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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solve equations having term $ xy$ [closed]

I want to solve equations: $$x^3-3xy^2=-11$$ and $$y^3-3x^2y=-2$$ for $x$ and $y$. How can I do it?
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A tricky nonlinear pair of equations

For example, a question like this. Solve $$ \begin{cases} \frac1x+\frac1{2y}&=(x^2 + 3y^2)(3x^2 + y^2) \\ \frac1x-\frac1{2y}&=2(y^4 - x^4) \\ \end{cases} $$ I couldn't see any way to approach ...
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State propagation with uncertain control input

Consider a nonlinear system $x(k+1)=f(x(k),u(k))$, where $x(k)\in\mathbb{R}^{n}$ is the state, $u(k)\in\mathbb{R}^m$ is the control input. Here $u(k)$ is normally distributed RV with mean $\mu_u(k)$ ...
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19 views

Exponential Bound on Nonlinear Dynamic System

Consider the nonlinear system described by \begin{equation*} \dot{z_1}=-z_1, \quad \dot{z_2}=z_1^2+2z_1\gamma-z_2, \end{equation*} where $\gamma\in\Gamma=[\gamma_{min},\gamma_{max}]\subset(0,\infty)$ ...
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What is meant by $f : [a,b] \times D \rightarrow R^m$

I am working through Nonlinear Systems by Khalil. Lemma 3.1 states the following Let $f : [a,b] \times D \rightarrow R^m$ be continuous for some domain $D \subset R^n$. Suppose that $[\partial f/\...
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1answer
25 views

How to simplify conditions for conjugations on $\Bbb C^2$

Is there a way to simplify the set of equations\begin{align*} \lvert c_{11}\rvert^2 +\overline{c_{12}}c_{21} & = 1\\ \overline{c_{11}}c_{12} +\overline{c_{12}}c_{22} & = 0\\ \overline{c_{21}}...
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Dimension of solution space to nonlinear system of equations

My lecture notes leading up to the implicit function theorem state the following: We investigate underdetermined systems of equations $f(x) = b$ where $f$ : $\mathbb{R}^{n+k} \to \mathbb{R}^k$ and $b ...
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27 views

Banach fixed-point theorem: Prove that given non-linear system has exactly one solution

Question: For $1\leq i, k \leq n$ you are given some real numbers $b_i$ and $c_{ik}$ so that: $$\sum_{i,k=1}^{n} c^2_{ik} < 1$$ Show, using Banach fixed-point theorem, that the following non-linear ...
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How to find the roots of a non-linear equation?

I am trying to solve the following system of non-linear equations: $$\left(\frac{M^2(x)}{2}+3\right)a^2(x)=g(x)$$ $$M(x)=\frac{p}{a^8(x)f(x)}$$ where $M(x),a(x)$ are the variables and $p$ is a ...
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Consistency of the system of quadratic equations

How to find the conditions for coefficients $a_1,a_2,b_1,b_2,k,d,c,k,s,w,m_1,m_2,m_3,e_1, e_2 \in \mathbb{R}$ when two following systems in $x,y,z \in \mathbb{R}$ are consistent? I tried to solve ...
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1answer
28 views

Determine the parameter which minimizes the index performance

Please help me solve this. If the system is with an initial condition , determine the parameter which minimizes the performance index
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Problem understanding bifurcations

I am beginning to study bifurcations, and I have some preoblems understanding some concepts. I have understood that a Bifurcation can be defined has the change of behaviour of a dynamical system as a ...
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1answer
28 views

Determine stability of the equilibrium state

Please help me. I am struggling to determine the stability of the equilibrium state $x = 0$ of the system $$x_1' = x_1(x_1^2 + x_2^2 - \beta^2) + x_2 \\ x_2' = x_2(x_1^2 + x_2^2 - \beta^2) - x_1$$ ...
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Finding $y(x)$ using a minimization problem.

I am going to post a problem that I've tried to solve and I feel like I've been banging my head against the wall. Any kind of hint or advice are more than welcome. The Problem: Consider a 3 meter ...
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1answer
44 views

How to solve nonlinear systems of equation with summation

I'm stuck with this $$ \sum _{n=0}^{80}\:0.1+b^{n+k}=100 $$ $$ 0.1+b^{80+k}=10 $$ Is it possible to solve?
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How can I use Lyapunov Control for nonlinear system $\dot x = f (x, u) $

Recently I made my system identification algorithm SINDY to work. Not it can estimate a nonlinear model from measurement data that comes from a very nonlinear hydraulic system. The input signal ...
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Strange ODE and approximate solutions: transforming the coordinate system and representing the solution in the form of a Fourier series

In the last topic, we considered the following differential equation: $\frac{dx}{dt}=f(x) \cdot a \cdot sin(\omega \cdot t)-a \cdot sin(\omega \cdot t + \frac{\pi}{2})$ In this topic we considering: $\...
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Derivative of Input in nonlinear State Space representation

