Skip to main content
SpringerLink
Log in

An exponential jerk system, its fractional-order form with dynamical analysis and engineering application

  • Methodologies and Application
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

A simple jerk system with only one exponential nonlinearity is proposed and discussed. Dynamic analysis of the integer-order jerk system shows the existence of chaotic oscillations. A model for the fractional-order jerk system is derived. The Adomian decomposition method is used to analyse the fractional-order jerk system. Stability analysis of the fractional-order jerk system shows that chaotic oscillations exist in orders less than one and bifurcation analysis shows the range of fractional orders for periodic and chaotic oscillations. To show the randomness of the fractional-order jerk system, a pseudorandom number generator is designed and tested. The NIST-800-22 tests show that the proposed fractional-order jerk system is effective in showing randomness. Finally, an image hiding application to the audio data has been realized by using the developed RNG algorithm. The encrypted image is hidden by being embedded in the audio data, and then, on the receiver side, the data are recovered by taking the image data from the hidden audio file.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Adomian G (1990) A review of the decomposition method and some recent results for nonlinear equations. Math Comput Modell 13(7):17–43

    MathSciNet  MATH  Google Scholar 

  • Agaian SS, Akopian D, Caglayan O, D’Souza SA (2005) Lossless adaptive digital audio steganography. In: Conference record of the thirty-ninth asilomar conference on signals, systems and computers 2005. IEEE, pp 903–906

  • Akgül A, Kaçar S, Pehlivan İ (2015) An audio data encryption with single and double dimension discrete-time chaotic systems. TOJSAT 5(3):14–23

    Google Scholar 

  • Bailey K, Curran K (2006) An evaluation of image based steganography methods. Multimed Tools Appl 30(1):55–88

    Google Scholar 

  • Baleanu D, Diethelm K, Scalas E, Trujillo JJ (2016) Fractional calculus: models and numerical methods, vol 5. World Scientific, Singapore

    MATH  Google Scholar 

  • Boroujeni EA, Momeni HR (2012) Non-fragile nonlinear fractional order observer design for a class of nonlinear fractional order systems. Signal Process 92(10):2365–2370

    Google Scholar 

  • Cafagna D, Grassi G (2015) Fractional-order systems without equilibria: the first example of hyperchaos and its application to synchronization. Chin Phys B 24(8):080502

    Google Scholar 

  • Caponetto R, Fazzino S (2013) An application of Adomian decomposition for analysis of fractional-order chaotic systems. Int J Bifurc Chaos 23(03):1350050

    MathSciNet  MATH  Google Scholar 

  • Chang C-C, Tseng H-W (2004) A steganographic method for digital images using side match. Pattern Recognit Lett 25(12):1431–1437

    Google Scholar 

  • Chang C-C, Lin C-Y, Wang Y-Z (2006) New image steganographic methods using run-length approach. Inf Sci 176(22):3393–3408

    MathSciNet  Google Scholar 

  • Charef A, Sun HH, Tsao YY, Onaral B (1992) Fractal system as represented by singularity function. IEEE Trans Autom Control 37(9):1465–1470

    MathSciNet  MATH  Google Scholar 

  • Chen W-Y (2007) Color image steganography scheme using set partitioning in hierarchical trees coding, digital Fourier transform and adaptive phase modulation. Appl Math Comput 185(1):432–448

    MathSciNet  MATH  Google Scholar 

  • Christos V, Akif A, Viet-Thanh P, Ioannis S, Ioannis K (2017) A simple chaotic circuit with a hyperbolic sine function and its use in a sound encryption scheme. Nonlinear Dyn 89:1–15

    Google Scholar 

  • Chunxia L, Jie Y, Xiangchun X, Limin A, Yan Q, Yongqing F (2012) Research on the multi-scroll chaos generation based on jerk mode. Procedia Eng 29:957–961

    Google Scholar 

  • Danca M-F, Tang WKS, Chen G (2016) Suppressing chaos in a simplest autonomous memristor-based circuit of fractional order by periodic impulses. Chaos Solitons Fractals 84:31–40

    MathSciNet  MATH  Google Scholar 

  • Delforouzi A, Pooyan M (2007) Adaptive digital audio steganography based on integer wavelet transform. In: Third international conference on intelligent information hiding and multimedia signal processing 2007, IIHMSP 2007, vol 2. IEEE, pp 283–286

  • Diethelm K (2010) The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type. Springer, Berlin

