  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.
