Global Stability Index for an Attractor with Riddled Basin in a Two-Species Competition System

  • Ummu Atiqah Mohd RoslanEmail author
  • Mohd Tirmizi Mohd Lutfi
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 295)


We consider a competition system between two-species containing riddled basin and second basin attractors. To characterize local geometry of riddled basin, we compute a global stability index for the attractor in the system. Our results show that the index varies from \(\infty \) down to positive values within a parameter region. The changes of the index indicates that the attractor looses its stability from asymptotically stable attractor to riddled basin attractor. Thus, the stability index has a great potential to become a new study on bifurcation of dynamical system since it is able to characterize different types of geometry of basins of attraction.


Riddled basin Stability index Dynamical system 


  1. 1.
    Podvigina, O., Ashwin, P.: On local attraction properties and a stability index for heteroclinic connections. Nonlinearity 24, 887–929 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Castro, S.B.S.D., Lohse, A.: Stability in simple heteroclinic networks in \(\mathbb{R}^4\). Dyn. Syst. 29(4), 451–481 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Keller, G.: Stability index for chaotically driven concave maps. J. Lond. Math. Soc. 29(4), 451–481 (2015)Google Scholar
  4. 4.
    Mohd Roslan, U.A., Ashwin, P.: Local and global stability indices for a riddled basin attractor of a piecewise linear map. Dyn. Syst. 31, 375–392 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Mohd Roslan, U.A.: Stability index for an attractor with riddled basin in dynamical systems. In: AIP Conference Proceedings, vol. 1870, pp. 0400071–0400076 (2017).
  6. 6.
    Ashwin, P., Buescu, J., Stewart, I.: From attractor to chaotic saddle: a tale of transverse instability. Nonlinearity 9, 703–737 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Viana, R.L., Camargo, S., Pereira, R.F., Verges, M.C., Lopes, S.R., Pinto, S.E.S.: Riddled basins in complex physical and biological systems. J. Comput. Interdiscip. Sci. 1(2), 73–82 (2009)Google Scholar
  8. 8.
    Glendinning, P.: Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations. Cambridge University Press, UK (1994)CrossRefGoogle Scholar
  9. 9.
    Alexander, J.C., Yorke, J.A., You, Z., Kan, I.: Riddled basin. Int. J. Bifurc. Chaos 2, 795–813 (1992)MathSciNetGoogle Scholar
  10. 10.
    Devaney, R.L.: An Introduction to Chaotic Dynamical Systems, 2nd edn. Addison-Wesley, USA (1989)zbMATHGoogle Scholar
  11. 11.
    Ott, E.: Chaos in Dynamical Systems, 2nd edn. Cambridge University Press, UK (2002)CrossRefGoogle Scholar
  12. 12.
    Buescu, J.: Exotic Attractors: from Liapunov Stability to Riddled Basins. Birkhäuser Verlag, Switzerland (1997)CrossRefGoogle Scholar
  13. 13.
    Milnor, J.: On the concept of attractor. Commun. Math. Phys. 99, 177–195 (1985)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dai, J., He, D.-R., Xu, X.-L., Hu, C.K.: A riddled basin escaping crisis and the universality in an integrate-and-fire circuit. Phys. A Stat. Mech. Appl. 500, 72–79 (2018). Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Ummu Atiqah Mohd Roslan
    • 1
    Email author
  • Mohd Tirmizi Mohd Lutfi
    • 1
  1. 1.Faculty of Ocean Engineering Technology and InformaticsUniversiti Malaysia TerengganuKuala NerusMalaysia

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