The paper is concerned with the fuzzy Conley index. Starting with fuzzy dynamical systems, we define isolated invariant sets and their isolating neighborhoods. The concept of fuzzy Conley index related to fuzzy dynamical systems is introduced and examples are given.

Conley index, invariant sets, isolating neighborhood, exit sets, attractor, repeller.

Received: August 9, 2021; Accepted: September 15, 2021; Published: October 22, 2021

How to cite this article: Rebecca Walo Omana, Fuzzy conley index and invariant fuzzy sets, Advances in Fuzzy Sets and Systems 26(2) (2021), 175-209. DOI: 10.17654/FS026020175

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License 

References

[1] G. Curvalcioglu and M. Citil, On fuzzy homotopy theory, Adv. Stud. Contemp. Math. (Kyungshang) 12 (2006), 163-166.
[2] M. S. Cecconello, J. Leite, R. C. Bassanezi and A. J. V. Brandao, Invariant and attractor sets for fuzzy dynamical systems, Fuzzy sets and Systems 265 (2015), 99-109.
[3] J. Guckenheimer and P. Holmes, Nonlinear Oscillation Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, N.Y., Berlin, 1984.
[4] K. Mischailov and M. Mrozek, Isolating neighborhood and Chaos, Japan J. Indust. Appl. Math. 12 (1995), 205-236.
[5] M. T. Mizukoshi, L. C. Barros and R. C. Bassanezi, Stability of fuzzy dynamical systems, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 17 (2009), 69-83. DOI: 10.1142/SO 218488509005747.
[6] M. T. Mizukoshi, L. C. Barros, Y. Chalco-Cano, H. Roman-Flores and R. C. Bassanezi, Fuzzy differential equations and the extension principle, Inform. Sci. 177 (2007), 3627-3635.
[7] V. Lakshmikanthan and R. Mohaptra, Theory of fuzzy differential and inclusions, Series in Mathematical Analysis and Applications, Taylor & Francis Group, London, 2003.
[8] D. Salamon, Connected simple systems and Conley Index of invariant sets, Trans. Amer. Math. Soc. 291 (1985), 1-41.
[9] J. Smoller, Shock Waves and Reaction Diffusion Equations, 2nd ed., Springer-Verlag, 1994.