Skip to main content
SpringerLink
Log in

A Novel Chaotic System With Boomerang-Shaped Equilibrium, Its Circuit Implementation and Application to Sound Encryption

  • Research Paper
  • Published:
Iranian Journal of Science and Technology, Transactions of Electrical Engineering Aims and scope Submit manuscript

Abstract

In the chaos literature, there has been much attention paid to chaotic systems with uncountable equilibrium points such as systems with line equilibrium, curve equilibrium. This paper reports a 3-D chaotic system with a closed curve of equilibrium points, which has the shape of a boomerang. Dynamics of the chaotic system with the boomerang equilibrium has been studied by using phase portraits, bifurcation diagram, Lyapunov exponents and Lyapunov dimension. Also, we design an electronic circuit implementation of the theoretical system to check its feasibility. As an application of the new chaotic system, we have derived new results for sound encryption with the new chaotic system.

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
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  • Ahmed M, Shamsi U, Khan IR (2015) An enhanced image encryption algorithm using fractional chaotic system. Procedia Comput Sci 57:852–859

    Article  Google Scholar 

  • Akgul A, Hussain S, Pehlivan I (2016) A new three-dimensional chaotic system, its dynamical analysis and electronic circuit applications. Optik 127:7062–7071

    Article  Google Scholar 

  • Akgul A, Moroz I, Pehlivan I, Vaidyanathan S (2016) A new four-scroll chaotic attractor and its engineering applications. Optik 127:5491–5499

    Article  Google Scholar 

  • Azar AT, Vaidyanathan S (2016) Advances in Chaos theory and intelligent control. Springer, Berlin

    Book  MATH  Google Scholar 

  • Bassham LE, Rukhin AL, Soto J, Nechvatal JR, Smid ME, Leigh SD, Levenson M, Vangel M, Heckert NA, Banks DL (2010) A statistical test suite for random and pseudorandom number generators for cryptographic applications. Technical report, National Institute of Standards and Technology

  • Chen Y, Yang Q (2015) A new Lorenz-type hyperchaotic system with a curve of equilibria. Math Comput Simul 112:40–55

    Article  MathSciNet  Google Scholar 

  • Danca MF, Kuznetsov F, Chen G (2017) Unusual dynamics and hidden attractors of the Rabinovich-Fabrikant system. Nonlinear Dyn 88(1):791–805

    Article  MathSciNet  Google Scholar 

  • Dudkowski D, Prasad A, Kapitaniak T (2015) Perpetual points and hidden attractors in dynamical systems. Phys Lett A 379(40–41):2591–2596

    Article  MathSciNet  MATH  Google Scholar 

  • Dudkowski D, Jafari S, Kapitaniak T, Kuznetsov NV, Leonov GA, Prasad A (2016) Hidden attractors in dynamical systems. Phys Rep 637:1–50

    Article  MathSciNet  MATH  Google Scholar 

  • Gotthans T, Petrzela J (2015) New class of chaotic systems with circular equilibrium. Nonlinear Dyn 81(3):1143–1149

    Article  MathSciNet  Google Scholar 

  • Gotthans T, Sprott JC, Petrzela J (2016) Simple chaotic flow with circle and square equilibrium. Int J Bifurc Chaos 26(8), Article ID 1650137

  • Guin A, Dandapathak M, Sarkar S, Sarkar BC (2017) Birth of oscillation in coupled non-oscillatory Rayleigh-Duffing oscillators. Commun Nonlinear Sci Numer Simul 42:420–436

    Article  MathSciNet  Google Scholar 

  • Hu W, Ding D, Zhang Y, Wang N, Liang D (2017) Hopf bifurcation and chaos in a fractional order delayed memristor-based chaotic circuit system. Optik 130:189–200

    Article  Google Scholar 

  • Hu X, Liu C, Liu L, Ni J, Li S (2016) Multi-scroll hidden attractors in improved sprott a system. Nonlinear Dyn 86(3):1725–1734

    Article  Google Scholar 

  • Jafari S, Sprott JC (2013) Simple chaotic flows with a line equilibrium. Chaos, Solitons Fractals 57:79–84

    Article  MathSciNet  MATH  Google Scholar 

  • Kacar S (2016) Analog circuit and microcontroller based RNG application of a new easy realizable 4D chaotic system. Optik 127:9551–9561

