Skip to main content
SpringerLink
Log in

Multi-scroll Chaotic Oscillator Based on a First-Order Delay Differential Equation

  • Chapter
  • First Online:
Chaos Modeling and Control Systems Design

Part of the book series: Studies in Computational Intelligence ((SCI,volume 581))

Abstract

After the discovery of the well-known chaotic Lorenz’s system, the study of chaos has received considerable attention due to its promising applications in a variety of fields, ranging from physics, economics, biology to engineering. Moreover, chaotic systems with multiple scrolls can exhibit more rich dynamics than the general chaotic ones with few attractors. This expansion of dynamics leads to multi-scroll chaotic oscillators showing better performance in several chaotic-based applications, such as secure communication, encrypting fingerprint image, controlling motion directions of autonomous mobile robots, or generating pseudo random numbers etc. As a result, investigating new chaotic oscillators with multiple scrolls has been become an attractive research direction of both theoretical and practical interest recently. Although numerous approaches for constructing multi-scroll attractors from conventional three-dimension chaotic systems have been reported intensively, there are few publications regarding the multi-scroll attractors from infinite dimensional time-delay systems. This work presents a new multi-scroll chaotic oscillator and its circuital design. This chaotic system is described by a first-order delay differential equation with piecewise linear function. It is shown through simulations that the proposed system can exhibit odd number of scrolls of chaotic attractors such as three-, five-, seven-, and nine-scroll attractors. In addition, the detailed implementation of the proposed multi-scroll oscillator using the electronic simulation package Multisim is also presented to show the feasibility of the oscillator. The Multisim results of the chaotic oscillator are well agree with the numerical simulation results. It is noting that the new multi-scroll chaotic circuit has been designed with simple common components, like resistors, capacitors, and operational amplifiers.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  Google Scholar 

  2. Hasselblatt, B., Katok, A.: A First Course in Dynamics: With a Panorama of Recent Developments. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  3. Strogatz, S.H.: Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering. Perseus Books, Massachusetts (1994)

    Google Scholar 

  4. Boccaletti, S., Kurths, J., Osipov, G., Valladares, D.L., Zhou, C.S.: The synchronization of chaotic systems. Phys. Rep. 366, 1–101 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen, G.R.: Controlling Chaos and Bifurcations in Engineering Systems. CRC Press, Boca Raton (1999)

    Google Scholar 

  6. Han, F., Hu, J., Yu, X., Wang, Y.: Fingerprint images encryption via multi-scroll chaotic attractors. Appl. Math. Comput. 185, 931–939 (2007)

    Article  MATH  Google Scholar 

  7. Yalcin, M.E.: Increasing the entropy of a random number generator using n-scroll chaotic attractors. Int. J. Bifurcat. Chaos 17, 4471–4479 (2007)

    Article  Google Scholar 

  8. Gamez-Guzman, L., Cruz-Hernandez, C., Lopez-Gutierrez, R., Garcia-Guerrero, E.E.: Synchronization of chua’s circuits with multi-scroll attractors: application to communication. Commun. Nonlinear Sci. Numer. Simul. 14, 2765–2775 (2009)

    Article  Google Scholar 

  9. Orue, A.B., Alvarez, G., Pastor, G., Romera, M., Montoya, F., Li, S.: A new parameter determination method for some double-scroll chaotic systems and its applications to chaotic cryptanalysis. Commun. Nonlinear Sci. Numer. Simul. 15, 3471–3483 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Lu, J.H., Chen, G.R.: Generating multiscroll chaotic attractors: theories, methods and applications. Int. J. Bifurcat. Chaos 16, 775–858 (2006)

    Article  Google Scholar 

  11. Ikeda, K., Daido, H., Akimoto, O.: Optical turbulence: chaotic behavior of transmitted light from a ring cavity. Phys. Rev. Lett. 45, 709–712 (1980)

    Article  Google Scholar 

  12. Mackey, M.C., Glass, L.: Oscillation and chaos in physiological control system. Science 197, 287–289 (1977)

    Article  Google Scholar 

  13. Ikeda, K., Matsumoto, K.: High-dimensional chaotic behaviour in system with time-delayed feedback. Physica D 29, 223–235 (1987)

    Article  MATH  Google Scholar 

  14. Xia, Y., Fu, M., Shi, P.: Analysis and synthesis of dynamical systems with time-delays. Springer, New York (2009)

    Book  MATH  Google Scholar 

  15. Lu, H., He, Z.: Chaotic behavior in first-order autonomous continuous-time systems with delay. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 43, 700–702 (1996)

    Article  Google Scholar 

  16. Kilinc, S., Yalcin, M., Ozoguz, S.: Multiscroll chaotic attractors from a hysteresis based time-delay differential equation. Int. J. Bifurcat. Chaos 20, 3275–3281 (2010)

