Right fractional calculus to inverse-time chaotic maps and asymptotic stability analysis

Abstract

Inverse or terminal value problems of fractional differential equations become popular recently. But memory effects or non-locality of fractional operators cause many difficulties for theoretical analysis. This study suggests a right fractional calculus method for inverse problem modelling and proposes a concept of inverse-time fractional chaotic maps. First, a simple right fractional linear differential equation's terminal value problem and the solutions are investigated. Then, some basics of the right discrete fractional calculus are introduced and the idea is extended to the discrete case. Right fractional sum equations are derived and numerical schemes are provided for dynamical analysis. Discrete chaos does exist in an inverse-time fractional logistic map and Hénon map, respectively. Their local stability conditions are given. It can be concluded that this right fractional calculus method is simpler but more efficient than the left one.

Keywords:

• Right fractional calculus
• inverse-time fractional chaotic maps
• inverse problems
• difference equations

Mathematics Subject classifications:

• 39A30
• 65Q10
• 26A33

Acknowledgments

The authors appreciate the Editor's valuable time and referees' sincere suggestions.

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No potential conflict of interest was reported by the author(s).

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All authors completed the paper together. All authors read and approved the final manuscript.

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Funding

This work is financially supported by the National Natural Science Foundation of China (NSFC) [grant numbers 62076141 and 12101338] and Sichuan Youth Science and Technology Foundation [grant number 2022JDJQ0046].