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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Showing existence and uniqueness in particular system of non-linear equations
I am trying to prove existence and uniqueness for the following system. Let $\theta$ be a constant in $(0,1)$, let $i,l=1,...,N$, let $a_{l}$ and $b_{i,l}$ be some constants with $a_l \in (0,1)$ and $...
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How to design PI controller for a non-linear dynamic system?
I have following dynamic system
$$
\frac{\mathrm{d}v_C}{\mathrm{d}t} = -\frac{1}{R_b\cdot C}\cdot v_C\cdot\alpha,
$$
where $v_C$ is the system state and output, $\alpha$ is the system input and $R_b, ...
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Solving for constant of a known nonlinear dynamic system, numerically
Given a dynamic system,
\begin{equation*}
\frac{d\vec{s}}{dt}=f(t,\vec{s},\vec{c})
\end{equation*}
where $\vec{s}$ are the state, and $\vec{c}$ are constants.
And $\vec{s_0},\vec{s_1},...\vec{s_n}$ ...
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Suggestion for the numerical solution of a nonlinear pde system.
I am faced with the following system of coupled nonlinear partial differential equations
$$
\begin{array}{lcccl}
\varphi_{tt} &-& a_1\varphi_{xx} &+& a_2\varphi_t &=& a_3\sin{\...
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Overdetermined nonlinear system of equations with measure zero set of solutions
Consider an overdetermined system of nonlinear equations of the following form:
$\pi_m(x)+g_m(y_1,\dots,y_N)=\pi_m(\tilde x)+g_m(\tilde y_1,\dots,\tilde y_N)$, where $m=1,\dots,M$ and $x,\tilde x, y_n,...
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Non linear equations
Does this system of non linear equations have a solution?
$$y_1 = x_1 \cdot (x_2)^2 \cdot (x_3)^{-1}$$
$$y_2 = x_2 \cdot (x_3)^{-1}$$
$$y_3 = x_1 \cdot (x_2)^2 \cdot (x_3)^{-2}$$
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How to Solve for rx in a Non-linear Equation in Python?
Question:
I'm working on a Python project and I have an equation that involves finding the value of rx, which is part of a larger expression. Here's the equation:
...
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1
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Trouble solving a non-linear first order PDE
I'm struggling to find solution of the below equation:
$$u_x^3 - u_y = 0$$
$$u(x,0) = 2x^{3/2}$$
I was taught about method of characteristics and how to parametrize the equations with $s$ and $t$ to ...
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Is this system of nonlinear equations well known?
Let let $i,l=1,...,N$, let $a_{i}$ and $b_{i,l}$ be some positive constants with $\sum_i b_{i,l} = \sum_l b_{i,l} = 1$ for all $i,l$. Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$. Consider the ...
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When are the zeroes from Bezout's theorem real/rational/integral
Suppose I have $n$ quadratic homogeneous polynomials $f_1, \cdots, f_n$ in $n+1$ variables over the field $\mathbb{C}$. Bezout's theorem says that generically there will be $2^n$ common zeroes.
I was ...
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What does it mean to take the Jacobian of a system of Differential Equations?
When solving nonlinear differential equations, we often use the "Jacobian of the system" to determine if fixed points are stable.
As an example, suppose I have a nonlinear system
$$x_{t} = f(...
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Zero set of system of two real quadratic forms
Background: Consider the equation $x^T A_1 x = 0$ where $x \in \mathbb{R}^\mu$ and $A_1 \in \mathbb{R}^{\mu \times \mu}$ is a symmetric matrix. Suppose we also demand the normalization $x^T x = 1$. ...
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Algorithm for non-linear system of equations
I would like some tips in figuring out a good algorithm to find the solution of the following system. Let $\theta$ be a constant in $(0,1)$, let $i,l=1,...,N$, let $a_{l}$ and $b_{i,l}$ be some ...
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Prove iterations of quadratic maps tends to $-\infty$
So far, I have to read about quadratic map in "Introduction to Dynamical Systems" by Brin and Stuck.
The quadratic map is
$$q_{\mu}(x) = \mu x(1-x), \quad \mu > 0.$$
Now, I need to show ...
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System of Quadratics Forms - when is a solution guaranteed to exist?
Suppose I have a system of $\nu$ quadratic forms,
\begin{align*}
x^T A_1 x &= 0 \\
x^T A_2 x &= 0 \\
&\vdots \\
x^T A_\nu x &= 0,
\end{align*}
where $x \in \mathbb{R}^{\mu}$ and each $...
- 551
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Preimage of a point under the Hopf map
Considering the Hopf map $S^3 \longrightarrow S^2$ given by $$(x,y,z,w) \mapsto \bigl(x^2+y^2-z^2-w^2,2(xw+yz),2(yw-xz)\bigr),$$
I know that the preimage of point is a great circle in $S^3$. For ...
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Is this chaotic behaviour? Weakly coupled Van der Pol oscillators
I was investigating the following system of two weakly coupled identical Van der Pol oscillators
$$\left\{\begin{array}{@{}l@{}}
\ddot{x}_1 + x_1 + \epsilon(x_1^2 - 1)\dot{x}_1 = \epsilon k(x_2 - x_1)...
