Commun Nonlinear Sci Numer Simulat 53 (2017) 31–43 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns Short communication Persistence of nonlinear hysteresis in fractional models of Josephson transmission lines J.E. Macías-Díaz ∗ Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes 20131, Mexico a r t i c l e i n f o Article history: Received 6 February 2017 Revised 17 April 2017 Accepted 27 April 2017 Available online 4 May 2017 Keywords: Nonlinear hysteresis Nonlinear supratransmission Fractional Josephson-junction chains Discrete Riesz derivatives a b s t r a c t In this note, we depart from a model describing the transmission of electric currents in Josephson-junction chains, and provide a fractional generalization using Riesz discrete differential operators. The fractional model considered has generalized Hamiltonian- and energy-like functionals. The model and the energy functionals are fully discretized in or- der to investigate numerically the complex dynamics of the system when a sinusoidal per- turbation at one end of the chain is imposed. As one of the most important results in this report, we establish the persistence of the nonlinear phenomena of supratransmission and infratransmission in Riesz fractional chains. Nonlinear hysteresis loops are obtained numerically for some values of the order of the fractional derivative, and numerical sim- ulations of the propagation of monochromatic wave signals through the transmission line are presented using the nonlinear bistability of the system. © 2017 Elsevier B.V. All rights reserved. 1. Introduction Nonlinear supratransmission is a phenomenon that was firstly investigated in chains of harmonic oscillators [1]. This process consists in the sudden increase in the amplitude of wave signals that propagate in a nonlinear chain driven at one end by a harmonic disturbance, and it has been found in Klein–Gordon [2] and Fermi–Pasta–Ulam chains [3] amongst many other nonlinear media [4–7]. Historically, the study of energy transmission in nonlinear wave equations has been an in- teresting topic of investigation [8]. These models have applications in the description of data transmission in optical fibers [9] and in the study of the self-induced transparency of systems subject to a high-energy incident laser pulse [10]. More generally, the behavior of continuous media subject to a wave radiation is a fundamental problem that has potential applica- tions in many nonlinear systems [11]. In the case of discrete nonlinear chains, supratransmission was observed in systems of pendula coupled through springs [12]. Later on, applications to the design of digital amplifiers of ultra weak signals [13] and light detectors sensitive to very weak excitations [14] have been proposed. Further applications to optical waveguide arrays using the discrete nonlinear Schrödinger equation [3], the realization of light filters [15] and the propagation of binary sig- nals in undamped or weakly damped mechanical chains of oscillators [16] have been realized. Some recent papers have been devoted to understand deeply this phenomenon under various physical scenarios, and to propose further applications [17]. ∗ Corresponding author. E-mail address: jemacias@correo.uaa.mx http://dx.doi.org/10.1016/j.cnsns.2017.04.030 1007-5704/© 2017 Elsevier B.V. All rights reserved.