    MATH  Google Scholar 

  • Du W-C, Hsu W-J (2003) Adaptive data hiding based on VQ compressed images. IEE Proc Vis Image Signal Process 150(4):233–238

    Google Scholar 

  • Ellner S, Gallant AR, McCaffrey D, Nychka D (1991) Convergence rates and data requirements for jacobian-based estimates of Lyapunov exponents from data. Phys Lett A 153(6–7):357–363

    MathSciNet  Google Scholar 

  • Frith D (2007) Steganography approaches, options, and implications. Netw Secur 2007(8):4–7

    Google Scholar 

  • He S-B, Sun K-H, Wang H-H (2014) Solution of the fractional-order chaotic system based on adomian decomposition algorithm and its complexity analysis

  • He S, Sun K, Wang H (2015) Complexity analysis and dsp implementation of the fractional-order Lorenz hyperchaotic system. Entropy 17(12):8299–8311

    Google Scholar 

  • Kilbas AAA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations, vol 204. Elsevier Science Limited

  • Lakshmikantham V, Vatsala AS (2008) Basic theory of fractional differential equations. Nonlinear Anal Theory Methods Appl 69(8):2677–2682

    MathSciNet  MATH  Google Scholar 

  • Li RH, Chen WS (2013) Fractional order systems without equilibria. Chin Phys B 22:040503

    Google Scholar 

  • Li C, Gong Z, Qian D, Chen YQ (2010) On the bound of the Lyapunov exponents for the fractional differential systems. Chaos Interdiscip J Nonlinear Sci 20(1):013127

    MathSciNet  MATH  Google Scholar 

  • Lorenz EN (1963) Deterministic nonperiodic flow. J Atmos Sci 20(2):130–141

    MathSciNet  MATH  Google Scholar 

  • Mahajan M, Kaur N (2012) Adaptive steganography: a survey of recent statistical aware steganography techniques. Int J Comput Netw Inf Secur 4(10):76

    Google Scholar 

  • Matsuoka H (2006) Spread spectrum audio steganography using sub-band phase shifting. In: International conference on intelligent information hiding and multimedia signal processing 2006, IIH-MSP’06. IEEE, pp 3–6

  • Maus A, Sprott JC (2013) Evaluating Lyapunov exponent spectra with neural networks. Chaos Solitons Fractals 51:13–21

    MathSciNet  MATH  Google Scholar 

  • Munmuangsaen B, Srisuchinwong B (2011) Elementary chaotic snap flows. Chaos Solitons Fractals 44(11):995–1003

    Google Scholar 

  • Munmuangsaen B, Srisuchinwong B, Sprott JC (2011) Generalization of the simplest autonomous chaotic system. Phys Lett A 375(12):1445–1450

    MATH  Google Scholar 

  • National institute of standards and 2001 tech., NIST-800-22. A statistical test suite for random and pseudo rngs for cryptographic applications. http://csrc.nist.gov/publications/nistpubs/800-22/sp-800-22-051501.pdf

  • Petráš I (2006) Method for simulation of the fractional order chaotic systems. Acta Montan Slovaca 11(4):273–277

    Google Scholar 

  • Pooyan M, Delforouzi A (2007) Lsb-based audio steganography method based on lifting wavelet transform. In: IEEE international symposium on signal processing and information technology 2007. IEEE, pp 600–603

  • Pourmahmood Aghababa M (2012) Robust finite-time stabilization of fractional-order chaotic systems based on fractional Lyapunov stability theory. J Comput Nonlinear Dyn 7(2):021010

    MathSciNet  Google Scholar 

  • Rajagopal K, Karthikeyan A, Duraisamy P (2017a) Hyperchaotic chameleon: fractional order FPGA implementation. Complexity. https://doi.org/10.1155/2017/8979408

  • Rajagopal K, Karthikeyan A, Srinivasan AK (2017b) FPGA implementation of novel fractional-order chaotic systems with two equilibriums and no equilibrium and its adaptive sliding mode synchronization. Nonlinear Dyn 87(4):2281–2304

  • Rajagopal K, Guessas L, Karthikeyan A, Srinivasan A, Adam G (2017c) Fractional order memristor no equilibrium chaotic system with its adaptive sliding mode synchronization and genetically optimized fractional order pid synchronization. Complexity. https://doi.org/10.1155/2017/1892618

