Global stability problem for quadratic stochastic operators

The classical Perron-Frobenius theorem for a positive (primitive) square stochastic matrix plays an important role in addressing the consensus problem within multi-agent systems governed by linear protocols. According to this theorem, a linear stochastic operator associated with the positive (primitive) square stochastic matrix possesses a unique stationary distribution (fixed point) within the simplex. Furthermore, its trajectory, starting from any point within the simplex, always converges to this unique stationary distribution (fixed point). An important question arises regarding the feasibility of generalizing the classical Perron-Frobenius theorem, extending from positive (primitive) square stochastic matrices to positive (diagonally primitive) cubic stochastic matrices. The result of this nature is significantly important in the nonlinear context of consensus problems within multi-agent systems, as we discussed in the previous talk. In the context of cubic stochastic matrices, the situation is more complex than one might anticipate. In this regard, some supporting examples will be provided during the talk. The primary goal of this talk is to extend the classical Perron-Frobenius theorem from square doubly stochastic matrices to cubic doubly stochastic matrices. Additionally, we also discuss a related global stability problem in the context of both autonomous and non-autonomous dynamical systems.