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On a one-parameter discrete-time \(Z_4\)-equivariant cubic dynamical system. (English) Zbl 1411.37027

Summary: The lemniscate sine and cosine are solutions of a \(Z_4\)-equivariant planar Hamiltonian system for all of which nontrivial solutions are nonhyperbolic periodic orbits. The forward Euler scheme is applied to this system and the one-parameter discrete-time \(Z_4\)-equivariant cubic dynamical system is obtained. The discrete-time system depending upon a parameter exhibits rich dynamics: numerical simulation shows that the system has attracting closed invariant curves, multiple periodic orbits and attracting sets exhibiting chaotic behavior. The approximating system of ordinary differential equations is constructed. We discuss the existence of closed invariant curves for the discrete-time system.

MSC:

37C80 Symmetries, equivariant dynamical systems (MSC2010)
39A12 Discrete version of topics in analysis
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37D10 Invariant manifold theory for dynamical systems

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[1] Arnol’d, V. I. [1988] Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd edition (Springer-Verlag, NY). · Zbl 0648.34002
[2] Arnol’d, V. I., Afrajmovich, V. S., Il’yashenko, Yu. S. & Shil’nikov, L. P. [1994] “ Bifurcation theory,” Dynamical Systems V. Encyclopedia of Mathematical Sciences, ed. Arnol’d, V. I. (Springer-Verlag, NY), pp. 1-205. · Zbl 0797.58003
[3] Barreira, L. & Valls, C. [2012] Dynamical Systems: An Introduction (Springer-Verlag, London). · Zbl 1269.37001
[4] Beyn, W.-J. [1987] “ On invariant closed curves for one-step methods,” Numer. Math.51, 103-122. · Zbl 0617.65082
[5] Easton, R. W. [1998] Geometric Methods for Discrete Dynamical Systems (Oxford University Press, NY).
[6] Field, M. & Golubitsky, M. [1995] “ Symmetric chaos: How and why,” Notices Amer. Math. Soc.42, 240-244. · Zbl 1042.37507
[7] Ghaziani, R. K., Govaerts, W. & Sonck, C. [2011] “ Codimension-two bifurcations of fixed points in a class of discrete prey-predator systems,” Discr. Dyn. Nat. Soc.2011, 1-27. · Zbl 1229.92072
[8] Gielis, J. [2003] “ A generic geometric transformation that unifies a wide range of natural and abstract shapes,” Amer. J. Botany90, 333-338.
[9] Golubitsky, M., Stewart, I. & Schaeffer, D. G. [1988] Singularities and Groups in Bifurcation Theory, Vol. 2 (Springer-Verlag, NY). · Zbl 0691.58003
[10] Golubitsky, M. & Stewart, I. [2002] The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space (Birkhäuser-Verlag, Basel). · Zbl 1031.37001
[11] Gritsans, A. & Sadyrbaev, F. [2006] “ Remarks on lemniscatic functions,” Acta Universitatis Latviensis. Mathematics688, 39-50.
[12] Hale, J. K. [1988] Asymptotic Behavior of Dissipative Systems (American Mathematical Society, Providence, RI). · Zbl 0642.58013
[13] Higham, N. J. [2008] Functions of Matrices: Theory and Computation (Society for Industrial and Applied Mathematics, Philadelphia). · Zbl 1167.15001
[14] Ilhem, D. & Amel, K. [2006] “ One-dimensional and two-dimensional dynamics of cubic maps,” Discr. Dyn. Nat. Soc.2006, 1-13. · Zbl 1111.37031
[15] Irwin, M. C. [2006] Smooth Dynamical Systems (World Scientific, Singapore).
[16] Kuznetsov, Y. A. [2004] Elements of Applied Bifurcation Theory, 3rd edition (Springer, NY). · Zbl 1082.37002
[17] Lynch, S. [2017] Dynamical Systems with Applications Using \(Mathematica^Ⓡ, 2\) nd edition (Birkhäuser, Switzerland). · Zbl 1388.37001
[18] Markushevich, A. I. [1966] The Remarkable Sine Functions (American Elsevier Publishing Company Inc., NY). · Zbl 0154.33701
[19] Melbourne, I., Dellnitz, M. & Golubitsky, M. [1993] “ The structure of symmetric attractors,” Arch. Rat. Mech. Anal.123, 75-98. · Zbl 0805.58043
[20] Milnor, J. [1992] “ Remarks on iterated cubic maps,” Exp. Math.1, 5-24. · Zbl 0762.58018
[21] Mira, C., Gardini, L., Barugola, A. & Cathala, J. C. [1996] Chaotic Dynamics in Two-Dimensional Noninvertible Maps (World Scientific, Singapore). · Zbl 0906.58027
[22] Pesin, Y. & Climenhaga, V. [2009] Lectures on Fractal Geometry and Dynamical Systems (American Mathematical Society, USA). · Zbl 1186.37003
[23] Petrişor, E. [1999] “ Symmetric attractors and symmetric fractals,” European Women in Mathematics Proc. 8th General Meeting, 12-17 December 1997 (Trieste, Italy, ), eds. Fainsilber, L. & Hobbs, C. (Hindawi, USA), pp. 169-176.
[24] Whittaker, E. T. & Watson, G. N. [1996] A Course of Modern Analysis, 4th edition (Cambridge University Press, Cambridge). · Zbl 0951.30002
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