Periodically Forced Ghost Cycles

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Overview

This repository contains the code and data to reproduce the results and figures from the study:

Daniel Koch, Ulrike Feudel, Aneta Koseska (2025).
Criticality governs response dynamics and entrainment of periodically forced ghost cycles.
Physical Review E, XX: XXXX–XXXX.

Link to publication (coming soon)

If you use this repository in your research, please cite the original paper.

Background

Ghost cycles are a class of limit cycles characterized by the presence of one or more ghost attractors that are visited sequentially along the orbit and which transiently capture trajectories, leading to a separation of timescales—reminiscent of slow-fast dynamical systems. The concept of ghost cycles was first introduced in: Koch D. et al., Phys. Rev. Lett. 133, 047202 (2024).

This study explores the dynamics of ghost cycles under periodic forcing and compares them with classical slow-fast oscillators. The findings highlight how criticality—the proximity to a bifurcation point—governs the response dynamics of periodically forced ghost cycles.

Getting Started

Dependencies

To recreate the figures and simulations, you’ll need:

• Python 3.x
• Required Python packages (installable via the environment ghostCycles_pythonEnvironment.yaml)

Running Code

• Main Figures: Run any of the FigureX.py scripts (e.g. Figure1.py, Figure2.py, etc.) to generate the main figures from the paper.

Supplementary Figures & Videos:

• Use SupplementaryFigureX.py and SupplementaryVideos.py to reproduce additional analyses and visuals.

Ensure that models.py and functions.py are located in the parent directory of the script you're executing.

Bifurcation Analysis

Bifurcation diagrams can be generated using the models provided in the XPP files directory. For running the .ode files you need to install XPP/XPPAUT.

Contact

For questions or additional information, please contact Daniel Koch: dkoch.research@protonmail.com