Dynamics analysis and cryptographic application of fractional logistic map

  • Liguo YuanEmail author
  • Song Zheng
  • Zeeshan Alam
Original Paper


Based on Lyapunov exponent, Schwarzian derivative, Shannon entropy and Kolmogorov entropy, we will firstly study chaos and bifurcation of fractional (semi-) logistic map (FLM). It is derived from fractional integration (not fractional derivative) of the classical logistic map. Then, this paper put forward a new accumulated coupled fractional (semi-) logistic map lattice (ACFLML) whose lattice function is the FLM. Local stability, pattern information, high-dimensional chaos and bifurcation of the ACFLML are analyzed based on stability theory, pattern simulation, 0–1 test for chaos, Lyapunov exponent spectrum and Kolmogorov entropy. Finally, the chaotic ACFLML is successfully applied to encryption and decryption of digital image. To evaluate security, histogram analysis, correlation analysis, differential attack, key space, key sensitivity, encryption time, computational complexity and chosen/known-plaintext attacks, analysis is performed. Simulation analysis shows that this encryption scheme is effective and has good statistical effect.


Fractional logistic map Chaos Coupled map lattice Chaotic dynamics Image encryption 



The research is supported by the Open Project of Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing (No. 2017CSOBDP0302), the Science and Technology Program of Guangzhou (No. 201707010031), the National Natural Science Foundation of China (Nos. 11671149, 51777180, 11402226), the Natural Science Foundation of Zhejiang Province (No. LY17A020007), First Class Discipline of Zhejiang−A (Zhejiang University of Finance and Economics-Statistics) and the Preeminent Youth Fund of Zhejiang University of Finance and Economics and Higher Education Commission of Pakistan under Start-up Research Grant Project (SRGP) (\(\#\)1419). L. G. Yuan’s research is partially supported by China Scholarship Council.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsSouth China Agricultural UniversityGuangzhouChina
  2. 2.Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data ProcessingYulin Normal UniversityYulinChina
  3. 3.Institute of Applied MathematicsZhejiang University of Finance and EconomicsHangzhouChina
  4. 4.Department of Mathematics, Statistics and Computer ScienceThe University of AgriculturePeshawarPakistan

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