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Global dynamics of fractional-order systems with an extended generalized cell mapping method

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Abstract

Global dynamics of fractional-order systems is studied with an extended generalized cell mapping (EGCM) method. The one-step transition probability matrix of Markov chain of the EGCM is generated by means of the improved predictor–corrector approach for fractional-order systems. The one-step mapping time of the proposed method is evaluated with the help of the short memory principle for fractional derivatives to deal with its non-local property and to properly define a bound of the truncation error and a function M by considering the features of cell mapping. In this way, a method of generalized cell mapping for global dynamics of a fractional-order system is developed. Three examples of global analysis on fractional-order systems are given to demonstrate the validity and efficiency of the proposed method. And attractors, boundaries, basins of attraction, and saddles are obtained by the EGCM.

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (NSFC) under the Grant Nos. 11332008, 11172223 and 11172224. Discussions and suggestions by Professor Y. Xu and Professor W. Xu and their students from Northwestern Polytechnical University are highly appreciated. The authors are very grateful to the anonymous reviewers for their insightful comments leading to the improvement of the paper.

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Authors and Affiliations

  1. State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an, 710049, China

    Xiaojun Liu, Ling Hong, Jun Jiang, Dafeng Tang & Lixin Yang

  2. School of Mathematics and Statistics, Tianshui Normal University, Tianshui, 741001, China

    Xiaojun Liu & Lixin Yang

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  1. Xiaojun Liu
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  2. Ling Hong
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  3. Jun Jiang
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  4. Dafeng Tang
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  5. Lixin Yang
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Correspondence to Ling Hong.

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Liu, X., Hong, L., Jiang, J. et al. Global dynamics of fractional-order systems with an extended generalized cell mapping method. Nonlinear Dyn 83, 1419–1428 (2016). https://doi.org/10.1007/s11071-015-2414-5

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