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In recent years, some efforts have been devoted to nonlinear dynamics of fractional discrete-time systems. A number of papers have so far discussed results related to the presence of chaos in fractional maps. However, less results have been published to date regarding the presence of hyperchaos in fractional discrete-time systems. This paper aims to bridge the gap by introducing a new three-dimensional fractional map that shows, for the first time, complex hyperchaotic behaviors. A detailed analysis of the map dynamics is conducted via computation of Lyapunov exponents, bifurcation diagrams, phase portraits, approximated entropy and C 0 complexity. Simulation results confirm the effectiveness of the approach illustrated herein. [ABSTRACT FROM AUTHOR]

This paper studies the dynamics of two fractional-order chaotic maps based on two standard chaotic maps with sine terms. The dynamic behavior of this map is analyzed using numerical tools such as phase plots, bifurcation diagrams, Lyapunov exponents and 0–1 test. With the change of fractional-order, it is shown that the proposed fractional maps exhibit a range of different dynamical behaviors including coexisting attractors. The existence of coexistence attractors is depicted by plotting bifurcation diagram for two symmetrical initial conditions. In addition, three control schemes are introduced. The first two controllers stabilize the states of the proposed maps and ensure their convergence to zero asymptotically whereas the last synchronizes a pair of non-identical fractional maps. Numerical results are used to verify the findings. [ABSTRACT FROM AUTHOR]

In this paper, we are particularly interested in the fractional form of the Hénon-Lozi type map. Using discrete fractional calculus, we show that the general behavior of the proposed fractional order map depends on the fractional order. The dynamical properties of the new generalized map are investigated by applying numerical tools such as: phase portrait, bifurcation diagram, largest Lyapunov exponent, and 0-1 test. It shows that the fractional order Hénon-Lozi map exhibits a range of different dynamical behaviors including chaos and coexisting attractors. Furthermore, a one-dimensional control law is proposed to stabilize the states of the fractional order map. Numerical results are presented to illustrate the findings. [ABSTRACT FROM AUTHOR]

Recently, hidden attractors with stable equilibria have received considerable attention in chaos theory and nonlinear dynamical systems. Based on discrete fractional calculus, this paper proposes a simple two-dimensional and three-dimensional fractional maps. Both fractional maps are chaotic and have a unique equilibrium point. Results show that the dynamics of the proposed fractional maps are sensitive to both initial conditions and fractional order. There are coexisting attractors which have been displayed in terms of bifurcation diagrams, phase portraits and a 0-1 test. Furthermore, control schemes are introduced to stabilize the chaotic trajectories of the two novel systems. [ABSTRACT FROM AUTHOR]

In this paper, we propose three fractional chaotic maps based on the well known 3D Stefanski, Rössler, and Wang maps. The dynamics of the proposed fractional maps are investigated experimentally by means of phase portraits, bifurcation diagrams, and Lyapunov exponents. In addition, three control laws are introduced for these fractional maps and the convergence of the controlled states towards zero is guaranteed by means of the stability theory of linear fractional discrete systems. Furthermore, a combined synchronization scheme is introduced whereby the fractional Rössler map is considered as a drive system with the response system being a combination of the remaining two maps. Numerical results are presented throughout the paper to illustrate the findings. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fractional form of a new three-dimensional generalized Hénon map and study the existence of chaos and its control. Using bifurcation diagrams, phase portraits and Lyapunov exponents, we show that the general behavior of the proposed fractional map depends on the fractional order. We also present two control schemes for the proposed map, one that adaptively stabilizes the fractional map, and another to achieve the synchronization of the proposed fractional map. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of a fractional order chaotic map corresponding to a recently developed standard map that exhibits a chaotic behavior with no fixed point. This is the first study to explore a fractional chaotic map without a fixed point. In our investigation, we use phase plots and bifurcation diagrams to examine the dynamics of the fractional map and assess the effect of varying the fractional order. We also use the approximate entropy measure to quantify the level of chaos in the fractional map. In addition, we propose a one-dimensional stabilization controller and establish its asymptotic convergence by means of the linearization method. [ABSTRACT FROM AUTHOR]

In this study, we expand a 2D sine map via adding the discrete memristor to introduce a new 3D fractional-order sine-based memristor map. Under commensurate and incommensurate orders, we conduct an extensive exploration and analysis of its nonlinear dynamic behaviors, employing diverse numerical techniques, such as analyzing Lyapunov exponents, visualizing phase portraits, and plotting bifurcation diagrams. The results emphasize the sine-based memristor map's sensitivity to fractional-order parameters, resulting in the emergence of distinct and diverse dynamic patterns. In addition, we employ the sample entropy ( S a m p E n ) method and C 0 complexity to quantitatively measure complexity, and we also utilize the 0–1 test to validate the presence of chaos in the proposed fractional-order sine-based memristor map. Finally, MATLAB simulations are be executed to confirm the results provided. [ABSTRACT FROM AUTHOR]

This paper introduces a novel fractional Ikeda-based memristor map and investigates its non-linear dynamics under commensurate and incommensurate orders using various numerical techniques, including Lyapunov exponent analysis, phase portraits, and bifurcation diagrams. The results reveal diverse and complex system behaviors arising from the interplay of different fractional orders in the proposed map. Furthermore, the study employs the sample entropy test to quantify complexity and validate the presence of chaos. Non-linear controllers are also presented to stabilize and synchronize the model. The research emphasizes the system's sensitivity to the fractional order parameters, leading to distinct dynamic patterns and stability regimes. The memristor-based chaotic map exhibits rich and intricate behavior, making it an interesting and important area of research. [ABSTRACT FROM AUTHOR]

In this paper, we present an innovative 3D fractional Hénon-based memristor map and conduct an extensive exploration and analysis of its dynamic behaviors under commensurate and incommensurate orders. The study employs diverse numerical techniques, such as visualizing phase portraits, analyzing Lyapunov exponents, plotting bifurcation diagrams, and applying the sample entropy test to assess the complexity and validate the chaotic characteristics. However, since the proposed fractional map has no fixed points, the outcomes reveal that the map can exhibit a wide range of hidden dynamical behaviors. This phenomenon significantly augments the complexity of the fractal structure inherent to the chaotic attractors. Moreover, we introduce nonlinear controllers designed for stabilizing and synchronizing the proposed fractional Hénon-based memristor map. The research emphasizes the system's sensitivity to fractional-order parameters, resulting in the emergence of distinct dynamic patterns. The memristor-based chaotic map exhibits rich and intricate behavior, making it a captivating and significant area of investigation. [ABSTRACT FROM AUTHOR]