single-valued map $f: {\mathbb{R}}^{N} \to {\mathbb{R}}^{N}$ defining the flows.

-   *Jump set:* a set $D \subset {\mathbb{R}}^{N}$ specifying the points where jumps are possible (or discrete evolution).

-   *Jump map:* a set-valued map $G: {\mathbb{R}}^{N} \rightrightarrows{\mathbb{R}}^{N}$ defining the jumps.

A hybrid system capturing the dynamics of the impulse-coupled oscillators is denoted as ${{{\mathcal}H}}_N := (C,f,D,G)$ and can be written in the compact form $${{{\mathcal}H}}_N: \qquad \tau \in {\mathbb{R}}^{N} \qquad \left\{ \begin{array}{llll} \dot{\tau} &=& f(\tau) &\quad \tau \in C \\ \tau^{+} &\in& G(\tau) & \quad \tau \in D \end{array}\right. , \label{eqn:HS}$$ where $N \in {\mathbb{N}}\setminus \{0,1\}$ is the number of impulse-coupled oscillators. The state of ${{{\mathcal}H}}_{N}$ is given by The flow and jump sets are defined to constrain the evolution of the timers. The flow set is defined by $$\startmodif C := P_N, \stopmodif
\label{eqn:flawiest}$$ where $I := \{1,2, \ldots , N\}$ and ${\bar{\tau}}> 0$ is the threshold. During flows, an internal clock gradually increases based on the homogeneous rate, $\omega$. Then, the flow map is defined as with $\omega > 0$ [[ defining the natural frequency of each impulse-coupled oscillator]{}]{}. The impulsive events are captured by a jump [[ set $D$ and a jump]{}]{} map $G$. Jumps occur when the 