Eur. Phys. J. Special Topics 229, 1083-1093 (2020)
https://doi.org/10.1140/epjst/e2020-900177-6

Regular Article

A fractional map with hidden attractors: chaos and control

¹ Laboratory of dynamical systems and control, University Larbi Ben M’hidi, Oum El Bouaghi, Algeria
² Laboratory of Mathematics, Informatics and Systems (LAMIS), University of Laarbi Tebessi, Tebessa 12002, Algeria
³ College of Humanities and Sciences, Ajman University, Ajman, UAE
⁴ Department of Mathematics, College of Sciences and Arts, Qassim University, Al-Rass, Buraydah, Kingdom of Saudi Arabia
⁵ Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO) University of Oran 1, Ahmed Benbella, Oran, Algeria
⁶ Modeling Evolutionary Algorithms Simulation and Artificial Intelligent, Faculty of Electrical & Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
⁷ College of Engineering, Prince Sultan University, Riyadh, Kingdom of Saudi Arabia
⁸ Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt

^a e-mail: kamina_aicha@yahoo.fr

Received: 25 August 2019
Received in final form: 2 October 2019
Published online: 26 March 2020

Abstract

This paper studies the dynamics of a new fractional-order map with no fixed points. Through phase plots, bifurcation diagrams, largest Lyapunov exponent, it is shown that the proposed fractional map exhibit chaotic and periodic behavior. New Hidden chaotic attractors are observed, and transient state is found to exist. Complexity of the new map is also analyzed by employing approximate entropy. Results, show that the fractional map without fixed point have high complexity for certain fractional order. In addition, a control scheme is introduced. The controllers stabilize the states of the fractional map and ensure their convergence to zero asymptotically. Numerical results are used to verify the findings.