Harvesting Dynamic Aspects from a Real Rational Map: The Bulb of Period 5
Keywords:Dynamics, Real Rational Maps, Stability, Fixed Points, Bifurcation Theory
AbstractThis work is a small, but strong, contribute to understand better the behavior, under iteration, of the Real Rational Map, f(x)=(xÂ²-a)/(xÂ²-b), with 0<b<a<1, with a and b real parameters. So, we show in this work how to choose some suitable regions of the parameter space (a,b), that allows a controlled and efficient study of this kind of maps, using some classical dynamical systems tools as the critical point orbit, Â Lyapunov exponents and their relation with the bifurcation diagram of the map.
AlsedÃ L., Llibre, J., Misiurewicz, M., â€œCombinatorial Dynamics and Entropy in Dimension Oneâ€, Second Edition, Advanced Series in Nonlinear Dynamics, vol.5, World Scientific, 2000.
Cabral, J., â€œHunting Admissible Kneading pairs of a real rational mapâ€, Asian Journal of Fuzzy and Applied Mathematics, Vol. 1, number 4, December 2013.
Cabral, J., â€œSymbolic Dynamics for Real Rational Mapsâ€, Asian Journal of Fuzzy and Applied Mathematics, Vol.1, number 4, December 2013.
Devaney, R., â€œAn introduction to chaotic dynamical systemsâ€, 2nd edition, Perseus Books, 1989.
Katok, A., â€œLyapunov exponents, entropy and periodic orbits for diffeomorphismsâ€. Publications MathÃ©matiques de lâ€™IHÃ‰S, 51, p.137-173.
Milnor J., Thurston W., â€œOn iterated maps of the intervalâ€, In Dynamical Systems proc. Univ. Maryland 1986-87, pp. 465-563, Lect. Notes in Math., Springer-Verlag, 1988.
Sousa Ramos, J. â€œHiperbolicidade e BifurcaÃ§Ã£o de Sistemas SimbÃ³licosâ€, Phd Thesis, Technical University of Lisbon, available at http://purl.pt/7368, 1989.
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