Generalized saddle-node ghosts and their composite structures in dynamical systems
The study of dynamical systems has long focused on the characterization of their asymptotic dynamics such as fixed points, limit cycles and other types of attractors and how these invariant sets change their properties as systems parameters change. More recently, however, the importance of transient dynamics, especially of long transients and sequential transitions between them, has been increasingly recognized in various fields including ecology, neuroscience and cell biology. Among several possible origins of long transients, ghost attractors have received particular attention due to interesting dynamical properties in non-autonomous settings, new theoretical developments, and an increasing number of systems that empirically show dynamics consistent with ghost attractors. Despite this growing interest in transient dynamics generally and ghost attractors in particular, there are significantly fewer theoretical concepts and software tools available to researchers to classify and characterize their underlying mechanisms compared to asymptotic dynamics. To address this gap, we generalize saddle-nodes to account for higher-dimensional center manifolds and provide a definition for their ghost attractors. We then introduce algorithms to specifically identify and characterize ghost attractors and their composite structures such as ghost channels and ghost cycles and show how these concepts and algorithms can be used to gain new insights into the transient dynamics of a wide range of system models focusing on living systems, allowing, e.g., to describe bifurcations of ghosts. The algorithms are implemented in Python and available as PyGhostID, a user-friendly open-source software package.