I am dealing with obtaining an space state representation of a nonlinear differential equation that arises from an inverted pendulum. It includes some terms that reflect the fact pendulum is ...
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ODE with complex harmonic oscillations and phasors

We have a differential equation of the following form: $\frac{dx}{dt}=f(x) \cdot a \cdot sin(\omega \cdot t)-a \cdot sin(\omega \cdot t + \frac{\pi}{2})$ where $f(x)$ - arbitrary function from state ...
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nonlinear systems of differential equations stable points with complex coordinates

I am trying to solve this system of differential equations. \begin{cases} x_1'= -x_1+2x_1^3+x_2,\\ x_2'= x_1+x_1x_2. \end{cases} *by abusing the notations, I assumed $x(t) = x$ Among fixed points, ...
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Lyapunov Function and Comparison Lemma

I am studying the book Nonlinear Systems from H.K. Khalil and I am stuck with a proof. I have a Lyapunov function V that satisfies the condition $$c_1\|y\|^2\leq V(t,x,y)\leq c_2\|y\|^2,$$ for all $y\...
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Extreme sensitivity to Initial guess of Non-linear system of Eqs

I'm solving a 3$\times$ 3 system of non-linear equations using Newton's method. The system is quite involved (see the graphic below) and not amenable to the sort of simple substitution and/or ...
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Trying to analytically find number of solutions to a system of nonlinear equations for physics application.

Is there a way to derive an analytical expression for the NUMBER of solutions to the following equation (corresponding to a hypersphere in N-dimensions): $\begin{equation} \displaystyle \sum^N_i P_i^2 ...
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1answer
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Real positive solution to specific system of equations

In the course of an optimization problem, I have encountered the following system of equations for $\alpha, \beta > 0$: $$ \begin{align} a_1 + a_2 + b_1 + b_2 &= \alpha + \beta\\ a_1b_1 + \frac{...
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1answer
51 views

construction Lyapunov function for global asymptotic stable systems

We know that converse Lyapunov theorems for the conditions that a system is golobal asymptotic stable insure diferentiable Lyapunov functions. In the below I have a nonlinear system that is global ...
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35 views

Linearizing a non-linear model

I have a non-linear programming model as below: $x_{ij}^2=k_{ij}(y_i-y_j), x_{ij}\in\mathbb{R}, y_i\in\mathbb{R^+}, y_j\in\mathbb{R^+}, \forall$ $i,j\in {[1,2,...,N]}$ $\sum_{j=1}^N a_{ij}.x_{ij}=b_i$ ...
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1answer
15 views

Show that the orbits are given by ellipses $\omega^{2}x^{2}+v^{2}=C$, where $C$ is any non- negative constant.

I am working through a text book by Strogatz Nonlinear dynamics and chaos . In chapter 5 question 5.1.1 (a). I have answered the question but would like to check if I have performed the integration ...
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33 views

Classifying equilibrium point (0,0) with eigenvalues $\lambda_{1,2}=0,-1$

I am given a nonlinear dynamical system: $\dot x = x^2y-x$ $\dot y = 4xy^2+4y^2-x^2y-xy$ I was able to find all the equilibrium points and classify them, expect $e_0=(0,0)$. I found that the Jacobian ...
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1answer
16 views

How to determine whether given nonlinear equation system cannot be solved analytically?

I am currently studying nonlinear equations that require numerical analysis methods to solve them. But I could not understand why can't I solve some equations analytically? For example: x^2 + 4y^2 - ...
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1answer
23 views

How to choose an initial value for a multidimensional equation while using Newton-Raphson method to solve it?

I am trying to solve the following nonlinear equation for $f \in \mathbb{R}^3$ using the Newton-Raphson method, $$G(f) = {\sin{(||f||)} \over ||f|| }Jf + {1-\cos(||f||) \over ||f||^2} f \times Jf = g$$...
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Change inlet boundary condition of non-linear system

This is a problem in modeling in hydraulic fracturing field. It's quite long so hopefully someone can patiently read and help me. The equation numbers are match those of the reference paper by Adachi,...
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27 views

Real Solutions for an Underdetermined Polynomial System

In dealing with some quantum-chemistry issues I have to face the following problem: *Find $N$ distinct real numbers in the interval $[0,1)$ such that: their average is $1/3$ the average of their ...
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28 views

Method to solve a system of nonlinear equations

Suppose that I have the following system, $$ \begin{equation} \left\{\begin{array}{lcl} a + b + c + d & = & \alpha \\ a^2 + b^2 + c^2 + d^2 & = & \beta \\ a^3 + b^3 + ...
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1answer
34 views

Newton-Coulomb differential equation solution

I am trying to solve the following DE: $$ m \mathbf{\ddot{r}}=\frac{Q_1Q_2}{4\pi\epsilon_0\mathbf{r}\cdot\mathbf{r}}\mathbf{\hat{r}} $$ Which I know is a tough one, so I settled for the simpler ...
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1answer
59 views

