# 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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### Understanding the Definition of the Koopman Operator

Consider a continuous time dynamical system $$\dot x(t) = F(x(t)),$$ where $x(t)$ is a coordinate vector of state and the right side of the equation $F$ is a non-linear smooth function. Let the state ...
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### Euler integration solution from system of ODE's - already estimated values

I am currently completing an investigation assignment on modelling the growth of a virus inside of the host. There are 3 ODEs that I am using in the system, all determined by change in t. The ...
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### Input for stopping in Dubin's path dynamics?

This question may seem pretty dumb. But I really want to know. I have the following linear dynamical system for Dubin's path, \begin{align*} \phi &= \begin{bmatrix} ...
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### How to prove Boundedness of two 3D coupled chaotic c=system resulting into 6D system? [closed]

I have a two 3D chaotic system and I couple them to make a 6D system. How could I prove boundedness of the coupled nonlinear differential equation? The system description is like this: xdot(t)=f1(x(t))...
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### Necessity of the hypotheses of Lyapunov asymptotic stability theorem

In my ordinary differential equations course we saw Liapunov's theorem for asymptotic stability. I have a doubt about the necessity of the "negative definite" assumption. The statement we ...
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### Is there a closed-form solution by using talor series approximation?

I wanna find a closed-form solution for $\mu$ from this expression in terms of other variables: $$BK_L\alpha(K_S+(1-\mu)I_0)^{\alpha-1}=\alpha(K_L+\mu I_0)^{\alpha-1}+s$$ Could Taylor approximation ...
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### Does a spiral fixed point going from stable to unstable always indicate Hopf bifurcation?

Does a spiral fixed point going from stable to unstable (with change in some parameter) always indicate Hopf bifurcation? Could it be a homoclinic bifurcation instead? (Homoclinic bifurcation keeps ...
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### First order trigonometric differential equation

How can I proceed to solve the following differential equation? $$\frac{dy}{dt} + a\cos(y) = u, \qquad y(0)=0$$ Where $a$ and $u$ are positive constants. Any help will ...
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### Space and time inversion of PDEs

I am unsure whether I am reversing the $z$ and $t$ dimensions in the following set of PDEs correctly, \begin{align} &\begin{aligned} \partial_{z} \mathcal{E}(z, t) = i \sqrt{d} P(z, t) ...
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### Prerequisite mathematics for nonlinear systems [closed]

I have a background in electrical engineering and linear control systems. I want to learn nonlinear systems. There is a book Nonlinear systems by Hassan K. Khalil. The book has a lot of advanced ...
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### Convergence Check for $2D$ Problem solved Numerically Using Different Methods

I have solved a $2D$ PDE for $u(x,z,t)$ using both COMSOL and Matlab. I want to check the convergence of these two methods and try to compare them to each other. To do this, I ran my Matlab script for ...
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Assume that $$\nabla H_1\times\nabla H_2 \:=\: V\quad\text{and}\quad V \:=\: \big(σy, x(r − z), xy\big)\,.$$ My question: If $V$ is given, is there any way to find out what $\,\nabla H_1\,$ and $\,\... • 11 0 votes 1 answer 52 views ### How many equations are required to solve$n$many variables? If we have consistent system of linear equations, each of which has variables$x_1, x_2, \ldots, x_n$, then we will require having$n$many equations in order to find the one-true single solutions set ... • 385 0 votes 1 answer 43 views ### A question about Comparison Principle in Nonlinear Systems? A question about Comparison Principle For a general system, we have $$V=x^{2}+y^{2}$$ where$x \in \mathbb{R}$and$y \in \mathbb{R}$are two independent states, and$V$is a Lyapunov function. ... 1 vote 2 answers 45 views ### Linear and nonlinear systems I get confused about the difference between linear and nonlinear system. Suppose that we have a linear system $$\label{1} Ax=b \tag{1}$$ with$A \in \mathbb{R}^{n\times n}$... • 23 0 votes 1 answer 65 views ### Dynamical systems with control input Please I have been trying to write the mathematical formulation of my nonlinear dynamical system for quite some time and I will appreciate any input. ** Problem Description** Assuming, I am traveling ... • 21 0 votes 1 answer 40 views ### Find a point on triangle and interpolated triangle normal which points to specific point in 3D Suppose i have triangle in 3d with vertices$v1, v2, v3 \in R^3$. Each triangle vertex has associated normal vector$ n1, n2, n3 \in R^3, ||n_i|| = 1$. In computer graphics such vectors sometimes ... • 235 0 votes 0 answers 28 views ### How to transform problem to convex optimization problem? I have the following problem $$min \{ x^TAx + 2b^Tx + c : \|x\|_2 <=1 \}$$ where $$A \in R^{n x n}, b \in R^n, c\in R$$ I need to show that this problem can be transformed to convex optimization ... 3 votes 0 answers 72 views ### Show that the image of the given function is convex Define a function$f: [-b,b]^d\mapsto \mathbb{R}^d (d>1)$by $$f(x)=\int_{[-a, a]^d} \frac{u}{1 + \exp(u^\top x)}du$$ Can we show that the image of$f$is a convex set? I did some simulations ... • 1,293 2 votes 2 answers 73 views ### Polynomial$x^2-x-1$exactly divides Polynomial$a_1x^{17}+a_2x^{16}+1$. Calculate$a_1*a_2$Polynomial$x^2-x-1$exactly divides Polynomial$a_1x^{17}+a_2x^{16}+1$. Calculate$a_1*a_2$My initial thought was to substitute root from the quadratic equation into the bigger polynomial but seems ... • 161 0 votes 0 answers 9 views ### Probabilistic bounds on approximation of nonlinear functions via Volterra functionals (and related methods) I'm working on a nonlinear systems identification problem, and as far as I can tell variations on Volterra functionals are the best approach known for this problem - barring deep learning. The latter ... 1 vote 1 answer 41 views ### how to prove$x(k) \leq x^*$? [closed] A is a positive n x n matrix and b is a positive n-vector. suppose the system x(k+1) = Ax(k) + b has an equilibirum point x*. ... 0 votes 0 answers 12 views ### Inference of unknown Vandermonde-Matrix Given some$\{y_m\}_{m=1}^\infty$and$n$what is the minimum$d>0$in order to deduce$x \in \mathbb{R}^n$, where$x$solves: $$y_m = \sum_{j=1}^n x_j^m \quad \forall m \in 1\dots d$$ or in ... • 1 0 votes 0 answers 10 views ### Numerical Stability of Non-Linear, Autonomous, PDEs on a Lie Algebra Background: The dynamics of my system are modeled by a system of PDE's. I adopt some finite difference approximations to simplify into a system of ODE's and want to study its numerical stability to ... 1 vote 0 answers 28 views ### Identifying Non Linear Equations The definition to identify a non linear equation, as mentioned in the book 'M. Lakshmanan and S. Rajasekar, Nonlinear Dynamics: Integrability, Chaos and Patterns (Springer - Verlag, Berlin, 2003)' is ... • 11 1 vote 0 answers 58 views ### Newton's Method for Nonlinear System with Constraints I have a local solution of a dynamical system$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x})$: $$\mathbf{x}(t) = \mathbf{g}(t;\mathbf{A}),$$ where$\bf{f},\bf{g}:\mathbb{R}^n\...
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Consider the following coupled system of differential equations: Our unknowns are the functions $u_1(t)$, $u_2(t)$ and $u_{GV}(t)$ Let's say $k_{GV}$ is a nonlinear function of the form \$k_3=f(u_{GV})...