Given the following dynamical system $$\begin{aligned} \dot x &= f(x,y) = -x + \alpha y \\ \dot y &= g(x,y) = -y\end{aligned}$$ for which values of $\alpha$ is the disk $B = \{(x, y)\mid x^2+y^2 \leq 1\}$ positively invariant? Now what I have done is that I have taken the orbital ...

Consider the following differential equations $${dS \over dt} = \lambda-\beta SI-\mu S+\theta I$$ $${dI \over dt} =\beta SI-(\mu +d)I-\theta I$$ In all papers that I have read it is only mentioned that $$\Omega = \left\{ (S,I) : I\geq 0, S \geq 0, S+I \leq {\lambda \over \mu} \right\}$$ is ...

I have just started a dynamical systems course and I am a bit confused as to how to determine if something is positively or negatively invariant, or just invariant. I know the defintions for invariance are as follows: Let $B \subset X$. $B$ is called Positively Invariant if $S_tB\subset B$ f...

Definition: Invariant set A set $S \subseteq \mathbb{M}$ is invariant if $\Psi_{t}\left ( S \right )\subseteq S,\forall t$ -if $\forall x \in S$, $\Psi_{t}\left ( x \right )\subseteq S,\forall t$ -if $\forall x\in S,q\left ( t,x \right )\subseteq S,\forall t$ Definition: ...

Suppose $f := R^n -> R^n$ is continuously differentiable. Given $\phi(t,y)$ is the solution to the IVP: $\dot{x} = f(x), \ x(0) = y$. Assume $x_0$ is an asymptotically stable equilibrium of the system $\dot{x} = f(x)$. Show that $D = \left\{y\in R^n: lim_{t\rightarrow \infty} \phi(t,y) = x_0\righ...

Given the planar dynamical system $$\dot x = x - y - x^3, \\ \dot y = x + y - y^3$$ show this is positively invariant in the annulus $1 \leq x^2 + y^2 \leq 3$. Hints: $$x^4 + y^4 = (x^2 + y^2)^2 − 2x^2y^2$$ $$x^4 + y^4 = \frac 12 (x^2 + y^2)^2 + \frac 12 (x^2 - y^2)^2$$ M...

Posting a possible proposal so that we can discuss in the comments (or at least show by voting) whether this would be the way to go. The tag path-length could remain, and it could be used for questions concerning length of a path in the graph-theoretic sense. This should be clearly explained in ...