Articles of nonlinear system

Lyapunov Stability of Non-autonomous Nonlinear Dynamical Systems

Let $\mathbf{F}:X\times\mathbb{R}^{+}\to X$ be a non-autonomous dynamical system, which is governed by $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, t, u)$, viz, \begin{equation} \begin{split} \dot{x}_1 &= x_2 – 314.2 \\ \dot{x}_2 &= 122\sin{x_1}\cos{x_1} – 154.8x_3\sin{x_1} – 0.5x_2 + 201.2 \\ \dot{x}_3 &= 0.64\cos{x_1} – 0.8x_3 + 0.2u \end{split} \end{equation} where $\mathbf{x} = \mathbf{x}(t) = [x_1(t), x_2(t), x_3(t)]^T \in X […]

Why Rotations (in CG) are not linear?

One of my teachers told me that a fundamental problem in Computer Graphics is that the rotations are not linear. The transations and scalar are linear, but not the rotations. He tried to explain to me why rotations were not linear, but I didn’t understand at all. Not to mention my undertanding about the linearity […]

Solving second order nonlinear ODE

Having the following second order ordinary differential equation: $$ \ddot{x} = a \cos(x) $$ where, $a$ is a constant. What’s an approach to solve this kind of equation?

Integer solutions to nonlinear system of equations $(x+1)^2+y^2 = (x+2)^2+z^2$ and $(x+2)^2+z^2 = (x+3)^2+w^2$

Do there exist integers $x,y,z,w$ that satisfy \begin{align*}(x+1)^2+y^2 &= (x+2)^2+z^2\\(x+2)^2+z^2 &= (x+3)^2+w^2?\end{align*} I was thinking about trying to show by contradiction that no such integers exist. The first equation gives $y^2 = 2x+3+z^2$ while the second gives $z^2 = 2x+5+w^2$. How can we find a contradiction from here?

Determining stability of ODE

I’m working on a prey-predator model. I’m using the following system of differential equations for it: \begin{align} x’&=-a_1x+a_2xy+a_3xz\\ y’&=b_1y-b_2xy\\ z’&=c_1z-c_2xz \end{align} Where $a_i, b_i, c_i >0$. One of the stationary points is $P=(\frac{b_1}{b_2},\frac{a_1}{a_2},0)$. Question: How can I determine the stability of this point $P$? Attempt: First I wrote the equation as: \begin{align} \frac{\mathrm d \underline{v}}{\mathrm […]

Cylinder in 3D from five points?

I have five points in 3D space which I want to use to form a cylinder in 3D. I assume that these points are exactly on the cylinder. I know the equation for a cylinder is given by an origin and a normal to define the center axis of the cylinder, along with the cylinder’s […]

On a system of equations with $x^{k} + y^{k} + z^{k}=3$ revisited

This 2014 post asks for triples $x,y,z$ such that $$\begin{aligned}x^{k} + y^{k} + z^{k} &= 3\\x^{k+1} + y^{k+1} + z^{k+1} &= 3\\x^{k+2} + y^{k+2} + z^{k+2} &= 3\end{aligned}\tag1$$ for $k=2012$. Of course, the choice of exponent was related to the year it was posted, and the nature of the $x,y,z$ was undefined. However, I wondered […]

System of non-linear equations.

I have to find all triplets $(x,y,z)$ that satisfy: $$x^{2012} + y^{2012} + z^{2012} = 3\\x^{2013} + y^{2013} + z^{2013} = 3\\x^{2014} + y^{2014} + z^{2014} = 3$$ I’ve found the trivial solution $(1,1,1)$ but I don’t know how to start looking for more… Does this system have an infinite amount of solutions?

The system of equations

Let $a,b,c\in \mathbb{R}$ that $a^{2}+b^{2}+c^{2}=1$. We want to find $x,y,z,w$ in the following equations: $$\begin{align} x^{2}+y^{2}+z^{2}+w^{2}&=1 \tag{1}\\ x^{2}+y^{2}-z^{2}-w^{2}&=a \tag{2}\\ xw+yz&=2b \tag{3}\\ yw-xz&=2c.\tag{4} \end{align}$$

Is there any guaranteed stability margins for Extended Kalman Filter (EKF)?

LQR controllers have guaranteed stability margins, but LQG controllers has not guaranteed stability margins, due to the linear kalman filter. But what will happen if I replace the linear kalman filter with the Extended Kalman Filter(EKF), which is a nonlinear kalman filter? Do I receive guaranteed stability margins then?