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Difference map and its electronic circuit realization

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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.

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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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Authors and Affiliations

  1. Division de Matemáticas Aplicadas, Instituto Potosino de Investigación Científica y Tecnológica, Camino a la Presa San José 2055, 78216, México, México

    M. García-Martínez & E. Campos-Cantón

  2. Facultad de Ciencias, Universidad Autónoma de San Luis Potosí, Zona universitaria, Avenida Salvador Nava S/N, 78290, México, México

    I. Campos-Cantón

  3. Departamento de Físico Matemáticas, Universidad Autónoma de San Luis Potosí, Zona universitaria, Avenida Niño Artillero S/N, 78290, México, México

    E. Campos-Cantón

  4. Department of Control Theory, Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 4, 18208, Prague, Czech Republic

    S. Čelikovský

Authors
  1. M. García-Martínez
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  2. I. Campos-Cantón
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  3. E. Campos-Cantón
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  4. S. Čelikovský
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Correspondence to M. García-Martínez.

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García-Martínez, M., Campos-Cantón, I., Campos-Cantón, E. et al. Difference map and its electronic circuit realization. Nonlinear Dyn 74, 819–830 (2013). https://doi.org/10.1007/s11071-013-1007-4

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  • DOI: https://doi.org/10.1007/s11071-013-1007-4

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