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

TL;DR

This study compares ghost cycles and slow-fast systems, showing ghost cycles have enhanced entrainment and complex dynamics in response to external forcing, suggesting greater environmental responsiveness.

Contribution

It introduces ghost cycles as an alternative to slow-fast systems, demonstrating their increased entrainment regions and richer dynamics under external forcing.

Findings

Ghost cycles show larger 1:1 entrainment regions.

Ghost cycles exhibit bursting and chaotic behaviors.

Attractor landscapes remodel continuously in ghost cycles.

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.