Dynamics of composition of two quadratic stochastic Volterra operators

• Main Authors: Ganikhodzaev, Rasul, Mukhamedov, Farrukh, Saburov, Mansoor, Qaralleh, Izzat
• Format: Conference or Workshop Item
• Language: English
• Published: 2013
• Subjects: QA Mathematics
• Online Access: http://irep.iium.edu.my/28933/1/Composition_of_QSO--ICMSS2013.pdf; http://irep.iium.edu.my/28933/; http://math.upm.edu.my/icmss2013/
• Institution: Universiti Islam Antarabangsa Malaysia

One of the simplest unsolved problems in the theory of nonlinear dynamical systems is to study asymptotic behaviors of quadratic stochastic operators (in short QSO) on the finite dimensional simplex. However, this problem is not fully finished even in 2D case. The reason is that, in such systems, we may observe some chaotic behaviors (more precisely, Li-Yorke chaotic behaviors). It is worth mentioning that the square of some quadratic homeomorphisms of the simplex are topologically conjugate to compositions of two distinct Volterra QSO. Therefore, it is of independent interest to study the asymptotical behavior of the composition of two Volterra QSO. In this paper, we shall present some unexpected asymptotical behaviors. For example, we can see some interesting asymptotical behavior of composition of regular (stable) and chaotic (or unstable) Volterra quadratic stochastic operators. Our obtained results enable to study general problems in the theory of quadratic stochastic operators.