This paper presents a three-dimensional topological horseshoe in the hyperchaotic generalized Hénon map. It looks like a planar Smale horseshoe with an additional vertical expansion, so we call it 3D Smale horseshoe. In this way, a computer assisted verification of existence of hyperchaos is provided by means of interval analysis.
A. Szymczak, “The Conley index and symbolic dynamics,” Topology, vol. 35, no. 2, pp. 287–299, 1996.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetJ. Kennedy and J. A. Yorke, “Topological horseshoes,” Transactions of the American Mathematical Society, vol. 353, no. 6, pp. 2513–2530, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetG. Baier and M. Klein, “Maximum hyperchaos in generalized Hénon maps,” Physics Letters. A, vol. 151, no. 6-7, pp. 281–284, 1990.
View at: Publisher Site | Google Scholar | MathSciNetG. Grassi and D. A. Miller, “Theory and experimental realization of observer-based discrete-time hyperchaos synchronization,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 49, no. 3, pp. 373–378, 2002.
View at: Publisher Site | Google ScholarD. A. Miller and G. Grassi, “A discrete generalized hyperchaotic Henon map circuit,” in Proceedings of the 44th IEEE Midwest Symposium on Circuits and Systems (MWSCAS '01), vol. 1, pp. 328–331, Dayton, Ohio, USA, August 2001.
View at: Publisher Site | Google ScholarX.-S. Yang and Y. Tang, “Horseshoes in piecewise continuous maps,” Chaos, Solitons and Fractals, vol. 19, no. 4, pp. 841–845, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetS. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1990.
View at: Zentralblatt MATH | MathSciNetG. Alefeld and G. Mayer, “Interval analysis: theory and applications,” Journal of Computational and Applied Mathematics, vol. 121, no. 1-2, pp. 421–464, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetS. M. Rump, “Intlab - interval laboratory,” 2006, http://www.ti3.tu-harburg.de/rump/intlab/.
View at: Google Scholar