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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How to linearize a state space equation with higher order $>2$?

Let us consider the following nonlinear polynomial system $$\dot{x} = f(x,u),$$ where $x=[x_1, ... , x_n]$. A Taylor expansion about $(x_0,u_0)$ gives $$f(x,u) = f(x_0,u_0) + \frac{\partial f}{\...
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linearization of non-linear ODE

We have a non-linear ODE of the form $ \dot x= f(x, h(x))$ where $g(x, h(x))=0$, $h: X\subseteq\mathbb R^n\to \mathbb R^m$, $g:\mathbb R^n\times \mathbb R^m\to \mathbb R^m$, $f:\mathbb R^n\times \...
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Upper bound on stepsize for guaranteeing convergence of forward euler method applied to nonlinear ODEs?

Consider the autonomous $d$-dimensional ODE given by $$ \dot{x}(t)=f(x(t)), \quad t\in [0,T], \quad x(0) = x_0 \in \mathbb{R}^d, $$ where $f$ is, in general, nonlinear. Consider now a discretized grid ...
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How do I solve this nonlinear ODE given the asymptotic series solutions as follows?

Differential Equation: $$-{\frac { \left( {\frac {\rm d}{{\rm d}R}}f \left( R \right) \right) ^{2}}{2\,f \left( R \right) }}+{\frac {{\rm d}^{2}}{{\rm d}{R}^{2}}}f \left( R \right) +{\frac {{\frac {...
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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? \begin{equation} \frac{dy}{dt} + a\cos(y) = u, \qquad y(0)=0 \end{equation} 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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Can I (almost) always transform a nonliear system of ODEs to a conditionally linear one, at least numerically?

There are some methods for solving a class of non-linear systems, so-called conditionally linear which posit the following form: $$\dot y_i = a(y_{\sim i}) y_i +b(y_{\sim i})\hspace{1cm} \forall i={1,....
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PSD of a stochastic Duffin - van der Pol equation

I consider a Van der Pol-like equation that reads $$ \ddot{x}(t)+(1+\alpha x(t)^2)\dot{x}(t) +\beta x(t) = \xi(t) $$, in which $\xi$ is a white noise gaussian like process and both $\alpha$ and $\beta$...
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N lousy shooters in a gunfight

$N$ players are in a gunfight. Starting from player 1, each player takes turns to act in the order of $1,2,...,N,1,2,...$ In their turn, a player randomly chooses one of the other remaining players as ...
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ODE on Circle and torus

How can we define an ODE on unit Circle $S^1=x^2+y^2=1$ and torus $S^1*S^1$? Every ODE on circle must satisfies $x \dot{x}+y \dot{y}=0$. Therefor $\dot{x}=-y , \dot{y}=x$ is a system on Circle. but ...
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2 votes
1 answer
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Solve this system of coupled non-linear equations

I have the following two equations with complex $\alpha, \beta$: $$0 = -i \omega_m \beta - ig|\alpha|^2 - \frac{\gamma}{2}\beta$$ $$0 = \alpha(i\Delta - \frac{\kappa}{2}) - i\epsilon - ig\alpha(\beta +...
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If steady states of a dynamic system exist only as limits, are they actually steady states?

I have a nonlinear dynamic model in discrete time. A simplified version of my dynamic system is: \begin{equation} x_{t+1} = \frac{1}{1 + \exp(f(x_t))} \end{equation} where $$f(x_t) = −\beta \left(2d \...
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Are these steady states of non linear dynamic system actually steady states?

I have the following non linear dynamic system in discrete time: \begin{equation} x_{t+1} = \frac{1}{1 + \exp\left(- \beta \left( 2 d \left(c + \frac{(1 - c)}{1 + a (1 - x_{t}) d}\right) - b - d \...
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Sufficient conditions for the uniqueness of the N-system of nonlinear equations

Consider $N$ nonlinear equations for $N$-dimensional vector ${x}$: $F({x})=0$. $F:\mathbb{R}_{++}^{N}\rightarrow \mathbb{R}^{N}$ is continuous and differentiable, and specifically, following form: \...
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I think there is a significant issue with Strogatz's working definition of attractors.

