Difference map and its electronic circuit realization
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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.
KeywordsChaotic behavior Lyapunov exponent Bifurcation parameter Bifurcation diagram Stability analysis
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.
- 14.Kahng, B.: Redefining chaos: Devaney-chaos for piecewise continuous dynamical systems. Int. J. Math. Models Methods Appl. Sci. 3(4), 317–326 (2009) Google Scholar
- 25.Chua, L., Komuro, M., Matsumoto, T.: The double scroll family: parts I and II. IEEE Trans. Circuits Syst. I 33, 1073–1118 (1986) Google Scholar
- 37.Verhulst, P.F.: Notice sur la loi que la population poursuit dans son accroissement. Corresp. Math. Phys. 10, 113–121 (1838) Google Scholar