Best way to numerically solve a nonlinear system $f(x)=0$, $f:\mathbb{R}^n\to\mathbb{R}^m$

I am aware that that there are a lot of methods known to solving a nonlinear system $\mathbf{f}(\mathbf{x})=0$, if $\mathbf{f}:\mathbb{R}^n\to\mathbb{R}^n$, assuming the Jacobian is non-singular. ...
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1answer
79 views

Global Asymptotic Stability of a System

I have a system $V(x)$, in $R^2$, and I've calculated that $V(x) \geq 0$ for all $x$ not equal to zero and that $V(0,0) = 0$ I've also calculated that $V'(x) \leq 0$ Since $V'(x)$ is NSD and not ...
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0answers
30 views

Linear stability analysis of a dynamical systems based only on symmetric part of Jacobian

Consider an autonomous dynamical system: $$\frac{dx}{dt} = f(x)$$ where $x\in\mathbb R^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$. Suppose 0 is a stationary point, $f(0)=0$. We Taylor expand to ...
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19 views

Is there a general procedure to solve/analyze systems of non-linear equations?

Consider a system of nonlinear equations, $$ x_i = f(x_{/i}) \, \\i \in \{1,...n\}$$ where one is looking for points $x^* = (x_1,...,x_n)$ that fulfill all the given requirements. I've seen some ...
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26 views

Charpit's method for a system of nonlinear partial differential equations

I have a highly nonlinear set of coupled PDEs. I know I will eventually need to solve them numerically, but first I need to understand how equations of this type should be solved. Essentially, I have ...
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15 views

Existence of bounded derivative inverse (Deuflhard Exercise 1.1)

The following exercise is from Deuflhard's "Newton Methods for Nonlinear Problems" : Given a nonlinear $C^1$-mapping $F:X\to Y$ over some domain $D\subset X$ for Banach spaces $X$, $Y$, each endowed ...
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Approach for non linear ODE with quadratic derivative.

What is the approach to solve the following nonlinear ODE: $$\ddot{x}(t) + a(\dot{x}(t))^2+b\dot{x}(t)+kx(t) = 0$$ where $a$ is positive when $\dot{x}<0$ and negative otherwise. I am trying to ...
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0answers
15 views

Shooting Method for non-linear coupled ODE DAE

I have a 2 coupled non-linear ODEs that make up a Differential Algebraic Equation. $$V' = -\frac{V}{2\delta}\delta ' - \frac{6}{\delta ^2 } + \frac{6}{V}\Delta T$$ $$\delta ' = \frac{6}{\delta V}-\...
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28 views

Plotting Phase Portrait for Nonlinear Damped Pendulum for larger damping

I'm asked to sketch the phase diagram near the equilibrium points of the nonlinear damped equation: $x ^ { \prime \prime } + k x ^ { \prime } + \sin x = 0$. I've found that for any integer $n$, $( n \...
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0answers
8 views

Propagating in time a Nonlinear Dynamically Inverted (NDI) System

Background Suppose I have a nonlinear system given by $\dot{x}=f(x)+G(x)u$ $y=Hx$ where $x$ is the state, $y$ is the output, $G$ is a control matrix. This form is identical to how one would ...
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30 views

Find optimal solution for this system of quadratic equations

I would like to find an optimal (e.g. like in a least square manner) solution for the following system of equations (noisy distance measurements between 4 points): $ (\color{green}{x_1}-x_2)^2+(\...
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34 views

Dynamical systems described by coupled nonlinear differential equations

Suppose a dynamical system is described by two variables, $x$ and $y$, and they change over time according to the following two coupled differential equations: \begin{equation} \begin{split} &\...
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3answers
114 views

Solving a nonlinear periodic ODE

Is it possible to solve the below ODE for arbitrary real values of $c$? $$2 (\frac{d \theta}{dx})^2 [\sin(2\theta) -c \cos(2 \theta)]-[\cos(2\theta) +c \sin(2 \theta)]\frac{d^2}{dx^2}\theta=0,$$ where ...
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1answer
23 views

Nonlinear vector calculus problem

Let $A$ be a vector field on $\mathbb{R}^3$. I am interested in finding solutions of $$ \nabla^2 A \times {\rm curl} A = 0,\\ \quad {\rm div} A = 0. $$ Are there any exact solutions with nonzero $\...
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1answer
46 views

A nonlinear system of 3 equations in 3 unknowns

Given that: $$2c-3bt=-358,\\2c+3b+4t=-102,\\-2ct+b=-318$$ find the value of $2(c+bt)$: is it required to find $c,b,$ and $t$ individually? If we must find them separately, then how? By WA, this system ...
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0answers
46 views

Analytic estimates of limit cycle parameters

Suppose we have a two-dimensional system of differential equations, say, the well-known Van der Pol oscillator: $$ \dot{x}=y, \dot{y}=\mu (1-x^2)y-x $$ Everyone knows that the study of limit cycles ...

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