The Fixed Points of b-Bistochastic-Volterra Quadratic Stochastic Operators On S 1 × S 1

Abstract

The main focus of this paper is to investigate the simplest non-linear Markov operators which is quadratic one. Study of quadratic stochastic operators (QSOs) is not an easy task as linear operators. Thus, researchers introduced classes of QSOs such as Volterra QSOs, strictly non-Volterra QSOs, Orthogonal preserving QSOs, Centered QSOs and etc. However, all the introduced classes were not yet cover the whole set of QSOs. Thus, we introduce a new class of QSOs, namely b-bistochastic-Volterra QSOs or simply bV QSOs. In this paper, the canonical form of bV-QSO defined on one dimensional simplex is provided. We note that, the main problem in the nonlinear operator theory is to study their dynamics. Thus, the set of all fixed points of bV-QSOs are then obtained and classified into attracting, repelling, saddle and non-hyperbolic by applying Jacobian matrix. This helps understanding the dynamical behaviours of bV-QSOs.