Criticality governs response dynamics and entrainment of periodically forced ghost cycles

Many natural and engineered systems display oscillations that are characterized by multiple timescales. Typically, such systems are described as slow-fast systems, where the slow dynamics result from a hyperbolic slow manifold that guides the movement of the system's trajectories. Recently, we have provided an alternative description in which the slow timescale results from Lyapunov-unstable transient dynamics of connected dynamical ghosts that form a closed orbit termed ghost cycle. Here, we investigate the response properties of both types of systems to external forcing. Using the classical Van der Pol oscillator and modified versions of this model that correspond to a one-ghost and a two-ghost cycle, respectively, we find significant differences in the responses of slow-fast systems and ghost cycles, including increased entrainment regions of the latter. Nonautonomous model analysis reveals that the differences stem from a continuous remodeling of the attractor landscape of the ghost cycle models, enabled by being organized close to saddle-node on invariant cycle bifurcations, in contrast to a qualitatively unaltered attractor landscape of the slow-fast system. We further demonstrate that the observed features occur in various systems with ghost cycles regardless of the exact mathematical model formulation leading to those ghost cycles, making them likely to apply to many other models with ghost cycles across different disciplines and contexts. We thus propose that systems containing ghost cycles display increased flexibility and responsiveness to continuous environmental changes.