    Article  Google Scholar 

  • Kengne J, Tabekoueng ZN, Fotsin HB (2016) Coexistence of multiple attractors and crisis route to chaos in autonomous third order Duffing-Holmes type chaotic oscillators. Commun Nonlinear Sci Numer Simul 36:29–44

    Article  MathSciNet  Google Scholar 

  • Kim D, Chang PH (2013) A new butterfly-shaped chaotic attractor. Results Phys 3:14–19

    Article  Google Scholar 

  • Kingni ST, Pham VT, Jafari S, Woafo P (2017) A chaotic system with an infinite number of equilibrium points located on a line and on a hyperbola and its fractional-order form. Chaos, Solitons Fractals 99(1):209–218

    Article  MathSciNet  MATH  Google Scholar 

  • Kuznetsov N (2016) The Lyapunov dimension and its estimation via the Leonov method. Phys Lett A 380(25–26):2142–2149

    Article  MathSciNet  MATH  Google Scholar 

  • Kuznetsov N, Leonov GA, Yuldashev MV, Yuldashev RV (2017) Hidden attractors in dynamical models of phase-locked loop circuits: limitations of simulation in MATLAB and SPICE. Commun Nonlinear Sci Numer Simul 51:39–49

    Article  Google Scholar 

  • Leonov G, Kuznetsov N, Mokaev T (2015) Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity. Commun Nonlinear Sci Numer Simul 28(1–3):166–174

    Article  MathSciNet  Google Scholar 

  • Leonov GA, Kuznetsov NV, Vagaitsev VI (2011) Localization of hidden Chuas attractors. Phys Lett A 375(23):2230–2233

    Article  MathSciNet  MATH  Google Scholar 

  • Leonov GA, Kuznetsov NV (2013) Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman and Kalman problems to hidden chaotic attractor in Chua circuits. Int J Bifurc Chaos 23(1), Article ID 1330002

  • Leonov GA, Kuznetsov NV, Kuznetsova OA, Seldedzhi AM, Vagaitsev VI (2011) Hidden oscillations in dynamical systems. Trans Syst Control 6:54–67

    Google Scholar 

  • Li C, Sprott JC, Yuan Z, Li H (2015) Constructing chaotic systems with total amplitude control. Int J Bifurc Chaos 25(10), Article ID 1530025

  • Li C, Sprott JC, Thioc W (2014) Bistability in a hyperchaotic system with a line equilibrium. J Exp Theor Phys 118(3):494–500

    Article  Google Scholar 

  • Lin Y, Wang C (2016) Current-mode multi-scroll chaos generator employing CCCII. Electron Lett 52(15):1295–1297

    Article  Google Scholar 

  • Liu H, Ren B, Zhao Q, Li N (2016) Characterizing the optical chaos in a special type of small networks of semiconductor lasers using permutation entropy. Opt Commun 359:79–84

    Article  Google Scholar 

  • Liu W, Sun K, Zhu C (2016) A fast image encryption algorithm based on chaotic map. Opt Lasers Eng 84:26–36

    Article  Google Scholar 

  • Liu X, Mei W, Du H (2016) Simultaneous image compression, fusion and encryption algorithm based on compressive sensing and chaos. Opt Commun 366:22–32

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Ma P (2014) Is there a name to this equation: \(\left( y - a | x |^b \right)^2 + (c x)^2 = d\)? https://math.stackexchange.com/questions/902120/is-there-a-name-to-this-equation-y-axb2-cx2-d. Accessed 1 Aug 2017

  • Misbah C (2017) Complex dynamics and morpohgenesis. Springer, Berlin

    Book  MATH  Google Scholar 

  • Nayfeh AH, Balachandran B (2008) Applied nonlinear dynamics: analytical, computational and experimental methods. Wiley, New Jersey

    MATH  Google Scholar 

  • Pehlivan I, Moroz IM, Vaidyanathan S (2014) Analysis, synchronization and circuit design of a novel butterfly attractor. J Sound Vib 333(20):5077–5096

    Article  Google Scholar 

  • Pham VT, Vaidyanathan S, Volos CK, Jafari S (2015) Hidden attractors in a chaotic system with an exponential nonlinear term. Eur Phys J Spec Top 224(8):1507–1517

