the proposed hybrid system model of the network of interest and several kinds of perturbations on the timer rates. Using the tools presented in [@teel2012hybrid; @Goebel.ea.09.CSM], we analytically characterize the effect of these perturbations on the already established asymptotic stability property of the desynchronization set. In particular, these perturbations capture situations where the agents in the network are heterogeneous due to having differing timer rates, threshold values, and update laws. To verify the analytical results, we simulate networks of impulse-coupled oscillators under several classes of perturbations. Specifically, we show numerical results when perturbations affect the update laws and the timer rates.

The remainder of this paper is organized as follows. Section \[sec:hs\] is devoted to hybrid modeling of networks of impulse-coupled oscillators. Section \[sec:AN\] introduces an algorithm to determine the desynchronization set. Section \[sec:lyapunov\] presents the stability results while the time to convergence is characterized in Section \[sec:timetoconverge\]. The robustness results are in Section \[sec:robustness\]. Section \[sec:numerics\] presents numerical results illustrating our results. Final remarks are given in Section \[sec:conclusion\].

[**Notation**]{}

Hybrid System Model of Impulse-Coupled Oscillators {#sec:hs}
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Mathematical Model {#sec:hybridmodel}
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In this paper, we consider a model of $N$ impulse-coupled oscillators. Each impulse-coupled oscillator has a continuous 