Energy-dynamics interplay in temporal networks triggers explosive synchronization

Synchronization in networks of coupled oscillators is a fundamental problem in the study of collective behavior. In this paper, we investigate the synchronization transition in networks of coupled dynamical systems from an energetic perspective. Interactions between systems/oscillators are assumed to be governed by one of the following mechanisms: (i) the intrinsic energy $\mathbf{H}$, describing the conservative internal dynamics of isolated systems, and (ii) the dissipative energy $\dot{\mathbf{H}}$, accounting for energy losses and exchanges due to interactions and damping. An energetic threshold is introduced to modulate the network connectivity, so that the topology evolves in time according to the instantaneous energetic similarity between systems, allowing us to analyze how the balance between intrinsic and dissipative energy shapes the transition to synchronization. Using the R\"ossler and Lorenz systems as representative examples, while keeping the framework general and applicable to other dynamical systems, we explore three representative dynamical regimes: periodic, multiperiodic, and chaotic. This reveals that, the nature of the synchronization transition strongly depends on the interplay between microscopic dynamics and the mesoscopic connectivity structure. In particular, chaotic oscillators coupled through intrinsic energy favor explosive synchronization, corresponding to a first-order transition, whereas periodic and multiperiodic dynamics lead to smooth second-order transitions. In contrast, dissipative-energy-based connectivity suppresses first-order transitions in chaotic networks but can induce second-order transition in multiperiodic systems.