Some Conceptual and Measurement Aspects of Complexity, Chaos, and Randomness from Mathematical Point of View

  • Fikri ÖztürkEmail author
Conference paper
Part of the Springer Proceedings in Complexity book series (SPCOM)


One of the main purposes of the mankind is to understand and explain the dynamics of real-world phenomena, i.e., modeling them, and also to build predictive models for their behaviors. The most powerful tool in modeling a deterministic dynamic is mathematics, and the most powerful tool in modeling a stochastic dynamic is statistics. Some characteristics of deterministic chaotic systems are well known, as well as of stochastic systems. Distinguishing deterministic dynamical systems from stochastic ones, based on observed data, is a difficult and yet unsolved statistical problem. Natural phenomena and human behaviors dynamics are very complex. If the existing complexity and chaos in natural dynamical systems, also inherited in their mathematical models, is not well understood, then management and control processes in such systems may result in catastrophes. This study aims to reveal and emphasize the role of mathematics in formulating the conceptual and measurement stages of complexity, chaos, and randomness.


Chaos Complexity Randomness Mathematics 


  1. Alligood, K. T., Sauer, T. D., & Yorke, J. A. (2000). Chaos, an introduction to dynamical systems (3rd ed.). New York: Springer.zbMATHGoogle Scholar
  2. Atakan, C., Dağalp, R., Potas, N., & Öztürk, F. (2017). Randomness and chaos. In Ş. Ş. Erçetin & N. Potas (Eds.), Chaos, complexity and leadership (pp. 621–646). Cham: Springer.Google Scholar
  3. Balibrea, F. (2006). Chaos, periodicity and complexity on dynamical systems. In A. Sengupta (Ed.), Chaos, nonlinearity, complexity, the dynamical paradigm of nature. Berlin: Springer.Google Scholar
  4. Barnsley, M. F. (1988). Fractals everywhere. Boston: Academic Press.zbMATHGoogle Scholar
  5. Bensoudane, H., Gentil, C., & Neveu, M. (2008). The local fractional derivative of fractal curves. IEEE International Conference on Signal Image Technology and Internet Based Systems, pp. 522–529.Google Scholar
  6. Cabello, A., Severini, S., & Winter, A. (2014). Graph-theoretic approach to quantum correlations. Physical Review Letters, 112, 040401.ADSCrossRefGoogle Scholar
  7. Cernenoks, J., Iraids, J., Opmanis, M., Opmanis, R., & Podnieks K. (2014). Integer complexity: Experimental and analytical results II. arXiv:1409.0446v1 [math.NT] 1 Sep 2014.Google Scholar
  8. Cordwell, K., Epstein, A., Hemmady, A., Miller, S. J., Palsson, E., Sharma, A., Steinerberger, S., & Vu, Y. N. T. (2018). On algorithms to calculate integer complexity. arXiv:1706.08424v3 [math.NT] 5 Aug 2018.Google Scholar
  9. Devaney, R. L. (2003). An introduction to chaotic dynamical systems. New York: Westview Press.zbMATHGoogle Scholar
  10. Falconer, K. J. (1982). Hausdorff dimension and the exceptional set of projections. Mathematika, 29, 109–115.MathSciNetCrossRefGoogle Scholar
  11. Falconer, K. J. (2003). Fractal geometry mathematical foundations and applications. Chichester: Wiley.CrossRefGoogle Scholar
  12. Iraids, J., Balodis, K., Cernenoks, J., Opmanis, M., Opmanis, R., & Podnieks, K. (2012). Integer complexity: Experimental and analytical results. Scientific Papers University of Latvia, Computer Science and Information Technologies, 787, 153–179.zbMATHGoogle Scholar
  13. Kolmogorov, A. N. (1956). Foundations on the theory of probability. New York: Chelsea Publishing Company.zbMATHGoogle Scholar
  14. Li, T. Y., & Yorke, J. (1975). Period three implies chaos. American Mathematical Monthly, 82, 985–992.MathSciNetCrossRefGoogle Scholar
  15. Lopez-Ruiz, R., & Fournier-Prunaret, D. (2004). Complex behavior in a discrete coupled logistic model for the symbiotic interaction of two species. Mathematical Biosciences and Engineering, 1(2), 307–324. Scholar
  16. Lopez-Ruiz, R., Mancini, H., & Calbet, X. (2002). A statistical measure of complexity. Physics Letters A Scholar
  17. Lowen, S., & Teich, M. C. (2005). Fractal-based point processes. New York: Willey-Interscience.CrossRefGoogle Scholar
  18. Mandelbrot, B. B. (1982). The fractal geometry of nature. San Francisco: WH Freeman & Co.zbMATHGoogle Scholar
  19. Murray, J. D. (1993). Mathematical biology. New York: Springer.CrossRefGoogle Scholar
  20. Rao, C. R. (1989). Statistics and truth Puting chance to work. Dordrecht: International Co-operative Publishing House.Google Scholar
  21. Rao, C. R. (1997). Statistics and truth Puting chance to work. Singapore: World Scientific Publishing Co.CrossRefGoogle Scholar
  22. Rovelli, C. (2017). Fizik Üzerine Yedi Kısa Ders. Can Sanat Yayınları. (Translation to Turkish from: Rovelli, C. (2014) Seven Brief Lessons on Physics.)Google Scholar
  23. Rovelli, C. (2018). Gerçeklik Göründüğü Gibi Değildir. Can Sanat Yayınları. (Translation to Turkish from: Rovelli, C. (2014) Reality Is Not What It Seems: The Journey to Quantum Gravity.)Google Scholar
  24. Tomé, L., & Açıkalın, Ş. N. (2017). Complexity theory as a new Lens in IR: System and change. In Ş. Ş. Erçetin & N. Potas (Eds.), Chaos, complexity and leadership (pp. 621–646). Cham: Springer.Google Scholar
  25. Wikipedia, free encyclopedia,

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Ankara UniversityAnkaraTurkey

Personalised recommendations