    Article  MATH  Google Scholar 

  17. Matsumoto, T.: A chaotic attractor from Chua’s circuit. IEEE Trans. Circuit Syst. I 31, 1055–1058 (1984)

    Article  MATH  Google Scholar 

  18. Fortuna, L., Frasca, M., Xibilia, M.G.: Chua’s circuit implementation: Yesterday, today and tomorrow. World Scientific, Singapore (2009)

    Google Scholar 

  19. Suykens, J.A.K., Vandewalle, J.: Generation of n-double scrolls (n = 1, 2, 3, 3, …). IEEE Trans. Circuit Syst. I 31, 1055–1058 (1984)

    Article  Google Scholar 

  20. Carbajal-Gomez, V.H., Tlelo-Cuautle, E., Fernandez, F.V.: Optimizing the positive Lyapunov exponent in multi-scroll chaotic oscillators with differential evolution algorithm. Appl. Math. Comput. 219, 8163–8168 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  21. Dadras, S., Momeni, H.R.: A novel three-dimensional autonomous chaotic system generating two, three and four-scroll attractors. Phys. Lett. A 373, 3637–3642 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. Liu, C., Yi, J., Xi, X., An, L., Fu, Y.: Research on the multi-scroll chaos generation based on Jerk mode. Procedia Eng. 29, 957–961 (2012)

    Article  Google Scholar 

  23. Luo, X.H., Lui, H.Q., Dai, X.G.: A family of multi-scroll chaotic attractors and its circuit design. Acta. Phys. Sin. 57, 7511–7516 (2008)

    MATH  Google Scholar 

  24. Yalcin, M.E., Suykens, J.A.K., Vandewalle, J.: Cellular Neural Networks, Multi-Scroll Chaos and Synchronization. World Scientific, Singapore (2005)

    MATH  Google Scholar 

  25. Srinivasan, K., Mohamed, I.R., Murali, K., Lakshmanan, M., Sinha, S.: Design of time delayed chaotic circuit with threshold controller. Int. J. Bifurcat. Chaos 20, 2185–2191 (2010)

    Article  Google Scholar 

  26. Wang, L., Yang, X.: Generation of multi-scroll delayed chaotic oscillator. Electron. Lett. 42, 1439–1441 (2006)

    Article  Google Scholar 

  27. Duan, S., Wang, L.: A novel delayed chaotic neural model and its circuitry implementation. Comput. Math. Appl. 57, 1736–1742 (2009)

    Article  MATH  Google Scholar 

  28. Horbelt, W., Timmer, J., Voss, H.U.: Parameter estimation in nonlinear delayed feedback systems from noisy data. Phys. Lett. A 299, 513–521 (2002)

    Article  MATH  Google Scholar 

  29. Le, L.B., Konishi, K., Hara, N.: Design and experimental verification of multiple delay feedback control for time-delay nonlinear oscillator. Nonlinear Dyn. 67, 1407–1418 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  30. Tamasevicius, A., Pyragine, T., Meskauskas, M.: Two scroll attractor in a delay dynamical system. Int. J. Bifurcat. Chaos 17, 3455–3460 (2007)

    Article  MATH  Google Scholar 

  31. Sprott, J.C.: A simple chaotic delay differential equation. Phys. Lett. A 366, 397–402 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  32. Wang, L., Duan, S., Yang, X. (2008). Generation and circuitry implementation of N-double scroll delayed chaotic attractors . In: Proceedings of the International Workshop on Nonlinear Dynamics and Synchronization

    Google Scholar 

  33. Bouali, S., Buscarino, A., Fortuna, L., Frasca, M., Gambuzza, L.V. (2012). Emulating complex business cycles by using an electronic analogue. Nonlinear Anal. Real World Appl. 13, 2459–2465

    Google Scholar 

  34. Sprott, J.C.: Simple chaotic systems and circuits. Am. J. Phys. 68, 758–763 (2013)

    Article  Google Scholar 

  35. Sprott, J.C.: A proposed standard for the publication of new chaotic systems. Int. J. Bifurcat. Chaos 21, 2391–2394 (2011)

    Article  MathSciNet  Google Scholar 

  36. Cuomo, K.M., Oppenheim, A.V.: Circuit implementation of synchronized chaos with applications to communications. Phys. Rev. Lett. 71, 65–68 (1993)

    Article  Google Scholar 

  37. Volos, C.K., Kyprianidis, I.M., Stouboulos, I.N.: Various synchronization phenomena in bidirectionally coupled double scroll circuits. Commun. Nonlinear Sci. Numer. Simul. 71, 3356–3366 (2011)

    Article  MathSciNet  Google Scholar 

  38. Barakat, M., Mansingka, A., Radwan, A.G., Salama, K.N.: Generalized hardware post processing technique for chaos-based pseudorandom number generators. ETRI J. 35, 448–458 (2013)

    Article  Google Scholar 

  39. Yalcin, M.E., Suykens, J.A.K., Vandewalle, J.: True random bit generation from a double-scroll attractor. IEEE Trans. Circuits Syst. I, Regular Papers 51, 1395–1404 (2004)

