Stronger Forms of Sensitivity for Induced Fuzzified Map

Abstract

Every dynamical system $(X,f)$ on a compact metric space X induces a fuzzy dynamical system $(\mathcal{F}(X),\hat{f}),$ on the space of fuzzy sets $\mathcal{F}(X)$, by Zadeh's extension principle. In this paper we consider stronger forms of sensitivity, viz. strong sensitivity, asymptotic sensitivity, syndetic sensitivity, multi-sensitivity and cofinite sensitivity. Some examples are given to expound the interrelation between them. Our main concern here is to find the relationship between f and $\hat{f}$ in terms of these forms of sensitivity. We Prove that these forms of sensitivity for f partially imply the same for $\hat{f}$ and in other way we also get partial induction.

Keywords:

• Zadeh's extension
• sensitivity
• strong sensitivity
• asymptotically sensitive
• cofinite sensitivity

Acknowledgements

We thank the referees for their valuable suggestions which helped us to improve this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Praveen Kumar

Praveen Kumar was born in Delhi, India. He received his B.Sc. Degree from Ramjas College, Delhi, India, in 2003 and M.Sc. in mathematics from IIT Delhi, India. He got his Ph.D Degree in Chaotic Dynamics and Induced Maps on Hyperspaces from University of Delhi, India, in 2017. He is currently an Assistant Professor in the department of Mathematics, Ramjas college, Delhi, India and having a teaching experience of more than 15 year. His research interest is to study topological dynamics.

Ayub Khan

Ayub Khan was born in Shaharanpur, India. He received his B.Sc. Degree from Ramjas college, Delhi, India, in 1979 and M.Sc, M. Phil degree in Linear and Non-Linear Stability of Dynamical Systems and Ph.D degree in Chaos in Non-Linear Planar Oscillations of a Satellite in Elliptic Orbits from University of Delhi, India in 1981, 1983 and 1995, respectively. He is currently a Professor and the Head of the Mathematics department of Jamia Millia Islamia University, Delhi, India. He has an teaching experience of more than 34 years. So far he has guided more than 13 Ph.D. students and 4 M.Phil students and have more than 150 research paper. He delivered more than 32 talks in various national and international Seminar/conferences. His research interest are analysis of chaos and synchronization for Non-linear dynamical system and Topological dynamics.