# 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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### 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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### 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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### 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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### 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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### 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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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: $\... 1answer 28 views ### Derivative of Input in nonlinear State Space representation I am dealing with obtaining an space state representation of a nonlinear differential equation that arises from an inverted pendulum. It includes some terms that reflect the fact pendulum is ... 0answers 16 views ### ODE with complex harmonic oscillations and phasors We have a differential equation of the following form:$\frac{dx}{dt}=f(x) \cdot a \cdot sin(\omega \cdot t)-a \cdot sin(\omega \cdot t + \frac{\pi}{2})$where$f(x)$- arbitrary function from state ... 1answer 73 views ### nonlinear systems of differential equations stable points with complex coordinates I am trying to solve this system of differential equations. \begin{cases} x_1'= -x_1+2x_1^3+x_2,\\ x_2'= x_1+x_1x_2. \end{cases} *by abusing the notations, I assumed$x(t) = x$Among fixed points, ... 0answers 13 views ### 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$y\...
I'm solving a 3$\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 ...