  • Rajagopal K, Guessas L, Vaidyanathan S, Karthikeyan A, Srinivasan A (2017d) Dynamical analysis and fpga implementation of a novel hyperchaotic system and its synchronization using adaptive sliding mode control and genetically optimized pid control. Math Prob Eng. https://doi.org/10.1155/2017/7307452

  • Rössler OE (1976) An equation for continuous chaos. Phys Lett A 57(5):397–398

    MATH  Google Scholar 

  • Rössler OE (1979) Continuous chaos—four prototype equations. Ann N Y Acad Sci 316(1):376–392

    MathSciNet  MATH  Google Scholar 

  • Schot SH (1978) Jerk: the time rate of change of acceleration. Am J Phys 46(11):1090–1094

    Google Scholar 

  • Shah P, Choudhari P, Sivaraman S (2008) Adaptive wavelet packet based audio steganography using data history. In: IEEE region 10 and the third international conference on industrial and information systems 2008, ICIIS 2008. IEEE, pp 1–5

  • Sprott JC (1996) Some simple chaotic flows. Phys Rev E 50(2):R647

    MathSciNet  Google Scholar 

  • Sprott JC (1997a) Some simple chaotic jerk functions. Am J Phys 65(6):537–543

    Google Scholar 

  • Sprott JC (1997b) Simplest dissipative chaotic flow. Phys Lett A 228(4–5):271–274

    MathSciNet  MATH  Google Scholar 

  • Sprott JC (2011) A new chaotic jerk circuit. IEEE Trans Circuits Syst II Express Briefs 58(4):240–243

    Google Scholar 

  • Srisuchinwong B, Nopchinda D (2013) Current-tunable chaotic jerk oscillator. Electron Lett 49(9):587–589

    Google Scholar 

  • Sun HH, Abdelwahab A, Onaral B (1984) Linear approximation of transfer function with a pole of fractional power. IEEE Trans Autom Control 29(5):441–444

    MATH  Google Scholar 

  • Sun W, Shen R-J, Yu F-X, Lu Z-M (2012) Data hiding in audio based on audio-to-image wavelet transform and vector quantization. In: Eighth international conference on intelligent information hiding and multimedia signal processing (IIH-MSP) 2012. IEEE, pp 313–316

  • Tavazoei MS, Haeri M (2007) Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems. IET Signal Process 1(4):171–181

    Google Scholar 

  • Trzaska Z (2011) Matlab solutions of chaotic fractional order circuits. In: Assi A (ed) Engineering education and research using MATLAB, chapter 19. Intech, Rijeka

    Google Scholar 

  • Vaidyanathan S, Volos C, Pham V-T, Madhavan K (2015) Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system and its spice implementation. Arch Control Sci 25(1):135–158

    MathSciNet  Google Scholar 

  • Wolf A, Swift JB, Swinney HL, Vastano JA (1985) Determining Lyapunov exponents from a time series. Phys D 16(3):285–317

    MathSciNet  MATH  Google Scholar 

  • Yu S, Lu J, Leung H, Chen G (2005) Design and implementation of \(n\)-scroll chaotic attractors from a general jerk circuit. IEEE Trans Circuits Syst I Regul Pap 52(7):1459–1476

    MathSciNet  MATH  Google Scholar 

  • Zhang R, Gong J (2014) Synchronization of the fractional-order chaotic system via adaptive observer. Syst Sci Control Eng Open Access J 2(1):751–754

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Non-Linear Dynamics, Defense University, Bishoftu, Ethiopia

    Karthikeyan Rajagopal & Anitha Karthikeyan

  2. Department of Electrical and Electronics Engineering, Faculty of Technology, Sakarya University of Applied Sciences, Serdivan, Sakarya, Turkey

    Akif Akgul & Sezgin Kacar

  3. Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Sajad Jafari

  4. Department of Computer Engineering, Faculty of Computer and Information Sciences, Sakarya University, 54187, Serdivan, Sakarya, Turkey

    Unal Cavusoglu

Authors
  1. Karthikeyan Rajagopal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Akif Akgul
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sajad Jafari
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Anitha Karthikeyan
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Unal Cavusoglu
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Sezgin Kacar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Akif Akgul.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by V. Loia.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rajagopal, K., Akgul, A., Jafari, S. et al. An exponential jerk system, its fractional-order form with dynamical analysis and engineering application. Soft Comput 24, 7469–7479 (2020). https://doi.org/10.1007/s00500-019-04373-w

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-019-04373-w

Keywords

Search

Navigation