A Fractional-Order Sinusoidal Discrete Map

Abstract

In this paper, a novel fractional-order discrete map with a sinusoidal function possessing typical nonlinear features, including chaos and bifurcations, is proposed. Firstly, the basic properties involving the stability of the equilibrium points and the symmetry of the map are studied by theoretical analysis. Secondly, the dynamics of the map in commensurate-order and incommensurate-order cases with initial conditions belonging to different basins of attraction is investigated by numerical simulations. The bifurcation types and influential parameters of the map are analyzed via nonlinear tools. Hopf, period-doubling, and symmetry-breaking bifurcations are observed when a parameter or an order is varied. Bifurcation diagrams and maximum Lyapunov exponent spectrums, with both a variation in a system parameter and an order or two orders, are shown in a three-dimensional space. A comparison of the bifurcations in fractional-order and integral-order cases shows that the variation in an order has no effect on the symmetry-breaking bifurcation point. Finally, the heterogeneous hybrid synchronization of the map is realized by designing suitable controllers. It is worth noting that the increase in a derivative order can promote the synchronization speed for the fractional-order discrete map.

Hidden multistability of fractional discrete non-equilibrium point memristor based map

“…Discrete fractional calculus has drawn the interest of a great number of researchers during the last several years [8][9][10][11][12], and they have been increasingly interested in its potential applications in neural networks, secure communication, biology, and other domains [13][14][15]. Recently numerous different dynamics including chaos, hyperchaos and coexisting attractors in fractional-order systems have been explored [16][17][18][19][20][21][22]. For example, the coexisting chaos and hyperchaos in the fractional-order discrete SIE epidemic model have been analyzed in [23].…”

“…A 3D fractional iterated map has been developed in [24], in which this fractional map was shown to have hidden attractors. In [25] Liu et al proposed a new fractional discrete sinusoidal map, whereas, the chaos in the fractional Hénon-Lozi type map has been examined in [26]. Gasri et al [27] revealed the rich chaotic dynamics of a novel fractional-order map with infinite line of equilibrium points, while In [28], Khennaoui et al have investigated the chaotic dynamics of a new 2D discrete system without equilibrium points.…”