views

How can you tell whether the equation of a non-linear relationship represents a parabola, a hyperbola or a circle?

How can you tell whether the equation of a non-linear relationship represents a parabola, a hyperbola or a circle?
2
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0answers
31 views

two variable perturbation analysis of nonlinear set of differential equations.

I have following set of equations, $\frac{dy(t)}{dt}=k z(t) - 3 k y(t) - y(t)^2 + \epsilon_1 (M-z(t))^2$ $\epsilon_2 \frac{dz(t)}{dt}=Mz(t) - z(t) y(t) - 2 \epsilon_2 y(t) + 2 \epsilon_1 \epsilon_2 ...
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0answers
25 views

non-linear PDE finite difference approach

How to approach this equation using finite difference method ...
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0answers
16 views

Numerically solve: system of non-linear complex valued equations

I need to solve a system of equations numerically. Definitions: $$ -1\leq \epsilon_1,\epsilon_2\leq 1 $$ $$ E_1, E_2, \omega\geq0 $$ $$ E_0 < 0 $$ $$ n=1,2,3,... $$ $$ \frac{1}{t_f-t_i} \left( ...
3
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1answer
96 views

Non linear second order ODE

I really need help solving this : $$y_{xx}-\left(y^{3}-y\right)-\varepsilon\frac{1}{2}\left(1-y^{2}\right)=0 $$ With boundary conditions : $$ y(\pm \infty )=-1 $$ I need to find a solution that ...
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0answers
43 views

How to show that a system $dx/dt = f(x,y)$, $dy/dt = f(x,y)$ has a stable limit cycle which lies in the anular region?

I know I need to convert it to polar system but I don't know how to do this My system is $$\dfrac{dx}{dt}= x - y -x(x^2 + 2y^2)$$ and $$\dfrac{dy}{dt} = x + y -y(x^2 + y^2).$$ The annular ...
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0answers
16 views

Nonlinear System of Equations: Criteria for Existence of Solution

Let $\Omega \subset \mathbb{R}^n$ and $F: \Omega \rightarrow \mathbb{R}^n$ is at least once continuously differentiable (but not necessarily a polynomial). we want to find a point $x^* \in \Omega$ ...
1
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1answer
21 views

non linear curve fitting with newton-gauß

Shape the following problem as a non linear curve fitting problem and write the first iteration step with the gauß newton method. On a map are n radiostations $S_1,...,S_n$ which coordinates are ...
9
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4answers
243 views

Solving a system of non-linear equations with 10 equations and 10 unknowns

I'm working on a problem where I seem to have run into a system of non-linear equations. I have ten equations and ten unknowns. In the equations below, all of the $\phi_{ij}$'s are known, but all of ...
0
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1answer
19 views

Derivation for state equation linearization

In the following notes, how to linearize a state equation is described. The part I don't understand is why you can just remove the $\delta$ like that. I think the state equation should be: ...
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39 views

Solution to a state-space equation

In my notes on non-linear linearization there is the following example. It asks to verify the solution to the state-space equation. My understanding is that the solution is where the equilibrium point ...
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2answers
45 views

Convergence of a particular fixed point iteration scheme

Setup I have the following non-linear system of equations: $$ \mathbf{x} P(\mathbf{x}) = 0 $$ where $\mathbf{x} \in \mathbb{R}_{>0}^n$ is a probability distribution, i.e., $\sum_i x_i = 1$, and ...
1
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0answers
28 views

A set of nonlinear coupled ODE

I have reached a set of ODE as \begin{align} &\ddot{\vec{a}}(t)+\omega_0^2\frac{\cos{(b(t))}\sin{(a(t))}}{a(t)}\vec{a}(t)=0\\ &\ddot{b}(t)+\omega_0^2\cos{(a(t))}\sin{(b(t))}=0 \end{align} ...
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3answers
63 views

In control theory, why do we linearize around the equilibrium for a nonlinear system?

For example, in these notes: In the first example with the pendulum, they define the equilibrium as where the pendulum is at the vertical position (x=0), with a angular velocity of 0 (x'=0) and the ...