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non-autonomous systems, criticality, ghost attractors, ghost cycles, slow-fast systems, relaxation oscillators
Abstract:
Synopsis
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.
Oscillations in natural systems often exhibit multiple time-scales, characterized by repeated switching between slow and fast motions. Such phenomena are typically modeled as slow-fast systems, whose dynamics formally corresponds to a type of stable limit cycle oscillators. We have recently proposed that similar dynamics can also arise by connecting unstable objects called ghost attractors forming a cycle, such that there is slow dynamics due to the ghosts, with fast switching between them. We show here that ghost cycles exhibit a higher flexibility in their response to periodic external inputs, allowing them to be more easily entrained in to a broad range of frequencies of the external input, but also to exhibit complex dynamics such as bursting and chaos. Our analyses show that this flexibility results from an organization at criticality, such that the ghost cycles can exploit different dynamical regimes present in the system under the influence of time-varying external inputs. We validate the results for different models and discuss the implications of these findings for information processing tasks in biological systems.
