set*]{}. We define desynchronization as the behavior in which the separation between all of the timers’ impulses is equal (and nonzero), see Figure \[fig:example\]. More specifically desynchronization is defined as follows:

\[def:desynch\] A solution $\tau$ to ${{{\mathcal}H}}_N$ is desynchronized if there exists $\Delta > 0$ and a sequence of non-repeated elements $\{i_m\}^N_{m = 1}$ of $I$ (that is, a reordering of the elements of the set $I = \{1,2, \ldots, N \}$) such that $\lim_{j \to \infty} (t_j^{i_m} - t_j^{i_{m+1}} )= \Delta$ for all $m \in \{1,2, \ldots, N-1\}$ and $\lim_{j \to \infty} (t_j^{N} - t_j^{i_{1}}) = \Delta,$ where $\{t_j^{i_m}\}_{j = 0}^\infty$ is the sequence of jump times of the state $\tau_{i_m}$.

In fact, this separation between impulses leads to an ordered sequence of impulse times with equal separation. The desynchronization set ${\mathcal{A}}$ for the hybrid system ${{{\mathcal}H}}_N$ captures such a behavior and is parameterized by $\varepsilon$, the threshold ${\bar{\tau}}$, and the number of impulse-coupled oscillators $N$.

To define this set, first we provide some basic intuition about the dynamics of ${{{\mathcal}H}}_N$ when desynchronized. The set ${\mathcal{A}}$ must be forward invariant and such that trajectories staying in it satisfy the property in Definition \[def:desynch\]. Due to the definition of the 