D$, there exists a solution and every maximal solution to ${{{\mathcal}H}}_{N}$ is complete and bounded.

Due to the jump map $G$, if the elements of the solution are initially equal (denote this set as ${\mathcal{S}}:= \{\tau \in P_N : \exists i,r \in I, i \neq r, \tau_{i} = \tau_r \}$) it is possible for them to remain equal for all time. Furthermore, it is also possible for solutions to be initialized on the jump set such that one element is at the threshold and another is equal to zero then after the jump they will be equal, e.g. let $\tau_{1} = {\bar{\tau}}$, $\tau_{2} = 0$ then $\tau_{1}^{+} = \tau_{2}^{+} = 0$. We denote this set as ${\mathcal{G}}:= \{\tau \in D\setminus {\mathcal{S}}: \exists i,r \in I, i \neq r, \tau_{i} = 0, \tau_r = {\bar{\tau}}\}$. The next result considers solutions initialized on the set ${\mathcal{X}}:= {\mathcal{S}}\cup {\mathcal{G}}$.

\[lem:defX\] For each $\tau(0,0) \in {\mathcal{X}}$, there exists a solution $\tau$ to ${{{\mathcal}H}}_{N}$ from $\tau(0,0)$ such that, for some $M \in \{0,1\}$, $\tau(t,j) \in {\mathcal{S}}$ for all $t+j \geq M$, $(t,j) \in {\mathop{\rm dom}\nolimits}\tau$.

Consider a solution $\tau$ to the hybrid system ${{{\mathcal}H}}_N$ with initial condition $\tau(0,0) \in \startmodif {\mathcal{S}}\stopmodif$. Due to the flow 