Ghost cycles exhibit increased entrainment and richer dynamics in response to external forcing compared to slow-fast systems

Abstract

Many natural, living 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 trajectories. Recently, we have provided an alternative description in which the slow dynamics result from a non-hyperbolic and Lyapunov-unstable attracting sets from connected dynamical ghosts that form a closed orbit (termed ghost cycles). Here we investigate the response properties of both type of systems to external forcing. Using the classical Van-der-Pol oscillator and two modified versions of this model that correspond to a 1-ghost and a 2-ghost cycle, respectively, we find that ghost cycles are characterized by significant increase especially in the 1:1 entrainment regions as demonstrated by the corresponding Arnold tongues and exhibit richer dynamics (bursting, chaos) in contrast to the classical slow-fast system. Phase plane analysis reveals that these features result from the continuous remodeling of the attractor landscape of the ghost cycles models characteristic for non-autonomous systems, whereas the attractor landscape of the corresponding slow-fast system remains qualitatively unaltered. We propose that systems containing ghost cycles display increased flexibility and responsiveness to continuous environmental changes.