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Why do two equivalent systems of equations produce a different result?
I've been reading a book about material and energy balance when I faced a confusing problem when solving a system of equations. I came to notice some new concepts (that were already there, but didn't ...
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Is it correct to exclude the nonlinear terms from the finite strains in linear elasticity, if their differentiation is needed in equilibrium equation?
It is clear that infinitesimal strains are obtained from finite strains (e.g. Green-Langragian or Eulerian-Almansi tensor) by removing nonlinear terms which smallness order is greater than that of ...
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Discontinuities in a system of equations solve - Hydraulic ram end stops
I am trying to implement hydraulic ram end stops/limit stops, this being one component within a wider hydraulic model. Each component contributes residual equations such as force equilibrium, mass ...
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Does this system have an analytical solution?
I am dealing with the system
$$\begin{cases}y'(x)=y(x)(1-z(x))\\ z'(x)=y(x)-z(x)\end{cases}$$
Maybe there is an analytical solution?
I write $e^x(z'+z)=e^xy$, so
$$z=e^{-x}\int_0^x e^t y(t) dt,$$
what ...
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Lyapunov Control vs Sliding Mode Control?
I think I have a good understanding of the idea behind each, i.e., Lyapunov control drives a system to an equilibrium and attempts to keep it there while sliding mode control drives a system to a ...
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Nonlinear-System
Solve the following system:
\begin{align*}
4^{-x} + 27^{-y} &= \frac{5}{6} \\
27^y - 4^x &\leq 1 \\
\log_{27}(y) - \log_{4}(x) &\geq \frac{1}{6}
\end{align*}
Source: Romanian National ...
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Find the general analytical solution for logistic growth differential equation inital value problem $\dot x(t) = \lambda[K-x(t)]x(t),\quad x(t_0)=x_0$
I have the following differential equation for logistic growth as an initial value problem:
$$\dot x(t) = \lambda[K-x(t)]x(t),\quad x(t_0)=x_0$$
I also was given the analytical solution to it and need ...
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Homotopy invariance seems to suggest 1 = 0 ? Please help correct my understanding.
I am studying the chapter on Degree theory and homotopy invariance theorem in the context of solving non-linear systems of equations from Iterative Solution of Nonlinear Equations in Several ...
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Generalized method of moments: necessary condition for identification
Consider a situation where we have a $K\times1$ vector of parameters $\theta$ and a set of restrictions$$E[\psi(X_{i};\theta)]=0$$where function $\psi$ is $M\times1$, with $M>K$, and $X_{i}$ ...
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First order non-linear ODE with quadratic and linear derivatives
I have the following ODE and I am not sure how to solve it. I have manipulated it a bit, but am not sure if my approach is the correct thing to do.
$$
0 = z^c-2(c-1)[1-z]y(z) + [2(1-z)-\mu]y'(z)-[\mu ...
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Finding fixed points of a specific constrained nonlinear PDE
I'm looking for attracting fixed points of the following differential equation in a vector $F$ of length $n$, and $M$ is a known $n \times n$ square matrix:
\begin{align}
\frac{dF}{dt} = I(F), \...
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Analytical solution to a nonlinear ODE $\dot x = cy(1-xy) - \beta cx$ and $\dot y = xc(1-xy)$
Is it possible to find an analytical solution to the following system of ODEs: $\dot x = cy(1-xy) - bcx$ and $\dot y = cx(1-xy)$? I can find fixed points and do stability analysis but I am looking for ...
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Proper linearization of ODEs of the form $\dot{x}(t) + f(x(t)) + \sigma(t) = 0$?
For a scalar ODE of the form $$\dot{x}(t) + f\left(x(t)\right) = 0 \label{1}\tag{1}$$ where $f \colon \mathbb R \to \mathbb R$ is some smooth function admitting a unique root $x^*$ such that $f(x^*) = ...
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How large the error ball of $\|x\| $is, when using the Lyapunov function $x^\top Px$?
I would like to know how large the error ball of $\|x\|$ is when using the Lyapunov function $x^\top Px$:
Assumption: I have an almost linear closed-loop system $\dot{x}=(A-BK)x+\epsilon(x)$ with ...
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When does higher-dimensional Newton-Raphson converge?
If we know objective function $f:\mathbb{R}\to\mathbb{R}$ is concave-up, decreasing, and has a solution $x^*$ on interval $I$, (or equivalently, $f$ is concave-down, increasing, and has a solution $x^{...
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Why is this equation non-linear
I came across this in my textbook. The author says this equation is nonlinear
$$m\frac{{{d^2}}}{{d{t^2}}}x = - {F_0}$$ for x<0
$$m\frac{{{d^2}}}{{d{t^2}}}x = {F_0}$$ for x>0
$$m\frac{{{d^2}}}{{...
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Spot redundant equations within nonlinear system of equations.
