Abstract
In this paper, Mira 2 map is investigated. The conditions of the existence for fold bifurcation, flip bifurcation and Naimark-Sacker bifurcation are derived by using center manifold theorem and bifurcation theory. And the conditions of the existence for chaos in the sense of Marroto are obtained. Numerical simulation results not only show the consistence with the theoretical analysis but also display complex dynamical behaviors, including period-n orbits, crisis, some chaotic attractors, period-doubling bifurcation to chaos, quasi-period behaviors to chaos, chaos to quasi-period behaviors, bubble and onset of chaos.
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Agiza, H.N., Elabbssy, E.M., EL-Metwally, H., Elsadany, A. A. Chaotic dynamics of a discrete prey-predator model with Holling type II. Nonlin. Anal.: Real World Appl., 10, 116–129 (2009)
Alligood, K. Sauer, T., Yorke, J.A. Chaos: an Introduction to Dynamical Systems. Springger-Verlag, New York, 1997
Carr, J. Application of center manifold theroem. Springer-Verlag, New York, 1981
Cartwright, J.H.E. Nonlinear stiffness, Lyapunov exponents, and attractor dimension. Phys. Lett. A, 264: 298–302 (1999)
Davies, B. Exploring chaos: theory and experiment. Westview Press, Boulder, 2003
Guckenheimer J., Holmes P. Nonlinear oscillations, dynamical systems and bifurcation of vector fields. Springer-Verlag, New York, 1986
Mira, C., Barugola, A., Gardini, L. Chaotic dynamics in two-dimensional nonvertible maps. World Scientific, Singapore, 1996
Marotto, F.R. Snap-back repellers imply chaos in R n. J. Math. Anal. Appl., 63: 199–223 (1978)
Marotto, F.R. On redefining a snap-back repeller. Chaos, Solitons & Fractals, 25: 25–28 (2005)
Nusse, H.E., Yorke, J.A. Dynamics: Numerical Explorations. Springer, New York, 1997
Styness, D., Hanan W.G., Pouryahya S., Herffernan, D.M. Scaling relations and critical exponents for two dimensional two parameter maps. The European Physycal Journal B, 77: 469–478 (2010)
Wiggins S. Introduction to applied nonlinear dynamical systems and chaos. Springger-Verlag, New York, 1990
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Supported by the National Science Foundations of China (10671063 and 61571052).
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Jiang, T., Yang, Zy. Bifurcations and chaos in Mira 2 map. Acta Math. Appl. Sin. Engl. Ser. 33, 967–978 (2017). https://doi.org/10.1007/s10255-017-0710-1
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DOI: https://doi.org/10.1007/s10255-017-0710-1