title: Basin stability in delayed dynamics.
abstract: Basin stability (BS) is a universal concept for complex systems studies, which focuses on the volume of the basin of attraction instead of the traditional linearization-based approach. It has a lot of applications in real-world systems especially in dynamical systems with a phenomenon of multi-stability, which is even more ubiquitous in delayed dynamics such as the firing neurons, the climatological processes, and the power grids. Due to the infinite dimensional property of the space for the initial values, how to properly define the basin's volume for delayed dynamics remains a fundamental problem. We propose here a technique which projects the infinite dimensional initial state space to a finite-dimensional Euclidean space by expanding the initial function along with different orthogonal or nonorthogonal basis. A generalized concept of basin's volume in delayed dynamics and a highly practicable calculating algorithm with a cross-validation procedure are provided to numerically estimate the basin of attraction in delayed dynamics. We show potential applicabilities of this approach by applying it to study several representative systems of biological or/and physical significance, including the delayed Hopfield neuronal model with multistability and delayed complex networks with synchronization dynamics.
authors.affiliation: School of Mathematical Sciences, LNSM and Centre for Computational Systems Biology, Fudan University, Shanghai 200433, China.|Potsdam Institute for Climate Impact Research (PIK), Potsdam 14473, Germany.
authors.forename: Siyang
authors.initials: S
authors.lastname: Leng
authors.affiliation: School of Mathematical Sciences, LNSM and Centre for Computational Systems Biology, Fudan University, Shanghai 200433, China.
authors.forename: Wei
authors.initials: W
authors.lastname: Lin
authors.affiliation: Potsdam Institute for Climate Impact Research (PIK), Potsdam 14473, Germany.|Department of Physics, Humboldt University, Berlin 12489, Germany.|Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom.
authors.forename: Jürgen
authors.initials: J
authors.lastname: Kurths