Is there a general procedure to detect if in a system of m-nonlinear equations (also non polynomial) of n-unknowns some of the equations are redundant?
Can the rank of the Jacobian matrix tell me ...
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Growth assumptions on nonlinear term in PDE
In Partial Differential Equations book by Evans they treat a nonlinear system of reaction-diffusion equations. The nonlinearity comes from the reaction term $f$
\begin{align*}
& \partial_t u - \...
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Non-standard finite difference for a reaction diffusion system
I want to discretize the following reaction diffusion system:
$\frac{\partial u(x,y,t)}{dt}=\nabla ^2u+ u(1-u)-\frac{uv}{u+\alpha v}$,
$\frac{\partial v(x,y,t)}{dt}=d\nabla ^2v+ \delta v\left(\beta-\...
- 3,795
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Find a Lyapunov function and prove that an almost linear closed-loop system is stable
I would like to find a Lyapunov function and prove the following closed-loop system is stable:
$\dot{x}=(A-BK)x+(z(u)+\epsilon)$,
where a function of control input $z(u)$ satisfies $\|z\| < \rho \|...
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Solving a quadratic vector equation
I'm interested in the vector equation
$$
a-Bx-2Cx-\left[\begin{array}{c}
x'C^{-1}A_{1}C^{-1}x\\
\vdots\\
x'C^{-1}A_{n}C^{-1}x
\end{array}\right]=0
$$
where $x$ is an unknown $n\times 1$ column vector, ...
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Solving $\partial_t \gamma_t(x) = - \gamma_t(x) + \frac{\gamma_t''(x)}{\gamma_t'(x)^2}$, a nonlinear PDE on quantile functions
While pondering Wasserstein-2 gradient flows of the Kullback-Leibler divergence functional $\text{KL}(\cdot \mid \nu)$, where $\nu \sim \mathcal N(0, 1)$ is the standard normal distribution (yes, I ...
- 4,683
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Critical Points of non linear differential equation
In this system of non-linear differential equation:
$$
x' = (x-y)(x+y) \\
y' = (2+x)(5+y)
$$
I'm getting four critical points $(-2,-2)$, $(-5,-5)$, $(-2,2)$ and $(5,-5)$ instead of three as suggested ...
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answer
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Real-Valued Error Function on SO(3)
In some geometric control papers, the author usually defines the real-valued error function to be:
$\Psi(R,R_d)$ = $\frac{1}{2} Trace[I - R_d^T R ]$. (1)
where $R_d$ is the arbitrary smooth attitude ...
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Globally exponentially stable point
consider this linear, non-autonomic system:
$x_1 ̇=-x_1-f(t)(x_2-x_3 )$,
$\ x_2 ̇= -x_2+x_1$,
$x_3 ̇=-x_3-x_1$
where $f(t)$ is continuously differentiable and satisfies
$0≤f'(t)≤f(t)≤k$ for all $0≤t ...
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answers
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Solve a system of 2 nonlinear equations with unknows inside integral expressions
I am trying to solve the following system of nonlinear equations
$$ N(x,y) = (b-x) \cdot y - \int_0^x f(\zeta(\bar{x},y)) \, d\bar{x} = A $$
$$ M(x,y) = C \cdot (b - x) \cdot y - \left( D - \frac{\...
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47
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non-linear system of ordinary differential equation
I have the following system of equations and aim to solve it analytically. My main problem is MATLAB code or function(s) to solve it. Any one can help me?
($$a_i, b_i, c_i$$ and $$d_i$$ are known ...
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29
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MATLAB for nonlinear algebraic equations solvers or numerical methods
Which gives much better results while solving system of non linear algebraic equations, fsolve or any other numerical method for example Newton's method? Can you suggest other numerical methods for ...
1
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0
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45
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Does this system evolve linearly?
Say I have discrete-time nonlinear dynamics:
$$ \mathbf{x}_{t+1}=f(\mathbf{x}_t) $$
where $\mathbf{x}\in\mathbb{R}^d$.
I have an observable function, as an example, defined as:
$$ g(\mathbf{x}) = \...
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1
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How to get the integral of the nonlinear diffusion equation [closed]
The nonlinear diffusion equation is
$$\frac{1}{r}\frac{\partial}{\partial r}(rh^n \frac{\partial h}{\partial r})=\frac{\partial h}{\partial t} \ (1)$$
where $h=h(r,t)$, $r\in[0,r_f]$, $t\geq0$. $r_f$ ...
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0
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15
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How to compute the condition number for non-linear systems?
I've been studying numerical methods for solving systems of equations, and I came across the concept of the condition number for linear systems. I understand that the condition number is used to ...
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Dynamic system, bifurcation and equilibrium points
I'm a bit stuck in solving the following exercise: consider the dynamical system
$$\dot{x} = 2 + 3\mu x - x^3$$
I have to find the equilibrium points and the stability type, and the the bifurcations ...
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Is state space representation useful for nonlinear control systems?
I understand that the state space representation is mathematically equivalent to the transfer function representation for linear systems, and that it allows us to solve the corresponding DE by finding ...
