Dynamics Analysis for a Generalized Approximate 3x + 1 Function

Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 113)


To study character of the generalized 3x + 1 function is a difficult problem in fractal. At first, we put forward a generalized approximate 3x + 1 function D(z) and point out fractal character of D(z) is similar to generalized 3x + 1 function. Secondly we execute dynamics analysis of D(z) and find the fixed point and periodic point. Thirdly, we point out all fixed points are at real axis of complex plane. We proved there is no other attract fixed point except zero and find the distribution of periodic point in complex plane. Then we proved that iteration of D(z) is not divergence at whole number. Finally, we draw fractal figures to validate the dynamics character of D(z).


Fractal Generalized 3x + 1 function Periodic point Fixed point Complex plane 


  1. 1.
    Alves JF, Graca MM, Dias MES et al (2005) A linear algebra approach to the conjecture of Collatz. Lin Alg Appl 394(1):277–289CrossRefMATHGoogle Scholar
  2. 2.
    Lagarias JC (1985) The 3x + l problem and its generalizations. Am Math Mon 92(1):3–23CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Wirsehing GJ (1998) The dynamical system generated by the 3n +1 function. Lect Notes Math 1681:153–159Google Scholar
  4. 4.
    Wu J, Hao S (2003) On equality of the adequate stopping time and the coefficient stopping time of n in the 3 N + 1 conjecture. J Huazhong Univ Sci Tech (Nature Science Edition) 31(5):114–116Google Scholar
  5. 5.
    Belaga E, Mignotte M (1998) Embedding the 3x + 1 Conjecture in a 3x + d context. Exp Math 7(2):145–151MATHMathSciNetGoogle Scholar
  6. 6.
    Simons J, de Weger B (2005) Theoretical and computational bounds for m-cycles of the 3n + 1 problem. Acta Arith 117(1):51–70CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Mandelbrot BB (1982) The fractal geometry of nature. Freeman W H, San Francisco, pp 1–122MATHGoogle Scholar
  8. 8.
    Pe JL (2004) The 3x + 1 fractal. Comput Graph 25(3):431–435CrossRefGoogle Scholar
  9. 9.
    Dumont JP, Reiter CA (2001) Visualizing generalized 3x + 1 function dynamics. Comput Graph 25(5):553–595CrossRefGoogle Scholar
  10. 10.
    Liu S, Wang Z (2009) Fixed point and fractal images for a generalized approximate 3x + 1 function. J Comput Aided Des Comput Graph 21(12):1740–1744Google Scholar
  11. 11.
    Liu S et al (2011) The existence of fixed point for a generalized 3x + 1 function. Appl Mech Mater 55–57:1341–1345Google Scholar
  12. 12.
    Liu S et al (2011) periodic point at real axis for a generalized 3x + 1 function. Appl Mech Mater 55–57:1670–1674Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Shuai Liu
    • 1
    • 2
  • Weina Fu
    • 2
  • Xiangjiu Che
    • 1
  • Zhengxuan Wang
    • 1
  1. 1.College of Computer Science and TechnologyJilin UniversityChangchunChina
  2. 2.Software CollegeChangchun Institute of TechnologyChangchunChina

Personalised recommendations