The inverse Frobenius-Perron problem: A survey of solutions to the original problem formulation

André M. McDonald; Michaël A. van Wyk; Guanrong Chen; André M. McDonald; Michaël A. van Wyk; Guanrong Chen

doi:10.3934/math.2021650

AIMS Mathematics

2021, Volume 6, Issue 10: 11200-11232. doi: 10.3934/math.2021650

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The inverse Frobenius-Perron problem: A survey of solutions to the original problem formulation

1.
Defence and Security Cluster at the Council for Scientific and Industrial Research, Mering Naude Rd., Brummeria 0184, Pretoria, South Africa
2.
School of Electrical and Information Engineering at the University of the Witwatersrand, 1 Jan Smuts Ave., Braamfontein 2000, Johannesburg, South Africa
3.
Department of Electrical Engineering at the City University of Hong Kong, 83 Tat Chee Ave., Kowloon Tong, Kowloon, Hong Kong SAR, China

Received: 27 February 2020 Accepted: 11 July 2021 Published: 03 August 2021
MSC : 37-02, 37A50, 37E05

The inverse Frobenius-Perron problem (IFPP) is a collective term for a family of problems that requires the construction of an ergodic dynamical system model with prescribed statistical characteristics. Solutions to this problem draw upon concepts from ergodic theory and are scattered throughout the literature across domains such as physics, engineering, biology and economics. This paper presents a survey of the original formulation of the IFPP, wherein the invariant probability density function of the system state is prescribed. The paper also reviews different strategies for solving this problem and demonstrates several of the techniques using examples. The purpose of this survey is to provide a unified source of information on the original formulation of the IFPP and its solutions, thereby improving accessibility to the associated modeling techniques and promoting their practical application. The paper is concluded by discussing possible avenues for future work.
- inverse Frobenius-Perron problem,
- ergodic map,
- invariant density,
- invariant measure,
- piecewise continuous maps,
- dynamical system
Citation: André M. McDonald, Michaël A. van Wyk, Guanrong Chen. The inverse Frobenius-Perron problem: A survey of solutions to the original problem formulation[J]. AIMS Mathematics, 2021, 6(10): 11200-11232. doi: 10.3934/math.2021650

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Abstract

The inverse Frobenius-Perron problem (IFPP) is a collective term for a family of problems that requires the construction of an ergodic dynamical system model with prescribed statistical characteristics. Solutions to this problem draw upon concepts from ergodic theory and are scattered throughout the literature across domains such as physics, engineering, biology and economics. This paper presents a survey of the original formulation of the IFPP, wherein the invariant probability density function of the system state is prescribed. The paper also reviews different strategies for solving this problem and demonstrates several of the techniques using examples. The purpose of this survey is to provide a unified source of information on the original formulation of the IFPP and its solutions, thereby improving accessibility to the associated modeling techniques and promoting their practical application. The paper is concluded by discussing possible avenues for future work.

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