    Article  Google Scholar 

  • Pham VT, Jafari S, Volos C, Giakoumis A, Vaidyanathan S, Kapitaniak T (2016) A chaotic system with equilibria located on the rounded square loop and its circuit implementation. IEEE Trans Circuits Syst II Express Briefs 63(9):878–882

    Article  Google Scholar 

  • Pham VT, Jafari S, Wang X, Ma J (2016) A chaotic system with different shapes of equilibria. Int J Bifurc Chaos 26, Article ID 1650069

  • Pham VT, Volos C, Jafari S, Vaidyanathan S, Kapitaniak T, Wang X (2016) A chaotic system with different families of hidden attractors. Int J Bifurc Chaos 26(8), Article ID 1650139

  • Pham VT, Vaidyanathan S, Volos CK, Jafari S, Gotthans T (2017) A three-dimensional chaotic system with square equilibrium and noequilibrium. In: Azar AT, Vaidyanathan S (eds) Fractional order control and synchronization of chaotic systems. Springer, Berlin, pp 613–635

    Chapter  Google Scholar 

  • Rasappan S, Vaidyanathan S (2012) Global chaos synchronization of WINDMI and Coullet chaotic systems by backstepping control. Far East J Math Sci 67(2):265–287

    MathSciNet  MATH  Google Scholar 

  • Sambas A, Sanjaya WS, Mamat M (2015) Bidirectional coupling scheme of chaotic systems and its application in secure communication system. J Eng Sci Technol Rev 8(2):89–95

    Article  Google Scholar 

  • Sambas A, Vaidyanathan S, Mamat M, Sanjaya WM, Rahayu DS (2016) A 3-D novel jerk chaotic system and its application in secure communication system and mobile robot navigation. In: Vaidyanathan S, Volos C (eds) Advances and applications in Chaotic systems. Springer, Berlin, pp 283–310

    Chapter  Google Scholar 

  • Senouci A, Boukabou A (2016) Fuzzy modeling, stabilization and synchronization of multi-scroll chaotic systems. Optik 127(13):5351–5358

    Article  Google Scholar 

  • Sun J, Shen Y (2016) Compound-combination anti-synchronization of five simplest memristor chaotic systems. Optik 127(20):9192–9200

    Article  Google Scholar 

  • Tacha OI, Volos CK, Kyprianidis IM, Stouboulos IN, Vaidyanathan S, Pham VT (2016) Analysis, adaptive control and circuit simulation of a novel nonlinear finance system. Appl Math Comput 276:200–217

    MathSciNet  MATH  Google Scholar 

  • Vaidyanathan S (2015) Adaptive synchronization of chemical chaotic reactors. Int J Chem Tech Res 8(2):612–621

    Google Scholar 

  • Vaidyanathan S (2016) A novel 3-D jerk chaotic system with three quadratic nonlinearities and its adaptive control. Arch Control Sci 26(1):19–47

    Article  MathSciNet  MATH  Google Scholar 

  • Vaidyanathan S (2016) Anti-synchronization of 3-cells cellular neural network attractors via integral sliding mode control. Int J PharmTech Res 9(1):193–205

    Google Scholar 

  • Vaidyanathan S (2016) Global chaos control of the FitzHugh-Nagumo chaotic neuron model via integral sliding mode control. Int J PharmTech Res 9(4):413–425

    Google Scholar 

  • Vaidyanathan S, Azar AT (2016) Takagi-Sugeno fuzzy logic controller for Liu-Chen four-scroll chaotic system. Int J Intell Eng Inform 4:135–150

    Google Scholar 

  • Vaidyanathan S, Rajagopal K (2017) LabVIEW implementation of chaotic masking with adaptively synchronised forced Van der Pol oscillators and its application in real-time image encryption. Int J Simul Process Model 12(2):165–178

    Article  Google Scholar 

  • Vaidyanathan S, Volos C (2015) Analysis and adaptive control of a novel 3-D conservative no-equilibrium chaotic system. Arch Control Sci 25(3):333–353

    Article  MathSciNet  Google Scholar 

  • Vaidyanathan S, Volos C (2017) Advances in memristors, memristive devices and systems. Springer, Berlin

