Complex nonlinear dynamics in fractional and integer order memristor-based systems

In this paper, a fractional-order (and an integer-order) memristor-based system with the flux-controlled memristor characterized by smooth quadratic nonlinearity is proposed and detailed dynamical analysis is carried out by means of theoretical and numerical methods. To be more specific, stability of each equilibrium point in the equilibrium set is analyzed for the integer-order memristive system. Meanwhile, dynamical behavior depending on the initial states of the memristor is investigated and dynamical bifurcation depending on the slope of the memductance function is also considered. The bi-furcation analysis is verified by numerical methods, including, phase portraits, bifurcation diagrams, Lyapunov exponents spectrum, and Poincare mappings. For the fractional-order case, based on the fractional-order stability theory, stability analysis is carried out just for a certain equilibrium point. Moreover, bifurcation behavior depending on the incommensurate order is discussed by virtue of numerical methods based on the Adams-Bashforth- Moulton algorithm. This paper indicates how the fractional order model and the initial state of the memristor extend the dynamical behaviors of the traditional chaotic systems. (C) 2016 Elsevier B.V. All rights reserved.