In Strogatz's book, Nonlinear Dynamics and Chaos, he gives the following working definition of attractors (page 690 of second edition): "More precisely, we define an attractor to be a closed set $...
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Feedback linearization with integral action - How?

Assume that you know sort of the dynamics of the system. It's not 100% perfect, but it's at least 90% perfect. $$\dot x = f(x, u)$$ I want to find a control law that suits this system. I have been ...
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A nonlinear system of 1st order ODEs

I have the following nonlinear system of 1st order ODEs: $e^p\dfrac{dy}{dp}=k(\cosh{w})^{2n}$ and $\dfrac{1}{y+1}\left(\dfrac{dy}{dp}+y\right)=\left(2(n+1)\dfrac{dw}{dp}\right)^2$ where $k$ is a non-...
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Determine the class of this system of partial differential equations

Suppose $x\in\mathbb{R}^n$, $V(x)$ and $\phi(x)$ are maps from $\mathbb{R}^n\rightarrow \mathbb{R}$, in which $\phi(x)$ is an auxiliary function. Could anyone help me to classify this system of ...
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Is it possible to obtain the solution to a networked ode system according to the solutions to the subsystems?

Given a networked ode system $\frac{dx_i}{dt}=f(x_i)+\sum_{j=1}^N A_{ij} g(x_i,x_j),\forall i\in\{1,\dots,N\}$ I am wondering whether the solution to this system can be somehow aggregated from the ...
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2 answers
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How to solve the system of nonlinear equations in N-R method or other numerical methods?

Consider the system of infinite series \begin{align} &F=x+\frac{y^{3^2}}{3}+\frac{x^{3^5}}{3^2}+\frac{y^{3^7}}{3^3}+\frac{x^{3^{10}}}{3^4}+\frac{y^{3^{12}}}{3^5}+\cdots=0 \\ &G=y+\frac{x^{3^3}}...
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Period doubling on Möbius strip

I have a hard time understanding a simple argument for the following. Quite often I find in papers that limit cycles before and after period-doubling bifurcations lie on the Möbius strip in the state-...
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Solving the equation system $\{x+z=-6,y+zx=-1,yz=30\}$?

I am interested in solving the following system of non-linear equations. I am looking for $$x,y,z \in \mathbb{C}$$ with $$x+z = -6$$ $$y+zx=-1$$ $$yz=30$$ In fact I am actually interested in just ...
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Adaptive step size for Euler Method - How to create?

I think Euler's Method is a great way to simulate ODE:s. It's not the most accurate, but it's the fastest and simplest. Euler's Method is usaly used with fixed step size, where ...
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How to solve the Ordinary differential equation with denominator having functional variable

How to solve the Second order Ordinary differential equation with denominator having functional variable x''(y)+x'(y)-(x(y)-z(y))/(x(y)+z(y))=0
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How to solve a system of pdes coupled in their boundary condition

Initially i had the following system of pdes to solve: $$\frac{\partial A}{\partial x} =\frac{1}{v_A(y)}.\frac{1}{Pe_A} .\frac{\partial^2A}{\partial y^2} \qquad for \quad 0\leq y\leq \alpha$$ $$\frac{...
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If the cross product result is known, then how to calculate the factors vector $A$ and vector $B\,$?

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 $\,\...
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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 ...
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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. ...
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1 vote
2 answers
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Linear and nonlinear systems

I get confused about the difference between linear and nonlinear system. Suppose that we have a linear system \begin{equation}\label{1} Ax=b \tag{1} \end{equation} with $A \in \mathbb{R}^{n\times n}$ ...
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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 ...
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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 ...
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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 ...
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3 votes
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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 ...
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2 answers
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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 ...
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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 ...
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1 answer
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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*. ...
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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 ...
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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 ...
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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 ...
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Newton's Method for Nonlinear System with Constraints

I have a local solution of a dynamical system $\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x})$: \begin{equation} \mathbf{x}(t) = \mathbf{g}(t;\mathbf{A}), \end{equation} where $\bf{f},\bf{g}:\mathbb{R}^n\...
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1 vote
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Why is this nonlinear system of differential equations giving me the exact same results using a coupled and uncoupled analysis?

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})...
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