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Hassan, Sk. Sarif

Discrete dynamics of one dimensional Collatz like integral value transformations. (English) Zbl 1358.37069

J. Appl. Math. Comput. 49, No. 1-2, 91-105 (2015).
Summary: In Discrete dynamics and , one of the most important challenging conjectures is Collatz conjecture. Some basic algebraic and analytical properties of Collatz like integral value transformations (IVTs) in the context of discrete dynamical system over \(\mathbb N_0\) are adumbrated. The dynamical maps associated to the dynamical systems are everywhere continuous but nowhere differentiable (smooth) in domain \(\mathbb N_0\). Under such Collatz like IVTs, for any initial point \(X_0\in\mathbb N_0\) the dynamical systems are convergent and converge eventually to a single point attractor, zero. In other words, the image sets associated to the Collatz like dynamical maps in each iteration converge to a single point image set consisting the point zero. The rate of convergence of the Collatz like dynamical system with respect to some property is also described. It is observed that the \(l_2\)-norm of the ith order derivative operator for all Collatz like map is a decreasing convergent sequence which converges to zero.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37M05 Simulation of dynamical systems

Keywords:

discrete dynamical systems; fractal dimension
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[1] Adrien Richarda, J.C.: Necessary conditions for multistationarity in discrete dynamical systems. Discret. Appl. Math. 155(18), 2403-2413 (2007) · Zbl 1125.37062 · doi:10.1016/j.dam.2007.04.019
[2] Hassan, Sk S., Choudhury, P.Pal, Singh, Rajneesh, Das, Snigdha, Nayak, B.K.: Collatz function like integral value transformations. Alexandria J. Math. 1(2), 30-35 (2010)
[3] Hassan, Sk S., Roy, A., Choudhury, P.Pal, Nayak, B.K.: Integral value transformations: a class of discrete dynamical systems. J. Orissa Math. Soc. 31(1), 113-126 (2012)
[4] Hassan, Sk. S., Nayak, B. K., Choudhury, P. Pal.: (2012) One Dimensional p-adic Integral Value Transformations, arXiv:1106.3586 (2012) (Under review).
[5] Hassan, Sk. S., Choudhury, P. Pal., Nayak, B. K., Ghosh, A. and Banerjee, J. (2013) Integral Value Transformations: A Class of Affine Discrete Dynamical Systems and an Application, Accepted for Publication. · Zbl 1039.37024
[6] Oded Galor: Discrete Dynamical Systems. Springer (2006), ISBN: 3540367756. · Zbl 1223.39002
[7] Chamberland, M.: A continuous extension of the 3x+1 problem to the real line dynamics of continuous. Discret. Impuls. Syst. 2, 495-509 (1996) · Zbl 0874.11017
[8] Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (1993) · Zbl 0792.58014
[9] Silverman, J.H.: The Arithmetic of Dynamical Systems. Springer, New York (2007) · Zbl 1130.37001 · doi:10.1007/978-0-387-69904-2
[10] Alligood, K., Sauer, T.D., Yorke, J.: Chaos: An Introduction to Dynamical Systems. Springer-Verlag, New York (1997) · Zbl 0867.58043 · doi:10.1007/978-3-642-59281-2
[11] Avnir, D. (1998). Is the geometry of nature fractal, Science. Vol. 279 (39). · Zbl 1225.37100
[12] Bisoi, A.M.: On calculations of fractal dimension of images. Pattern Recog. Lett. 22(6-7), 631-637 (2001) · Zbl 1010.68902
[13] Block, L., Coppel, W.: Dynamics in One Dimension. Lecture Notes in Mathematics. Springer Verlag, Berlin (1992) · Zbl 0746.58007
[14] Bransley, M. F.: Fractals Everywhere. Academic Press. (1988), ISBN 0-12-079062-9. · Zbl 0691.58001
[15] Casartelli, M.: Intermittency from Collatz’s itineraries and complexity indicators. J. Phys. A: Math. Gen. 35, 4501 (2002) · Zbl 1039.37024 · doi:10.1088/0305-4470/35/21/301
[16] Celso Grebogia, E.O.: Strange attractors that are not chaotic. Physica D: Nonlinear Phenom. 13(12), 261268 (1984) · Zbl 0588.58036
[17] Falconer, K. J.: Fractal Geometry: Mathematical Foundations and Applications. John-Wiley & Sons (1990). ISBN- 0-471-92287-0. · Zbl 0689.28003
[18] Holmgren, R.L.: A First Course in Discrete Dynamical Systems. Springer Verlag, New York (2006) · Zbl 0797.58001
[19] Lagarias, J.: The \[3x+13\] x+1 problem and its generalizations. Am. Math. Mon. 92, 3-23 (1985) · Zbl 0566.10007 · doi:10.2307/2322189
[20] Andrei, St, Masalagiu, C.: About the Collatz Conjecture. Acta Inf. 35, 167-179 (1998) · Zbl 0959.11008 · doi:10.1007/s002360050117
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