is, a reordering of the elements of the set $I = \{1,2,\ldots,N\}$) then, after $N$ jumps, it follows that

Let $\tau$ be a solution to ${{{\mathcal}H}}_{N}$ from $P_{N}\setminus {\mathcal{X}}$. There exists a sequence $i_{k}$ of distinct elements with $i_{k} \in I$ for each $k \in I$, such that $0 \leq \tau_{i_{1}}(t,j) < \tau_{i_{2}}(t,j) < \ldots < \tau_{i_{N}}(t,j) \leq {\bar{\tau}}$ over $[t_{0},t_{1}]\times \{0\}$. After the jump at $(t,j) = (t_{1},0)$ we have $0 = \tau_{i_{N}}(t,j+1) < \tau_{i_{1}}(t,j+1) < \tau_{i_{2}}(t,j+1) < \ldots <  \tau_{i_{N-1}}(t,j+1) < {\bar{\tau}}$. Continuing this way for each jump, it follows that after $N-1$ more jumps, the solution is such that $0 \leq \tau_{i_{1}}(t_{N},j+N) < \tau_{i_{2}}(t_{N},j+N) < \ldots < \tau_{i_{N}}(t_{N},j+N) \leq {\bar{\tau}}$ and the order at time $(t,j)$ is preserved.

Using these properties of solutions to ${{{\mathcal}H}}_N$, the next section defines the set to which these solutions converge and establishes its stability properties.

Dynamical Properties of ${{{\mathcal}H}}_{N}$ {#sec:analysis}
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Our goal is to show that the desynchronization configuration [[ of ${{{\mathcal}H}}_N$, which is defined in Section \[sec:AN\]]{}]{}, is asymptotically stable. We recall from [@teel2012hybrid; @Goebel.ea.09.CSM] the following definition of asymptotic stability for general hybrid systems with state $x \in {\mathbb{R}}^n$.

\[def:AS\] A closed set ${\mathcal{A}}\subset {\mathbb{R}}^n$ is said to 