    Book  Google Scholar 

  • Vaidyanathan S, Volos C, Thanh VT, Madhavan K, Idowu BA (2014) Adaptive backstepping control, synchronization and circuit simulation of a 3-D novel jerk chaotic system with two hyperbolic sinusoidal nonlinearities. Arch Control Sci 24(3):257–285

    Article  MathSciNet  MATH  Google Scholar 

  • Vaidyanathan S, Sambas A, Mamat M, Sanjaya WSM (2017) Analysis, synchronisation and circuit implementation of a novel jerk chaotic system and its application for voice encryption. Int J Model Ident Control 28(2):153–166

    Article  Google Scholar 

  • Volos C, Kyprianidis I, Stouboulos I, Tlelo-Cuautle E, Vaidyanathan S (2015) Memristor: a new concept in synchronization of coupled neuromorphic circuits. J Eng Sci Technol Rev 8(2):157–173

    Article  Google Scholar 

  • Volos CK, Kyprianidis IM, Stouboulos IN (2013) Image encryption process based on chaotic synchronization phenomena. Sig Process 93(5):1328–1340

    Article  Google Scholar 

  • Volos CK, Prousalis D, Kyprianidis I, Stouboulos I, Vaidyanathan S, Pham VT (2016) Synchronization and anti-synchronization of coupled Hindmarsh-Rose neuron models. Int J Control Theory Appl 9(1):101–114

    Google Scholar 

  • Wang C, Zhang H, Fan W, Fan X (2016) A new wind power prediction method based on chaotic theory and Bernstein neural network. Energy 117:259–271

    Article  Google Scholar 

  • Wang X, Pham VT, Volos C (2017) Dynamics, circuit design and synchronization of a new chaotic system with closed curve equilibrium. Complexity 2017, Article ID 7138971

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

    Article  MathSciNet  MATH  Google Scholar 

  • Wu T, Sun W, Zhang X, Zhang S (2016) Concealment of time delay signature of chaotic output in a slave semiconductor laser with chaos laser injection. Opt Commun 381:174–179

    Article  Google Scholar 

  • Xu C, Wu Y (2015) Bifurcation and control of chaos in a chemical system. Appl Math Model 39:2295–2310

    Article  MathSciNet  Google Scholar 

  • Zhang C (2016) Theoretical design and circuit realization of complex grid multi-wing chaotic system. Optik 127:4584–4589

    Article  Google Scholar 

  • Zhang C, Yu S (2010) Generation of grid multi-scroll chaotic attractors via switching piecewise linear controller. Phys Lett A 374(30):3029–3037

    Article  MATH  Google Scholar 

  • Zhang J, Gao Y (2017) Synchronization of coupled neural networks with time-varying delay. Neurocomputing 219:154–162

    Article  Google Scholar 

  • Zhusubaliyev ZT, Mosekilde E (2015) Multistability and hidden attractors in a multilevel DC/DC converter. Math Comput Simul 109:32–45

    Article  MathSciNet  Google Scholar 

  • Zhusubaliyev ZT, Mosekilde E, Rubanov VG, Nabokov R (2015) Multistability and hidden attractors in a relay system with hysteresis. Physica D 306:6–15

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Faculty of Engineering, University of Zanjan, Zanjan, Iran

    Saleh Mobayen

  2. Research and Development Centre, Vel Tech University, Chennai, Avadi, India

    Sundarapandian Vaidyanathan

  3. Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Tasikmalaya, Indonesia

    Aceng Sambas

  4. Department of Electrical and Electronics Engineering, Sakarya University, Sakarya, Turkey

    Sezgin Kaçar

  5. Department of Computer Engineering, Sakarya University, Sakarya, Turkey

    Ünal Çavuşoğlu

Authors
  1. Saleh Mobayen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sundarapandian Vaidyanathan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Aceng Sambas
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sezgin Kaçar
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Ünal Çavuşoğlu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sundarapandian Vaidyanathan.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants performed by any of the authors.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mobayen, S., Vaidyanathan, S., Sambas, A. et al. A Novel Chaotic System With Boomerang-Shaped Equilibrium, Its Circuit Implementation and Application to Sound Encryption. Iran J Sci Technol Trans Electr Eng 43, 1–12 (2019). https://doi.org/10.1007/s40998-018-0094-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40998-018-0094-0

Keywords

Search

Navigation