A sufficient condition for a cubic operator to preserve a simplex

Jaxongir N. Jumayev

Karshi State University, Kuchabag, Karshi, Uzbekistan

Abstract: There is a one-to-one correspondence between cubic operators that map $\mathbb R^m$ to itself and $m^4$-dimensional matrices. It is known that any cubic operator associated with a stochastic (in a fixed sense) $m^4$-dimensional matrix preserves the standard simplex. In this paper, we establish a sufficient conditions on the non-stochastic $m^4$-dimensional matrix that ensure the corresponding cubic operator also preserves the simplex. Furthermore, on the two-dimensional simplex we investigate the dynamics of a cubic operator defined by a non-stochastic matrix.

Keywords: cubic operator, simplex, non-stochastic matrix, fixed point, invariant set, trajectory.

UDC: 517.98

Received: 21.11.2024
Received in revised form: 29.01.2025
Accepted: 04.06.2025

Language: English