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Caputo-Hadamard fractional differential equations on time scales: Numerical scheme, asymptotic stability, and chaos
Guo-Cheng Wu1, Ting-Ting Song2, Shuqiang Wang3
1Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, People's Republic of China.
Abstract:
This study investigates Caputo-Hadamard fractional differential equations on time scales. The Hadamard fractional sum and difference are defined for the first time. A general logarithm function on time scales is used as a kernel function. New fractional difference equations and their equivalent fractional sum equations are presented by the use of fundamental theorems. Gronwall inequality, asymptotical stability conditions, and two discrete-time Mittag-Leffler functions of Hadamard type are obtained. Numerical schemes are provided and chaos in fractional discrete-time logistic equation and neural network equations are reported.
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