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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.

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. Reference: Wikipedia.

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 it (them).

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System of nonlinear equations that leads to cubic equation

The system of equations are: $$\begin{align}2x + 3y &= 6 + 5x\\x^2 - 2y^2 - (3x/4y) + 6xy &= 60\end{align}$$ I can solve it through substitution but it is an arduous process to reach this cubic equation: $$20x^3 + 56x^2 - 243x - 544 = 0$$ And I can…
joefieldsend
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When can a non-autonomous system NOT be re-written as an autonomous system?

Consider Duffing's equation $\ddot x + \delta \dot x + \alpha x + \beta x^3 = \gamma \cos{\omega t},$ where $\delta, \alpha, \beta, \gamma$ and $\omega$ are real parameters, $t$ represents time and $\dot x := dx/dt$. Since there is an explicit…
trolle3000
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System of non-linear equations.

I have to find all triplets $(x,y,z)$ that satisfy: $$x^{2012} + y^{2012} + z^{2012} = 3\\x^{2013} + y^{2013} + z^{2013} = 3\\x^{2014} + y^{2014} + z^{2014} = 3$$ I've found the trivial solution $(1,1,1)$ but I don't know how to start looking for…
Dos.SieteUnoOchoDosOchoEtc
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The Stable Manifold Theorem Applications

Definition: Let $\phi_t(x)$ be the flow of the nonlinear system $x'=f(x)$. The global stable manifold of $x'=f(x)$ at $0$ is defined by: $$W^s(0)=\bigcup_{t\leq 0}\phi_t(S)$$ Where $S$ is a $k$-dimensional differentiable manifold tangent to the…
riva
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Algorithms for factoring multivariate polynomials

I am wondering if there are any algorithms to factor polynomials in multiple variables, when you know that the factors are other polynomials with rational or integer coefficients. I know you have the rational root theorem, which helps out a lot, but…
Ruben
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Are continuous chaotic systems necessarily uncomputable?

I have seen the claim in a recent unpublished paper that chaotic dynamics are necessarily uncomputable. This follows, they argue, from the sensitivity to initial conditions shown in chaotic systems. This glib identity between chaos and computability…
eric
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Roots of a set of nonlinear equations $ax + yz = b_1; ay + xz = b_2; az + xy = b_3$

Let $a$ be a non-negative real number, $b_1, b_2, b_3$ be real numbers, and $x, y, z$ be variables. Is it possible to analytically find the root closest to origin $(0, 0, 0)$ of the set of nonlinear equations given by: $$\begin{cases} ax + yz = …
Ahmet Taha KORU
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Understanding the singularity in $f^{-1}(x)=\int_0^x f(t)dt$

Note that here $f^{-1}(x)$ is the functional inverse of $f$. Clearly from the definition of the equation $f^{-1}(0)=0\Rightarrow f(0)=0$. By setting $x\to f(x)$ and differentiating one finds $$f'(x)f(f(x))=1$$ and therefore $\lim_{x\to…
K. Grammatikos
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Finding the period of a nonlinear ODE

I am studying a periodic physical system with a nonlinear ODE $$x''=f(x)+g(x)x'^2$$ I think the periodicity comes from the $x'^2$ term because this provides two possible numbers to give a same right hand side value. The following shows three…
Shengkai Li
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Is there a connection between topological mixing and squashing functions used in neural networks?

Sigmoid, ReLU, tanh, logistic -type "squashing" functions are popular in neural networks to introduce nonlinearity into the transformations of the input vector, allowing the network to fit complex input-output surfaces. Deeper layers (stacking…
ejang
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What is meant when mathematicians or engineers say we cannot solve nonlinear systems?

I was watching a video on "system identification" in control theory, in which the creator says that we don't have solutions to nonlinear systems. And I have heard this many times in many contexts, related to control problems or nonlinear odes, etc.…
krishnab
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Eigenvalues of a $12 \times 12$ Jacobian matrix

Consider the set of coordinates $X_{i, j}^{(\ell)}$ and $Y_{i, j}^{(\ell)}$, where $i \in (1, 2, 3), j \in (1, 2, 3)$ and $i \neq j$ and $\ell = \pm 1$. The superscipt $(\ell)$ is an index. Consider the change of variables from $\mathbf{X}$ to…
Mr. G
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Are there Soliton Solutions for Maxwell's Equations?

Some non-linear differential equations (such as Korteweg–de Vries and Kadomtsev–Petviashvili equations) have "solitary waves" solutions (solitons). Does the set of partial differential equations known as "Maxwell's equations" theoretically admit…
user559615
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How to solve non-linear differential equation

How to solve non-linear differential equation $$y'(x) = y(y(x)), \quad y\colon\mathbb{R}\to\mathbb{R}?$$ Of course, $y(x)\not\equiv 0$. If we substitute $y(x) = Ax^n$, we get complex $n$ and $A$. Any numerical solution doesn't work because we can't…
Michael Galuza
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Solution of system of non-linear equations

Is there a general condition for the existence and uniqueness of solution of a system of simultaneous non-linear equations similar to the determinant test for a system of linear equations. What are the solution methods (theoretical and numerical)…
Julie Kirk
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