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<p>This study provides a thorough investigation of the commensurate fractional-order Rucklidge system, addressing key limitations in existing research. We validate the chaotic behavior of the system by calculating the Lyapunov exponents and characterizing the fractal nature of the attractor using the Kaplan–Yorke dimension. A bifurcation diagram is obtained to examine the transitions between periodic and chaotic regimes, while the stability of the system is assessed using the fractional Routh–Hurwitz criterion. Additionally, we evaluate the offset boosting capability of the system, which allows for controlled manipulation of attractor positions without altering the system’s inherent dynamics. Furthermore, we quantify signal complexity using spectral entropy and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11477_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(C_0\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> complexity metrics. To assess the randomness and cryptographic potential of the generated sequences, we perform validation with the NIST SP 800-22 statistical test suite, confirming their suitability for secure applications. By utilizing the high entropy and sensitivity to parameter variations inherent to the fractional order Rucklidge system, we propose a novel image encryption scheme that integrates chaotic keystreams into the ChaCha20 algorithm. Experimental results demonstrate significant improvements in security, highlighting the system as a powerful tool for both nonlinear system analysis and cryptographic applications.</p>

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Fractional chaotic dynamics in the rucklidge system and its application to image encryption

  • 期刊论文

  • 发表日期：

    2025年6月24日

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  • Sajad Iqbal,
  • Jun Wang,
  • Haris Calgan

摘要

This study provides a thorough investigation of the commensurate fractional-order Rucklidge system, addressing key limitations in existing research. We validate the chaotic behavior of the system by calculating the Lyapunov exponents and characterizing the fractal nature of the attractor using the Kaplan–Yorke dimension. A bifurcation diagram is obtained to examine the transitions between periodic and chaotic regimes, while the stability of the system is assessed using the fractional Routh–Hurwitz criterion. Additionally, we evaluate the offset boosting capability of the system, which allows for controlled manipulation of attractor positions without altering the system’s inherent dynamics. Furthermore, we quantify signal complexity using spectral entropy and \(C_0\) C 0 complexity metrics. To assess the randomness and cryptographic potential of the generated sequences, we perform validation with the NIST SP 800-22 statistical test suite, confirming their suitability for secure applications. By utilizing the high entropy and sensitivity to parameter variations inherent to the fractional order Rucklidge system, we propose a novel image encryption scheme that integrates chaotic keystreams into the ChaCha20 algorithm. Experimental results demonstrate significant improvements in security, highlighting the system as a powerful tool for both nonlinear system analysis and cryptographic applications.