We investigate the impact of network heterogeneity on synergistic contagion dynamics. By extending a synergistic contagion model to diverse heterogeneous network topologies, we uncover the emergence of novel dynamical regimes characterized by multiple stable states, separated by a rich set of bifurcations reaching up to codimension 4. Additionally, we demonstrate how synergy fundamentally reshapes the influence of nodes based on their degree. Unlike in nonsynergistic epidemics, low-degree nodes can play a pivotal role in enabling network invasion at the onset of spread, while high-degree nodes can trigger explosive contagion. These findings challenge conventional control strategies, highlighting the need for new approaches to enhance or suppress synergistic contagion.

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• Bifurcations
• Collective behavior in networks
• Collective dynamics
• Network phase transitions
• Spreading