A Simple Condition for Checking Non-vanishing Basin of Attraction Stability for a Class of Positive Nonlinear Systems

When dealing with positive nonlinear systems, conventional stability requires too much for the equilibrium points located on the boundary of the positive orthant, which encourages the consideration of ‘stability with respect to the positive orthant.´ In addition, since this case arises often when the bifurcation occurs, variation of stability property (regarding the size of basin of attraction) along the variation of the parameter becomes of interest. Motivated by this fact, NvBA(Non-vanishing Basin of Attraction)-stability was recently proposed and investigated. In particular, it was claimed that NvBA-stability holds if and only if the same property holds for the reduced order system on a parametrized center manifold. However, the verification of NvBA-stability is not easy in general, because a solution to the partial differential equation for the center manifold needs to be found. In this paper, we present a readily verifiable condition for the NvBA-stability by restricting the system structure and by utilizing the information about the location of another equilibria that split from or merge into the equilibrium of interest due to the parameter variation. The proposed condition requires neither the solution to the center manifold equation nor the construction of Lyapunov functions.