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.
Highlights
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
The linear stability theory provides a local method to determine whether a state is stable or not
Limited by the infinite dimension of the initial state space, it is still difficult to have a comprehensive understanding of attractors in delayed dynamics
Summary
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. It is possible to construct analytical measures, such as the Hausdorff measure, to assess the basin’s ‘volume’[20], such measures are less geometric intuitive or numerical computable as basin’s area or volume in the traditional BS theory of ordinary differential equations To overcome these difficulties, we articulate in the following the concept of the BS by a technique which projects the infinite dimensional initial state space of the DDE to a finite dimensional Euclidean space. Of the synchronization of delayed complex networks and multi-stability of equilibriums and periodic orbits in time-delayed oscillators, show the broad potential of applications of this developed approach
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