state is in the jump set $D$ defined as $$D := \left\{ \tau \in P_N : \ \exists i \in I \ \mbox{s.t.} \ \tau_i = {\bar{\tau}}\right\} . 
\label{eqn:D}$$ From such points, the $i$-th timer is reset to zero and forces a jump of all other timers. Such discrete dynamics are captured by the following jump map: for each $\tau \in D$ define $
G(\tau) = \left[ g_1(\tau) \ \ g_2(\tau)\ \ \ldots \ \ g_N(\tau)  \right]^\top,
$ where, for each $i \in I$, $$g_i (\tau) = \left\{ \begin{array}{l} 0 \qquad \qquad \ \ \ \ \mbox{if } \tau_{i} = {\bar{\tau}}, \tau_r < {\bar{\tau}}\ \ \forall r \in I\setminus \{i\} \\ 
\{0 , \tau_{i}(1+\varepsilon) \} \ \mbox{if } \tau_{i} = {\bar{\tau}}\ \exists r \in I \setminus \{i\} \  \mbox{s.t.} \  \tau_r = {\bar{\tau}}\\ 
(1+\varepsilon)\tau_{i} \qquad \  \mbox{if } \tau_i < {\bar{\tau}}\ \exists r \in I \setminus \{i\} \  \mbox{s.t.} \  \tau_r = {\bar{\tau}}\end{array} \right. \label{eqn:gi}$$ with parameters $\varepsilon \in (-1,0)$ and ${\bar{\tau}}> 0$; for $\tau \in D$, $g_i$ is not empty. When a jump is triggered, the state $\tau_i$ jumps according to the $i$-th component of the jump map $g_i$. When a state reaches the threshold ${\bar{\tau}}$, it is reset 