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Nonlinear Dynamics

, Volume 74, Issue 3, pp 819–830 | Cite as

Difference map and its electronic circuit realization

  • M. García-MartínezEmail author
  • I. Campos-Cantón
  • E. Campos-Cantón
  • S. Čelikovský
Original Paper

Abstract

In this paper we study the dynamical behavior of the one-dimensional discrete-time system, the so-called iterated map. Namely, a bimodal quadratic map is introduced which is obtained as an amplification of the difference between well-known logistic and tent maps. Thus, it is denoted as the so-called difference map. The difference map exhibits a variety of behaviors according to the selection of the bifurcation parameter. The corresponding bifurcations are studied by numerical simulations and experimentally. The stability of the difference map is studied by means of Lyapunov exponent and is proved to be chaotic according to Devaney’s definition of chaos. Later on, a design of the electronic implementation of the difference map is presented. The difference map electronic circuit is built using operational amplifiers, resistors and an analog multiplier. It turns out that this electronic circuit presents fixed points, periodicity, chaos and intermittency that match with high accuracy to the corresponding values predicted theoretically.

Keywords

Chaotic behavior Lyapunov exponent Bifurcation parameter Bifurcation diagram Stability analysis 

Notes

Acknowledgements

M. García-Martínez is a doctoral fellows of CONACYT (Mexico) in the Graduate Program on Control and Dynamical Systems at DMAp-IPICYT.

E. Campos-Cantón acknowledges CONACYT for the financial support through project No. 181002.

S. Čelikovský has been supported by the Czech Science Foundation through the research grant no. 13-20433S.

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • M. García-Martínez
    • 1
    Email author
  • I. Campos-Cantón
    • 2
  • E. Campos-Cantón
    • 1
    • 3
  • S. Čelikovský
    • 4
  1. 1.Division de Matemáticas AplicadasInstituto Potosino de Investigación Científica y TecnológicaMéxicoMéxico
  2. 2.Facultad de CienciasUniversidad Autónoma de San Luis PotosíMéxicoMéxico
  3. 3.Departamento de Físico MatemáticasUniversidad Autónoma de San Luis PotosíMéxicoMéxico
  4. 4.Department of Control Theory, Institute of Information Theory and AutomationAcademy of Sciences of the Czech RepublicPragueCzech Republic

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