  Understanding the basins of attraction (BoA) is often a paramount
consideration for nonlinear systems. Most existing approaches to determining a
high-resolution BoA require prior knowledge of the system's dynamical model
(e.g., differential equation or point mapping for continuous systems, cell
mapping for discrete systems, etc.), which allows derivation of approximate
analytical solutions or parallel computing on a multi-core computer to find the
BoA efficiently. However, these methods are typically impractical when the BoA
must be determined experimentally or when the system's model is unknown. This
paper introduces a model-free sampling method for BoA. The proposed method is
based upon hybrid active learning (HAL) and is designed to find and label the
"informative" samples, which efficiently determine the boundary of BoA. It
consists of three primary parts: 1) additional sampling on trajectories (AST)
to maximize the number of samples obtained from each simulation or experiment;
2) an active learning (AL) algorithm to exploit the local boundary of BoA; and
3) a density-based sampling (DBS) method to explore the global boundary of BoA.
An example of estimating the BoA for a bistable nonlinear system is presented
to show the high efficiency of our HAL sampling method.
