Heterogeneity enhances dynamic survivability of coupled oscillators

This study extends the concept of survivability from network security to coupled oscillators, introducing dynamic survivability as a novel notion to describe a system's capacity to sustain specific collective dynamics under attacks, which is crucial for ensuring functional integrity. A general analytical framework along with quantitative metrics is established to evaluate this capability. Focusing on synchronization as the key task, we theoretically analyze heterogeneous coupled oscillators under both the all-to-all network and complex networks, and derive closed-form expressions for critical attack cost. Numerical simulations show strong agreement with theoretical predictions, validating the proposed framework. Furthermore, we reveal a counterintuitive principle: increased heterogeneity in dynamical parameters-measured by the standard deviation of the Hopf bifurcation parameter-significantly enhances system survivability against attacks. This finding holds across the all-to-all network, Erdős-Rényi random network, and Barabási-Albert scale-free network, demonstrating generality beyond specific structures. Our work establishes a new analytical framework for dynamic survivability in oscillator networks and suggests that engineering parameter diversity rather than pursuing homogeneity offers a promising pathway for designing robust systems.