Han, D.D., Min, L.Q., Zang, H.Y. and Yang, X.P. (2019) Robust Chaos of Cubic Polynomial Discrete Maps with Application to Pseudorandom Number Generators. Mathematical Problems in Engineering, 2019, Article ID: 8250903.
https://doi.org/10.1155/2019/8250903

has been cited by the following article:

• TITLE: Carleman Linearization and Systems of Arbitrary Depth Polynomial Recursions

  AUTHORS: Mikołaj Myszkowski

  KEYWORDS: Polynomial Recursion, Carleman Linearization, Transfer Matrices

  JOURNAL NAME: Advances in Linear Algebra & Matrix Theory, Vol.12 No.1, January 19, 2022

  ABSTRACT: New approach to systems of polynomial recursions is developed based on the Carleman linearization procedure. The article is divided into two main sections: firstly, we focus on the case of uni-variable depth-one polynomial recurrences. Subsequently, the systems of depth-one polynomial recurrence relations are discussed. The corresponding transition matrix is constructed and upper triangularized. Furthermore, the powers of the transition matrix are calculated using the back substitution procedure. The explicit expression for a solution to a broad family of recurrence relations is obtained. We investigate to which recurrences the framework can be applied and construct sufficient conditions for the method to work. It is shown how introduction of auxiliary variables can be used to reduce arbitrary depth systems to the depth-one system of recurrences dealt with earlier. Finally, the limitations of the method are discussed, outlining possible directions for future research.