Windows of Opportunity for Synchronization in Non-Fast Switching Networks: Analytical Results

Many complex systems in science and engineering operate under on-off coupling between components, often modeled as temporal or evolving networks. A key subclass of these systems, blinking networks, captures on-off switching of connections, typically in the fast-switching regime, where analytical results can be obtained by averaging techniques. While synchronization in such fast-switching networks is well understood, non-fast switching remains analytically intractable despite its potential to produce counterintuitive dynamics. In particular, prior numerical studies have revealed ``windows of opportunity'' in which synchronization emerges at intermediate switching rates, even though it is absent in both the fast and slow switching regimes. In this thesis, we provide the first analytical framework for understanding synchronization in non-fast switching networks of saddle-focus oscillators. Focusing on Rössler networks, we exploit a structural property of the oscillator class to reduce the variational equations to a linear system with constant coefficients. Using Floquet theory, we identify switching intervals that stabilize synchronization, even when both switching modes are individually unstable. We extend our results to larger networks and piecewise-smooth chaotic systems, revealing how network topology interacts with switching frequency to support stable synchronization.