[go: up one dir, main page]
Skip to main content
SpringerLink
Log in

Hybrid Synchronization of Identical Chaotic Systems Using Sliding Mode Control and an Application to Vaidyanathan Chaotic Systems

  • Chapter
  • First Online:
Advances and Applications in Sliding Mode Control systems

Part of the book series: Studies in Computational Intelligence ((SCI,volume 576))

Abstract

Hybrid phase synchronization is a new type of synchronization of a pair of chaotic systems called the master and slave systems. In hybrid phase synchronization, the odd numbered states of the master and slave systems are completely synchronized (CS), while their even numbered states are anti-synchronized (AS). The hybrid phase synchronization has applications in secure communications and cryptosystems. This work derives a new result for the hybrid phase synchronization of identical chaotic systems using sliding mode control. The main result has been proved using Lyapunov stability theory. Sliding mode control (SMC) is well-known as a robust approach and useful for controller design in systems with parameter uncertainties. As an application of this general result, a sliding mode controller is derived for the hybrid phase synchronization of the identical 3-D Vaidyanathan chaotic systems (2014). MATLAB simulations have been provided to illustrate the Vaidyanathan system and the hybrid synchronizer results for the identical Vaidyanathan systems.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
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

Similar content being viewed by others

References

  • Arneodo, A., Coullet, P., Tresser, C.: Possible new strange attractors with spiral structure. Common. Math. Phys. 79(4), 573–576 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  • Bidarvatan, M., Shahbakhti, M., Jazayeri, S.A., Koch, C.R.: Cycle-to-cycle modeling and sliding mode control of blended-fuel HCCI engine. Control Eng. Pr 24, 79–91 (2014)

    Google Scholar 

  • Cai, G., Tan, Z.: Chaos synchronization of a new chaotic system via nonlinear control. J. Uncertain Syst. 1(3), 235–240 (2007)

    Google Scholar 

  • Carroll, T.L., Pecora, L.M.: Synchronizing chaotic circuits. IEEE Trans. Circuits Syst. 38(4), 453–456 (1991)

    Article  Google Scholar 

  • Chen, G., Ueta, T.: Yet another chaotic attractor. Intern. J. Bifurc. Chaos 9(7), 1465–1466 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, H.K., Lee, C.I.: Anti-control of chaos in rigid body motion. Chaos, Solitons Fractals 21(4), 957–965 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, W.-H., Wei, D., Lu, X.: Global exponential synchronization of nonlinear time-delay Lure systems via delayed impulsive control. Commun. Nonlinear Sci. Numer. Simul. 19(9), 3298–3312 (2014)

    Article  MathSciNet  Google Scholar 

  • Das, S., Goswami, D., Chatterjee, S., Mukherjee, S.: Stability and chaos analysis of a novel swarm dynamics with applications to multi-agent systems. Eng. Appl. Artif. Intell. 30, 189–198 (2014)

    Article  Google Scholar 

  • Feki, M.: An adaptive chaos synchronization scheme applied to secure communication. Chaos, Solitons Fractals 18(1), 141–148 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Feng, Y., Han, F., Yu, X.: Chattering free full-order sliding-mode control. Automatica 50(4), 1310–1314 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Gan, Q., Liang, Y.: Synchronization of chaotic neural networks with time delay in the leakage term and parametric uncertainties based on sampled-data control. J. Frankl. Inst. 349(6), 1955–1971 (2012)

    Article  MathSciNet  Google Scholar 

  • Gaspard, P.: Microscopic chaos and chemical reactions. Physica A: Stat. Mech. Appl 263(1–4), 315–328 (1999)

    Article  Google Scholar 

  • Gibson, W.T., Wilson, W.G.: Individual-based chaos: Extensions of the discrete logistic model. J. Theor. Biol. 339, 84–92 (2013)

    Google Scholar 

  • Guégan, D.: Chaos in economics and finance. Ann. Rev. Control 33(1), 89–93 (2009)

    Article  Google Scholar 

  • Hamayun, M.T., Edwards, C., Alwi, H.: A fault tolerant control allocation scheme with output integral sliding modes. Automatica 49(6), 1830–1837 (2013)

    Article  MathSciNet  Google Scholar 

  • Huang, J.: Adaptive synchronization between different hyperchaotic systems with fully uncertain parameters. Phys. Lett. A 372(27–28), 4799–4804 (2008)

    Article  MATH  Google Scholar 

  • Huang, X., Zhao, Z., Wang, Z., Li, Y.: Chaos and hyperchaos in fractional-order cellular neural networks. Neurocomputing 94, 13–21 (2012)

    Article  Google Scholar 

  • Jiang, G.-P., Zheng, W.X., Chen, G.: Global chaos synchronization with channel time-delay. Chaos, Solitons Fractals 20(2), 267–275 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Karthikeyan, R., Sundarapandian, V.: Hybrid chaos synchronization of four-scroll systems via active control. J. Elect. Eng. 65(2), 97–103 (2014)

    Google Scholar 

  • Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Netw 32, 245–256 (2012)

    Article  MATH  Google Scholar 

  • Kengne, J., Chedjou, J.C., Kenne, G., Kyamakya, K.: Dynamical properties and chaos synchronization of improved Colpitts oscillators. Commun. Nonlinear Sci. Numer Simul. 17(7), 2914–2923 (2012)

    Article  MathSciNet  Google Scholar 

  • Khalil, H.K.: Nonlinear Systems. Prentice Hall, Upper Saddle River (2001)

    Google Scholar 

  • Kyriazis, M.: Applications of chaos theory to the molecular biology of aging. Exp. Gerontol. 26(6), 569–572 (1991)

    Article  Google Scholar 

  • Li, N., Pan, W., Yan, L., Luo, B., Zou, X.: Enhanced chaos synchronization and communication in cascade-coupled semiconductor ring lasers. Commun. Nonlinear Sci. Numer. Simul. 19(6), 1874–1883 (2014)

    Article  Google Scholar 

  • Li, N., Zhang, Y., Nie, Z.: Synchronization for general complex dynamical networks with sampled-data. Neurocomputing 74(5), 805–811 (2011)

    Article  Google Scholar 

  • Lian, S., Chen, X.: Traceable content protection based on chaos and neural networks. Appl. Soft Comput. 11(7), 4293–4301 (2011)

    Article  Google Scholar 

  • Lin, W.: Adaptive chaos control and synchronization in only locally Lipschitz systems. Phys. Lett. A 372(18), 3195–3200 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, C., Liu, T., Liu, L., Liu, K.: A new chaotic attractor. Chaos, Solitions Fractals 22(5), 1031–1038 (2004)

    Article  MATH  Google Scholar 

  • Liu, L., Zhang, C., Guo, Z.A.: Synchronization between two different chaotic systems with nonlinear feedback control. Chin. Phys. 16(6), 1603–1607 (2007)

    Article  Google Scholar 

  • Lorenz, E.N.: Deterministic periodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)

    Article  Google Scholar 

  • Lü, J., Chen, G.: A new chaotic attractor coined. Intern. J. Bifurc. Chaos 12(3), 659–661 (2002)

    Article  MATH  Google Scholar 

  • Lu, W., Li, C., Xu, C.: Sliding mode control of a shunt hybrid active power filter based on the inverse system method. Intern. J. Elect. Power Energy Syst. 57, 39–48 (2014)

    Article  Google Scholar 

  • Mondal, S., Mahanta, C.: Adaptive second order terminal sliding mode controller for robotic manipulators. J. Franklin Inst. 351(4), 2356–2377 (2014)

    Article  MathSciNet  Google Scholar 

  • Murali, K., Lakshmanan, M.: Secure communication using a compound signal from generalized chaotic systems. Phys. Lett. A 241(6), 303–310 (1998)

    Article  MATH  Google Scholar 

  • Nehmzow, U., Walker, K.: Quantitative description of robotenvironment interaction using chaos theory. Robot. Auton. Syst. 53(3–4), 177–193 (2005)

    Article  Google Scholar 

  • Njah, A.N., Ojo, K.S., Adebayo, G.A., Obawole, A.O.: Generalized control and synchronization of chaos in RCL-shunted Josephson junction using backstepping design. Physica C 470(13–14), 558–564 (2010)

    Article  Google Scholar 

  • Ouyang, P.R., Acob, J., Pano, V.: PD with sliding mode control for trajectory tracking of robotic system. Robot. Comput. Integr. Manuf. 30(2), 189–200 (2014)

    Article  Google Scholar 

  • Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64(8), 821–824 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • Perruquetti, W., Barbot, J.P.: Sliding Mode Control in Engineering. Marcel Dekker, New York (2002)

    Google Scholar 

  • Petrov, V., Gaspar, V., Masere, J., Showalter, K.: Controlling chaos in Belousov-Zhabotinsky reaction. Nature 361, 240–243 (1993)

    Article  Google Scholar 

  • Qu, Z.: Chaos in the genesis and maintenance of cardiac arrhythmias. Prog. Biophys. Mol. Biol. 105(3), 247–257 (2011)

    Article  Google Scholar 

  • Rafikov, M., Balthazar, J.M.: On control and synchronization in chaotic and hyperchaotic systems via linear feedback control. Commun. Nonlinear Sci. Numer. Simul. 13(7), 1246–1255 (2007)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Rasappan, S., Vaidyanathan, S.: Hybrid synchronization of n-scroll Chua and Lur’e chaotic systems via backstepping control with novel feedback. Arch. Control Sci. 22(3), 343–365 (2012b)

    MATH  Google Scholar 

  • Rhouma, R., Belghith, S.: Cryptoanalysis of a chaos based cryptosystem on DSP. Commun. Nonlinear Sci. Numer. Simul. 16(2), 876–884 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  • Sarasu, P., Sundarapandian, V.: Adaptive controller design for the generalized projective synchronization of 4-scroll systems. Intern. J. Syst. Signal Control Eng. Appl. 5(2), 21–30 (2012a)

    Google Scholar 

  • Sarasu, P., Sundarapandian, V.: Generalized projective synchronization of two-scroll systems via adaptive control. Int. J. Soft Comput. 7(4), 146–156 (2012b)

    Article  Google Scholar 

  • Sarasu, P., Sundarapandian, V.: Generalized projective synchronization of two-scroll systems via adaptive control. Eur. J. Sci. Res. 72(4), 504–522 (2012c)

    Google Scholar 

  • Shahverdiev, E.M., Bayramov, P.A., Shore, K.A.: Cascaded and adaptive chaos synchronization in multiple time-delay laser systems. Chaos, Solitons Fractals 42(1), 180–186 (2009)

    Article  Google Scholar 

  • Shahverdiev, E.M., Shore, K.A.: Impact of modulated multiple optical feedback time delays on laser diode chaos synchronization. Optics Commun. 282(17), 3568–3572 (2009)

    Article  Google Scholar 

  • Sharma, A., Patidar, V., Purohit, G., Sud, K.K.: Effects on the bifurcation and chaos in forced Duffing oscillator due to nonlinear damping. Commun. Nonlinear Sci. Numer. Simul. 17(6), 2254–2269 (2012)

    Article  MathSciNet  Google Scholar 

  • Sprott, J.C.: Some simple chaotic flows. Phys. Rev. E 50(2), 647–650 (1994)

    Article  MathSciNet  Google Scholar 

  • Sprott, J.C.: Competition with evolution in ecology and finance. Phys. Lett. A 325(5–6), 329–333 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Suérez, I.: Mastering chaos in ecology. Ecol. Model. 117(2–3), 305–314 (1999)

    Article  Google Scholar 

  • Sundarapandian, V.: Output regulation of the Lorenz attractor. Far East J. Math. Sci. 42(2), 289–299 (2010)

    MathSciNet  MATH  Google Scholar 

  • Sundarapandian, V., Karthikeyan, R.: Hybrid synchronization of hyperchaotic Lorenz and hyperchaotic Chen systems via active control. J. Eng. Appl. Sci. 7(3), 254–264 (2012)

    Article  Google Scholar 

  • Sundarapandian, V., Pehlivan, I.: Analysis, control, synchronization, and circuit design of a novel chaotic system. Math. Comput. Model. 55(7–8), 1904–1915 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Suresh, R., Sundarapandian, V.: Global chaos synchronizatoin of a family of \(n\)-scroll hyperchaotic Chua circuits using backstepping control with recursive feedback. Far East J. Math. Sci. 73(1), 73–95 (2013)

    MATH  Google Scholar 

  • Tigan, G., Opris, D.: Analysis of a 3D chaotic system. Chaos, Solitons Fractals 36, 1315–1319 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Tu, J., He, H., Xiong, P.: Adaptive backstepping synchronization between chaotic systems with unknown Lipschitz constant. Appl. Math. Comput. 236, 10–18 (2014)

    Article  MathSciNet  Google Scholar 

  • Ucar, A., Lonngren, K.E., Bai, E.W.: Chaos synchronization in RCL-shunted Josephson junction via active control. Chaos, Solitons Fractals 31(1), 105–111 (2007)

    Article  Google Scholar 

  • Usama, M., Khan, M.K., Alghatbar, K., Lee, C.: Chaos-based secure satellite imagery cryptosystem. Comput. Math. Appl. 60(2), 326–337 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Utkin, V.I.: Sliding Modes in Control and Optimization. Springer, New York (1992)

    Google Scholar 

  • Vaidyanathan, S.: Anti-synchronization of Newton-Leipnik and Chen-Lee chaotic systems by active control. Intern. J. Control Theory Appl. 4(2), 131–141 (2011)

    Google Scholar 

  • Vaidyanathan, S.: Adaptive backstepping controller and synchronizer design for Arneodo chaotic system with unknown parameters. Intern. J. Comput. Sci. Inform. Technol. 4(6), 145–159 (2012a)

    Article  Google Scholar 

  • Vaidyanathan, S.: Anti-synchronization of Sprott-L and Sprott-M chaotic systems via adaptive control. Intern. J. Control Theory Appl. 5(1), 41–59 (2012b)

    MathSciNet  Google Scholar 

  • Vaidyanathan, S.: Output regulation of the Liu chaotic system. Appl. Mech. Mater. 110–116, 3982–3989 (2012c)

    Google Scholar 

  • Vaidyanathan, S.: A new six-term 3-D chaotic system with an exponential nonlinearity. Far East J. Math. Sci. 79(1), 135–143 (2013a)

    MATH  Google Scholar 

  • Vaidyanathan, S.: Analysis and adaptive synchronization of two novel chaotic systems with hyperbolic sinusoidal and cosinusoidal nonlinearity and unknown parameters. J. Eng. Sci. Technol. Rev. 6(4), 53–65 (2013b)

    MathSciNet  Google Scholar 

  • Vaidyanathan, S.: A new eight-term 3-D polynomial chaotic system with three quadratic nonlinearities. Far East J. Math. Sci. 84(2), 219–226 (2014)

    MathSciNet  MATH  Google Scholar 

  • Vaidyanathan, S., Rajagopal, K.: Global chaos synchronization of four-scroll chaotic systems by active nonlinear control. Intern. J. Control Theory Appl. 4(1), 73–83 (2011a)

    Google Scholar 

  • Vaidyanathan, S., Rajagopal, K.: Hybrid synchronization of hyperchaotic Wang-Chen and hyperchaotic Lorenz systems by active nonlinear control. Intern. J. Syst. Signal Control Eng. Appl. 4(3), 55–61 (2011b)

    Google Scholar 

  • Vaidyanathan, S., Sampath, S.: Anti-synchronization of four-scroll chaotic systems via sliding mode control. Int. J. Autom. Comput. 9(3), 274–279 (2012)

    Article  Google Scholar 

  • Volos, C.K., Kyprianidis, I.M., Stouboulos, I.N.: Experimental investigation on coverage performance of a chaotic autonomous mobile robot. Robot. Autonom. Syst. 61(12), 1314–1322 (2013)

    Article  Google Scholar 

  • Wang, F., Liu, C.: A new criterion for chaos and hyperchaos synchronization using linear feedback control. Phys. Lett. A 360(2), 274–278 (2006)

    Article  MATH  Google Scholar 

  • Witte, C.L., Witte, M.H.: Chaos and predicting varix hemorrhage. Med. Hypotheses 36(4), 312–317 (1991)

    Article  MathSciNet  Google Scholar 

  • Wu, X., Guan, Z.-H., Wu, Z.: Adaptive synchronization between two different hyperchaotic systems. Nonlinear Analysis: Theory, Methods Appl. 68(5), 1346–1351 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Xiao, X., Zhou, L., Zhang, Z.: Synchronization of chaotic Lure systems with quantized sampled-data controller. Commun. Nonlinear Sci. Numer. Simul. 19(6), 2039–2047 (2014)

    Article  MathSciNet  Google Scholar 

  • Yuan, G., Zhang, X., Wang, Z.: Generation and synchronization of feedback-induced chaos in semiconductor ring lasers by injection-locking. Optik Intern. J. Light Electron Optics 125(8), 1950–1953 (2014)

    Article  Google Scholar 

  • Zaher, A.A., Abu-Rezq, A.: On the design of chaos-based secure communication systems. Commun. Nonlinear Syst. Numer. Simul. 16(9), 3721–3727 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, H., Zhou, J.: Synchronization of sampled-data coupled harmonic oscillators with control inputs missing. Syst. Control Lett. 61(12), 1277–1285 (2012)

    Article  MATH  Google Scholar 

  • Zhang, J., Li, C., Zhang, H., Yu, J.: Chaos synchronization using single variable feedback based on backstepping method. Chaos, Solitons Fractals 21(5), 1183–1193 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, X., Liu, X., Zhu, Q.: Adaptive chatter free sliding mode control for a class of uncertain chaotic systems. Appl. Math. Comput. 232, 431–435 (2014)

    Article  MathSciNet  Google Scholar 

  • Zhou, W., Xu, Y., Lu, H., Pan, L.: On dynamics analysis of a new chaotic attractor. Phys. Lett. A 372(36), 5773–5777 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research and Development Centre, Vel Tech University, Avadi, Chennai, 600062, Tamil Nadu, India

    Sundarapandian Vaidyanathan

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

    Ahmad Taher Azar

Authors
  1. Sundarapandian Vaidyanathan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ahmad Taher Azar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sundarapandian Vaidyanathan .

Editor information

Editors and Affiliations

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

    Ahmad Taher Azar

  2. Department of Engineering Design and Mathematics, University of the West of England, Bristol, United Kingdom

    Quanmin Zhu

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Vaidyanathan, S., Azar, A.T. (2015). Hybrid Synchronization of Identical Chaotic Systems Using Sliding Mode Control and an Application to Vaidyanathan Chaotic Systems. In: Azar, A., Zhu, Q. (eds) Advances and Applications in Sliding Mode Control systems. Studies in Computational Intelligence, vol 576. Springer, Cham. https://doi.org/10.1007/978-3-319-11173-5_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-11173-5_20

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-11172-8

  • Online ISBN: 978-3-319-11173-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation