I'm having a difficult time understanding these terms when dealing with dynamical systems (e.g. visualizing what these terms mean for a dynamical system). The dynamical systems I'm dealing with involve heteroclinic networks. Some terminology from this paper I'm reading:
Definition 1: A heteroclinic cycle is a collection of equilibria $\left\lbrace Q_1,\cdots,Q_n \right\rbrace$ of some ordinary differential equations (let's suppose it's $\dot{x}_i = f_i(x_1,\cdots,x_n)$), together with a set of heteroclinic connections $\left\lbrace \gamma_1(t),\cdots,\gamma_n(t) \right\rbrace$ where $\gamma_j \rightarrow Q_j$ as $t \rightarrow -\infty$ and $\gamma_j \rightarrow Q_{j+1}$ as $t \rightarrow \infty$, and where $Q_{n+1} \equiv Q_1$.
Definition 2: Let $C_1, \cdots, C_m$ for $m\geq 2$ be a collection of heteroclinic cycles. We say that $N = \bigcup_{i=1}^n C_i$ forms a heteroclinic network if there do not exist networks (or cycles) $N_1$ and $N_2$ such that $N= N_1 \cup N_2$ where $N_1 \cap N_2 = \emptyset$
Definition 3: We call a flow invariant set $X$ essentially asymptotically stable (e.a.s) if there exists a set $C$ such that given any number $a \in (0,1)$ and any neighborhood $U$ of $X$, there is an open neighborhood $V \subset U$ of $X$ such that:
- all trajectories starting in $V \setminus C$ remain in $U$ and are asymptotic to $X$
- $\mu(V \setminus C)/\mu(V) >a$ where $\mu$ is Lebesgue measure
Definition 1 makes sense to me. Definition 2 basically says (to my understanding) nothing is disjoint and that there's always a way to essentially trace a line through heteroclinic connections to get from one equilibria to another. Definition $3$ to me is confusing. I don't understanding how to think of this definition conceptually in terms of a dynamical system. The paper says that e.a.s can be thought of as "not asymptotically stable but which attracts almost all trajectories that start nearby".
I don't understand how the measure-theoretic definition correlates with that interpretation, nor am I understanding the difference between asymptotically stable and e.a.s. I'm not experience in topology which might be why I'm struggling to wrap my head around this. One other quote I don't understand: "One necessary condition for a cycle to be asymptotically stable is that $W^u(Q_i)$, the unstable manifold of $Q_i$, lies in $W^s(Q_{i+1})$, the stable manifold of $Q_{i+1}$. All cycles within a heteroclinic network violate this condition so none are asymptotically stable." Why is this condition violated?
Condition 3 is saying that the measure of initial conditions NOT asymptotically stable (this set is called C) is small, since the condition 2 of definition 3 means that the ratio $\dfrac{\mu(V\backslash C)}{\mu(V)}$ can be made as arbitrarily close to 1 (by choosing a as close to 1 as desired). If set C was empty, then X will be asymptotically stable. But definition 3 allows for a nonempty but arbitrary small measure set C, and hence X is essentially asymptotically stable.