Eur. Phys. J. Special Topics 229, 2261–2273 (2020)
https://doi.org/10.1140/epjst/e2020-900193-4

Regular Article

Bifurcation and chaos in the fractional form of Hénon-Lozi type map

¹ Laboratory of Mathematics, Informatics and Systems (LAMIS), University of Larbi Tebessi, Tebessa 12002, Algeria
² College of Humanities and Sciences, Ajman University, Ajman, UAE
³ Laboratory of dynamical systems and control, University of Larbi Ben M’hidi, Oum El Bouaghi, Algeria
⁴ Institute for Advanced Study, Shenzhen University, Shenzhen, Guangdong 518060, P.R. China
⁵ Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
⁶ Department of Mathematics, College of Sciences and Arts, Al-Rass, Qassim University, Buraydah, Kingdom of Saudi Arabia
⁷ Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella, Oran, Algeria
⁸ Department of Mathematics, Faculty of Science, The University of Jordan, Amman, Jordan
⁹ Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia

^a e-mail: phamvietthanh@tdtu.edu.vn

Received: 7 September 2019
Accepted: 8 June 2020
Published online: 28 September 2020

Abstract

In this paper, we are particularly interested in the fractional form of the Hénon-Lozi type map. Using discrete fractional calculus, we show that the general behavior of the proposed fractional order map depends on the fractional order. The dynamical properties of the new generalized map are investigated by applying numerical tools such as: phase portrait, bifurcation diagram, largest Lyapunov exponent, and 0-1 test. It shows that the fractional order Hénon-Lozi map exhibits a range of different dynamical behaviors including chaos and coexisting attractors. Furthermore, a one-dimensional control law is proposed to stabilize the states of the fractional order map. Numerical results are presented to illustrate the findings.