ARTICLE IN PRESS JID: CHAOS [m5G;November 19, 2019;20:45] Chaos, Solitons and Fractals xxx (xxxx) xxx Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos Quasi-pinning synchronization and stabilization of fractional order BAM neural networks with delays and discontinuous neuron activations A. Pratap a , R. Raja b , J. Cao c,∗ , Fathalla A. Rihan d , Aly R. Seadawy e a Department of Mathematics, Veltech High Tech Engineering College, Avadi, Chennai 600 062, India b Ramanujan Centre for Higher Mathematics, Alagappa University, Karaikudi 630 004, India c School of Mathematics, Southeast University, Nanjing 211189, China d Department of Mathematical Sciences, UAE University, Al Ain, Abu Dhabi 15551, UAE e Department of Mathematics and Statistics, Taibah University, Medina 41 477, Saudi Arabia a r t i c l e i n f o Article history: Received 21 November 2018 Revised 4 July 2019 Accepted 11 October 2019 Available online xxx Keywords: Quasi-synchronization β-Exponential stabilization Discontinuous BAM-type neural networks Fractional order Time-varying delays Filippov’s solutions Pinning control a b s t r a c t This manuscript concerns quasi-pinning synchronization and β-exponential pinning stabilization for a class of fractional order BAM neural networks with time-varying delays and discontinuous neuron acti- vations (FBAMNNDDAs). Firstly, under the framework of Filippov solution and fractional-order differential inclusions analysis for the initial value problem of FBAMNNDDAs is presented. Secondly, two kinds of novel pinning controllers according to pinning control technique are designed. By means of fractional or- der Lyapunov method and designed pinning control strategy, the sufficient criteria is given first to ensure the quasi-synchronization for the dynamic behavior of FBAMNNDDAs. Furthermore, the error bound of pinning synchronization is explicitly evaluated. Thirdly, via Kakutani s fixed point theorem of set-valued map analysis, Razumikhin condition, and a nonlinear pinning controller, the existence and β-exponential stabilization of FBAMNNDDAs equilibrium point is obtained in the voice of linear matrix inequality (LMI) technique. Fourthly, based on as well as Mittag-Leffler function and growth condition, the global existence of a solution in the Filippov sense of such system is guaranteed with detailed proof. At last, a numerical example with computer simulations are performed to illustrate the effectiveness of proposed theoretical consequences. © 2019 Elsevier Ltd. All rights reserved. 1. Introduction In 1695, the idea of fractional order calculus becomes first off mentioned through German mathematician Leibniz and it failed to attract more attention for a long time since it lack of application background and the complexity. In past few decades, fractional or- der calculus has superior characteristics over traditional calculus [1,2], and some excellent results on fractional order systems based on fractional-order calculation have been demonstrated, see [3–6]. Within the field of electronics, the version of fractional capacitor, formally called the fractance, has been offered, which describes the fractional differentiation constitutive relationship I t = CD β V t be- tween V t and I t passing through it, where C is the capacitance of the capacitor, V t is input voltage, I t is current and the fractional order β is identified with the misfortunes of the capacitor. The integer-order capacitor (inductor) is in reality not existing, that’s ∗ Corresponding author. E-mail address: jdcao@seu.edu.cn (J. Cao). only an approximation of a fractional order capacitor (or inductor) C. The primary reason is that dielectric materials represent capac- itor (inductor) pondered the fractional order traits. Consequently, the fractional-order differential equation can accurately describe with a capacitance (or inductor) circuit system [7,8]. Neural net- works have found a wide scope of applications in automatic con- trol, combinatorial optimization, image processing and signal pro- cessing [9–12]. An electronic implementation of an artificial neural network model, many of the researchers attempted to update the normal capacitor by fractional capacitor, then it creates the frac- tional order neural network models. Until newly, it has got increas- ing interests of many researchers and it plays a vital role in syn- chronization [13,14], state estimation [15], dissipativity [16], pas- sivity [17], stability [18] and stabilization [19] of fractional order neural networks and the research of the fractional order dynami- cal system has been a hot spot. As a type of recurrent neural networks, BAM neural networks was firstly predicted by Kosko in 1987. As we recognize, a BAM type of neural network model is a nonlinear feedback network https://doi.org/10.1016/j.chaos.2019.109491 0960-0779/© 2019 Elsevier Ltd. All rights reserved. Please cite this article as: A. Pratap, R. Raja and J. Cao et al., Quasi-pinning synchronization and stabilization of fractional order BAM neu- ral networks with delays and discontinuous neuron activations, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109491