    Article  MathSciNet  Google Scholar 

  40. Volos, C.K., Kyprianidis, I.M., Stouboulos, I.N.: Image encryption process based on chaotic synchronization phenomena. Sig. Process. 93, 1328–1340 (2013)

    Article  Google Scholar 

  41. Hoang, T.-M., Nguyen, T., Nguyen, V., Chedjou, J., and Kyamakya, K. (2009). Design and simulation of circuit for synchronization of multidelay feeback systems. In: Proceedings of the International Symposium on Theoretical Engineering

    Google Scholar 

  42. Namajunas, A., Pyragas, K., Tamasevicius, A.: An electronic analog of the Mackey-Glass system. Phys. Lett. A 201, 42–46 (1995)

    Article  Google Scholar 

  43. Buscarino, A., Fortuna, L., Frasca, M., Sciuto, G.: Design of time-delay chaotic electronic circuits. IEEE Trans. Circuits Syst. I Regular Papers 58, 1888–1896 (2011)

    Article  MathSciNet  Google Scholar 

  44. Liao, X., Guo, S., Li, C.: Stability and bifurcation analysis in tri-neuron model with time delay. Nonlinear Dyn. 49, 319–345 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  45. Pei, L., Wang, Q., Shi, H.: Bifurcation dynamics of the modified physiological model of artificial pancreas with insulin secretion delay. Nonlinear Dyn. 63, 417–427 (2011)

    Article  MathSciNet  Google Scholar 

  46. Yongzhen, P., Shuping, L., Changguo, L.: Effect of delay on a predator-prey model with parasitic infection. Nonlinear Dyn. 63, 311–321 (2011)

    Article  MathSciNet  Google Scholar 

  47. Nagatani, T.: Chaos and dynamical transition of a single vehicle induced by traffic light and speedup. Phys. A 348, 561–571 (2005)

    Article  Google Scholar 

  48. Peil, M., Jacquot, M., Chembo, Y., Larger, L., Erneux, T.: Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators. Phys. Rev. E 79, 026208 (2009)

    Article  Google Scholar 

  49. Safonov, L.A., Tomer, E., Strygin, V., Ashkenazy, Y., Havlin, S.: Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic. Chaos 12, 1006–1014 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  50. Wang, X., Yu, H., Zhong, S., Agarwal, R.: Analysis of mathematics and dynamics in a food web system with impulsive perturbations and distributed time delay. Appl. Math. Model. 34, 3850–3863 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  51. Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170, 421–428 (1992)

    Article  Google Scholar 

  52. Sun, C., Lin, Y., Han, M.: Stability and Hopf bifurcation for an epidemic disease model with delay. Chaos, Solitons Fractals 30, 204–216 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  53. Ponomarenko, V., Prokhorov, M., Karavaev, A., Kulminskiy, D.: An experimental digital communication scheme based on chaotic time-delay system. Nonlinear Dyn. 74, 1013–1020 (2013)

    Article  MathSciNet  Google Scholar 

  54. Tang, Y., Wang, Z., Fang, J.A.: Image encryption using chaotic coupled map lattices with time-varying delays. Commun. Nonlin. Sci. Numer. Simul. 15, 2456–2468 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  55. Kwon, O., Park, J., Lee, S.: Secure communication based on chaotic synchronization via interval time varying delay feedback control. Nonlinear Dyn. 63, 239–252 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  56. Ponomarenko, V., Karavaev, A., Glukhovskaya, E., Prokhorov, M.: Hidden data transmission based on time delayed feedback system with switched delay time. Tech. Phys. Lett. 38, 51–54 (2012)

    Article  Google Scholar 

  57. Prokhorov, M.D., Ponomarenko, V.I.: Encryption and decryption of information in chaotic communication systems governed by delay-differential equations. Chaos, Solitons Fractals 63, 871–877 (2008)

    Article  Google Scholar 

Download references

Acknowledgments

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.99-2013.06.

Author information

Authors and Affiliations

  1. School of Electronics and Telecommunications, Hanoi University of Science and Technology, Hanoi, Vietnam

    Viet-Thanh Pham

  2. Physics Department, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Christos K. Volos

  3. Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India

    Sundarapandian Vaidyanathan

Authors
  1. Viet-Thanh Pham
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christos K. Volos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sundarapandian Vaidyanathan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Viet-Thanh Pham .

Editor information

Editors and Affiliations

  1. Faculty of Computers and Information, Benha University, Benha, Egypt

    Ahmad Taher Azar

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

    Sundarapandian Vaidyanathan

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Pham, VT., Volos, C.K., Vaidyanathan, S. (2015). Multi-scroll Chaotic Oscillator Based on a First-Order Delay Differential Equation. In: Azar, A., Vaidyanathan, S. (eds) Chaos Modeling and Control Systems Design. Studies in Computational Intelligence, vol 581. Springer, Cham. https://doi.org/10.1007/978-3-319-13132-0_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-13132-0_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-13131-3

  • Online ISBN: 978-3-319-13132-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation