Skip to main content
SpringerLink
Log in

Control Theory and Synchronization

  • Chapter
  • First Online:
Synchronization of Integral and Fractional Order Chaotic Systems

Part of the book series: Understanding Complex Systems ((UCS))

  • 1183 Accesses

Abstract

This chapter focuses on the main concepts necessary to a proper understanding of the topics developed in this book. In reading this chapter, the emphasis should be on definitions, and in addition, some examples will be given in order to clarify the concepts of control theory and synchronization. The most important issues are topics on differential algebra, differential geometry, and their applications.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 54.99
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. A. Balanov, N. Janson, D. Postnov, O. Sosnovtseva, Synchronization: From Simple to Complex (Springer, New York, 1965)

    Google Scholar 

  2. R.W. Brockett, Nonlinear control theory and differential geometry, in Proceedings of ICM, Warszawa, 1983, pp. 1357–1368

    Google Scholar 

  3. R. Caponetto, G. Dongola, L. Fortuna, I. Petrás, Fractional Order Systems: Modeling and Control Applications. World Scientific Series on Nonlinear Science Series A, vol. 72 (World Scientific Publishing, Singapore, 2010), pp. 1–4

    Google Scholar 

  4. R. Caponetto, G. Dongola, L. Fortuna, I. Petrás, Fractional Order Systems: Modeling and Control Applications. World Scientific Series on Nonlinear Science Series A, vol. 72 (World Scientific Publishing, Singapore, 2010), pp. 59–60

    Google Scholar 

  5. R. Caponetto, G. Dongola, L. Fortuna, I. Petrás, Fractional Order Systems: Modeling and Control Applications. World Scientific Series on Nonlinear Science Series A, vol. 72 (World Scientific Publishing, Singapore, 2010), pp. 62–65

    Google Scholar 

  6. J.-F. Chang, M.-L. Hung, Y.-S. Yang, T.-L. Liao, J.-J. Yan, Controlling chaos of the family of Rössler systems using sliding mode control. Chaos Solitons Fractals 37(2), 609–622 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. A. Charef, H.H. Sun, Y.Y. Tsao, B. Onaral, Fractal system as represented by singularity function. IEEE Trans. Autom. Control 37(9), 1465–1470 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  8. G. Duffing, Erzwungene Schwingung bei veränderlicher Eigenfrequenz und ihre technische Bedeutung (Vieweg, Braunschweig, 1918)

    Google Scholar 

  9. M. Fliess, Generalized controller canonical forms for linear and nonlinear dynamics. IEEE Trans. Autom. Control 35(9), 994–1001 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  10. A.L. Fradkov, Cybernetical Physics: From Control of Chaos to Quantum Control (Springer, Berlin, 2007), p. 213

    Google Scholar 

  11. A.L. Fradkov, B. Andrievsky, R.J. Evans, Adaptive observer-based synchronization of chaotic system with first-order coder in the presence of information constraints. IEEE Trans. Circ. Syst. I Reg. Pap. 55(6), 1685–1694 (2008)

    Article  MathSciNet  Google Scholar 

  12. P. Gaspard, in Encyclopedia of Nonlinear Science, ed. by A. Scott (Routledge, New York, 2005), pp. 808–811

    Google Scholar 

  13. A. Isidori, Nonlinear Control Systems, 3rd edn. Communications and Control Engineering (Springer, New York, 1995)

    Book  MATH  Google Scholar 

  14. M. Javidi, N. Nyamoradi, Numerical chaotic behaviour of the fractional Rikitake system. World Acad. Union 9, 120–129 (2013)

    Google Scholar 

  15. H.K. Khalil, Nonlinear Systems (Prentice Hall, Upper Saddle River, 2002), pp. 4803–4811

    MATH  Google Scholar 

  16. L.M. Kocié, S. Gegovska-Zajkova, S. Kostadinova, On Chua Dynamical System, vol. 2 (Scientific Publications of the State University of Novi Pazar, Novi Pazar, 2010), pp. 53–60

    Google Scholar 

  17. E.R. Kolchin, Differential Algebra and Algebraic Groups (Academic, New York, 1973)

    MATH  Google Scholar 

  18. R. Martínez Guerra, J.L. Mata-Machuca, Generalized synchronization via the differential primitive element. Appl. Math. Comput. 232, 848–857 (2014)

    Article  MathSciNet  Google Scholar 

  19. J.L. Mata Machuca, R. Martínez Guerra, R. López Aguilar, Observadores para Sincronización de Sistemas Caóticos: Un Enfoque Diferencial y Algebraico (Editorial Académica Española, Saarbrücken, 2013)

    Google Scholar 

  20. T. Matsumoto, L.O. Chua, M. Komuro, The double scroll. IEEE Trans. Circ. Syst. Cas-32(8), 798–818 (1985)

    Google Scholar 

  21. H. Nijmeijer, A. Van der Schaft, Nonlinear Dynamical Control Systems (Springer, New York, 1990)

    Book  MATH  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  23. A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, 2002)

    Google Scholar 

  24. J.F. Ritt, Differential Algebra (Dover, New York, 1966)

    Google Scholar 

  25. A.E. Siegman, Lasers (University Science Books, Mill Valley, 1986)

    Google Scholar 

  26. D.E. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edn. (Springer, New York, 1998)

    Book  MATH  Google Scholar 

  27. S.H. Strogatz, Nonlinear Dynamics and Chaos (Perseus Books, Cambridge, 1994), pp. 423–448

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Control Automatico, CINVESTAV-IPN, Mexico, D.F., Mexico

    Rafael Martínez-Guerra, Claudia A. Pérez-Pinacho & Gian Carlo Gómez-Cortés

Authors
  1. Rafael Martínez-Guerra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Claudia A. Pérez-Pinacho
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gian Carlo Gómez-Cortés
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Martínez-Guerra, R., Pérez-Pinacho, C.A., Gómez-Cortés, G.C. (2015). Control Theory and Synchronization. In: Synchronization of Integral and Fractional Order Chaotic Systems. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-15284-4_1

Download citation

Publish with us

Policies and ethics

Search

Navigation