Skip to main content

All Questions

Filter by
Sorted by
Tagged with
2 votes
0 answers
153 views

Stability test for LTV systems by differential Lyapunov inequalities

Consider a linear time-varying system: \begin{equation} \dot x(t) = A(t) x(t), \tag{$*$} \end{equation} where $A(t)$ is a time-varying block matrix defined as $$ A(t) = \begin{bmatrix} 0 & I\\ -\...
Evan's user avatar
  • 55
1 vote
0 answers
157 views

Selecting a suitable Lyapunov function for the following systems?

i) SI MODEL Consider \begin{align} \frac{dS}{dt} &= \mu N -\frac{\beta S I}{N} - \nu S\\[2ex] \frac{dI}{dt} &= \frac{\beta S I}{N} -\nu I \end{align} Where $N=S+I$ is the total population. If ...
Math's user avatar
  • 185
1 vote
0 answers
181 views

Subset of the domain of attraction

Let $x \in R^n$ and $f : R^n \to R^n$, $f\in C^1$ $$ \frac{\mathrm{d}}{\mathrm{d}t} x(t) = f(x(t)) $$ be such that $f(0) = 0$ is asymptotically stable. The domain of attraction is the set of initial ...
SampleTime's user avatar
3 votes
0 answers
82 views

conditions for asymptotic comparison to hold

I have the following simple dynamical system: \begin{align} x_1' &= a - f(x_2)x_1\\ x_2' &= bx_1 - cx_2, \end{align} where all parameters and initial conditions are positive. $f(x_2)$ is a ...
Paichu's user avatar
  • 513
3 votes
1 answer
176 views

Is the convergence of $\dot{x}=2A(t)x$ faster than that of $\dot{x}=A(t)x$?

Let $x \in \mathbb{R}^{n}$ and $A(t) \in \mathbb{R}^{n\times n}$. If $\dot{x}=A(t)x$ and $\dot{x}=cA(t)x$ with $c>1$ are exponentially stable. Is the convergence rate of $x$ to zero of $\dot{x}=cA(...
guluzhu's user avatar
  • 33
2 votes
1 answer
674 views

Center manifold theorem and case of all zero eigenvalues

Is the center manifold theorem applicable if say for a planar (2D) system of non-linear ODE, the stability matrix has both eigenvalues zero? Of course, there is only one eigenvector. If not, what is ...
Subho's user avatar
  • 121
7 votes
1 answer
930 views

(In)stability of a two-dimensional dynamical system

Consider the following system of coupled differential equations \begin{eqnarray*} \dot{x}_1(t) & = & -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\ \dot{x}_2(t)...
Ludwig's user avatar
  • 2,712
2 votes
1 answer
134 views

On local attractivity of a coupled non-linear differential equation

Consider a dynamical system described by the following coupled non-linear differential equation \begin{align} \dot{x}_1(t) &= v + a_{12}\sin(x_2(t)-x_1(t)) + a_{13}\sin(x_3(t)-x_1(t))\\ \dot{x}_2(...
Ludwig's user avatar
  • 2,712
2 votes
0 answers
65 views

Exponential Convergence Under a Lyapunov-Like Assumption

Consider $V=\mathbb{R}^d\times\mathbb{R}^n$ with coordinate $x^T=[\theta^T,\sigma^T]$. I have an ODE of the form: $\dot{x}=F(x)$, where $F$ is assumed to be sufficiently smooth. Suppose that there's ...
Miel Sharf's user avatar
3 votes
1 answer
202 views

Stability of nonsmooth, Lipschitz continuous, autonomous system of differential equations

Consider the following autonomous system of differential equations: $$\frac{\mathrm d\mathbf x}{\mathrm dt} = \mathbf v(\mathbf x)$$ where $\mathbf x, \mathbf v \in \mathbb R^n$. Assume that $\...
valle's user avatar
  • 884
5 votes
2 answers
828 views

Conditions for convergence to non-isolated fixed points

Consider a dynamical system of the form $$ \dot{x}=f(x), \quad x\in X, $$ and assume that the system possesses a set of non-isolated fixed points. Suppose moreover that there exists a Lyapunov $V(x)$ ...
Ludwig's user avatar
  • 2,712
0 votes
1 answer
285 views

Quadratic stability of linear time varying system

(This question was originally asked at Math.SE, where it didn't receive any answers.) Consider the linear time-varying system $$ \dot{x} = A(t) x, $$ where $x \in \mathbb{R}^n$ and $A: [0,+\infty) \...
Tadashi's user avatar
  • 1,590