be

-   [*stable*]{} if for each $\varepsilon>0$ there exists $\delta>0$ such that each solution $x$ with $|x(0,0)|_{{\mathcal{A}}}\leq \delta$ satisfies $|x(t,j)|_{{\mathcal{A}}} \leq \varepsilon$ for all $(t,j)\in{\mathop{\rm dom}\nolimits}x$;

-   [*attractive*]{} if there exists $\mu > 0$ such that every maximal solution $x$ with $|x(0,0)|_{{\mathcal{A}}}\leq \mu$ is complete and satisfies\
    $\lim_{(t,j) \in {\mathop{\rm dom}\nolimits}x, t+j\to\infty} |x(t,j)|_{{\mathcal{A}}}=0$;

-   [*asymptotically stable*]{} if stable and attractive;

-   [*weakly globally asymptotically stable*]{} if ${\mathcal{A}}$ is stable and if, for every initial condition, there exists a maximal solution that is complete and satisfies $\lim_{(t,j) \in {\mathop{\rm dom}\nolimits}x, t+j\to\infty} |x(t,j)|_{{\mathcal{A}}}=0$.

The set of points from where the attractivity property holds is the basin of attraction and excludes all points where the system trajectories may never converge to ${\mathcal{A}}$. In fact, it will be established in Section \[sec:lyapunov\] that the basin of attraction for asymptotic stability of desynchronization of ${{{\mathcal}H}}_N$ does not include any point $\tau$ such that any two or more timers are equal or become equal after a jump, which is the set ${\mathcal{X}}$ defined in Lemma \[lem:defX\].

Construction of the set ${\mathcal{A}}$ for ${{{\mathcal}H}}_N$ {#sec:AN}
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In this section, we identify the set of points corresponding to the impulse-coupled oscillators being desynchronized, namely, we define the [*desynchronization 