Chaos, Solitons and Fractals 128 (2019) 390–401 Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos Frontiers Chaotic dynamics and chaos control for the fractional-order geomagnetic field model A. Al-khedhairi a , A.E. Matouk b,c , I. Khan d,∗ a Department of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia b Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah, 11952, Saudi Arabia c College of Engineering, Majmaah University, Al-Majmaah 11952, Saudi Arabia d Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam a r t i c l e i n f o Article history: Received 19 April 2019 Revised 11 July 2019 Accepted 11 July 2019 Keywords: Fractional-order geomagnetic field model Continuous dependence on initial conditions Chaos Novel linear control scheme ABC a b s t r a c t Fractional-order Geomagnetic Field model is considered in this work. A sufficient condition is used to prove that the solution of the fractional-order Geomagnetic Field model exists and is unique in a specific region. Conditions for continuous dependence on initial conditions in our model are discussed. In addi- tion, the conditions of local stability of the model’s five equilibrium points are obtained. Chaotic attrac- tors are shown to exist in the proposed fractional model. Also, Lyapunov exponents of the fractional-order Geomagnetic Field model are calculated and computations of Lyapunov spectrum as functions of all the model’s parameters and fractional-order are performed. Moreover, a novel linear control technique based on Lyapunov stability theory is introduced here to stabilize the chaotic states of the fractional-order Geo- magnetic Field model to its five equilibrium points. Finally, to verify the validity of our theoretical results and the effectiveness of the control scheme, numerical simulations based on the Atangana–Baleanu frac- tional integral in Caputo-sense are done to produce the chaotic attractors. © 2019 Elsevier Ltd. All rights reserved. 1. Introduction Fractional-order calculus is considered to be the extension of classical integer-order differential systems. Therefore, it is mainly related with applications involving derivatives and integrals of non-integer order [1–8]. In fact, fractional modeling enables re- searchers to achieve higher degrees of freedom and more realis- tic description of the physical phenomenon. It therefore becomes a focal topic in many fields [9–20]. There are many potential definitions of the fractional deriva- tives. Among of them, the Riemann–Liouville (RL) fractional derivative [3], the Caputo (C) fractional derivative [1], the Caputo–Fabrizio (CF) fractional derivative [21–24], the Atangana– Baleanu (AB) fractional derivative [6,8,25], the Yang–Srivastava- Machado (YSM) fractional derivative [26,27], the Yang–Gao– Machado–Baleanu (YGMB) fractional derivative [28,29] and the Yang–Machado (YM) fractional derivatives of variable order [30]. Furthermore, the Yang local fractional derivative was developed ∗ Corresponding author. E-mail addresses: ae.mohamed@mu.edu.sa (A.E. Matouk), ilyaskhan@tdtu.edu.vn (I. Khan). for solving nonlinear local fractional PDEs [31,32]. Moreover, there are general fractional calculus operators involving special functions which were reported in [33–39]. If the fractional-order derivative involves integration then it has non-local operator which implies that the future dynamics of the fractional model also depends on its history states; For example the RL, C and AB fractional deriva- tives have non-local operators. Thus, non-local operators are better candidate to handle complex dynamics of the physical phenom- ena and they are sorted according to their kernels. The RL and C fractional operators have singular kernels. However, the AB and YGMB operators have non-singular kernels. In [34], Yang used frac- tional operators with non-singular power-law kernels to study new rheological problems. In heat transfer problems [36], Feng applied fractional operator with non-singular power-law kernel to describe anomalous diffusion phenomena. Also, the fractional operator with the decay exponential kernel was applied by Yang et al. to investi- gate anomalous diffusion equations [37]. The classical Caputo definition [1], is described as D α φ (t ) = I k−α φ (k) (t ), I ν ψ (t ) =  t  0 (t − σ ) ν−1 ψ (σ )dσ  /Ŵ(ν ), (1) https://doi.org/10.1016/j.chaos.2019.07.019 0960-0779/© 2019 Elsevier Ltd. All rights reserved.