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The theory of critical curves determines the main characteristics of a discrete dynamical system in two dimensions. One important property that has garnered recent attention is the problem of chaos synchronization, along with the location of its chaotic attractors, basin boundaries, and bifurcation mechanisms. Varying the parameters of the maps reveals the instrumental role that these curves play, where the bifurcation leads to complex topological structures of the basins occurs by contact with the basin boundaries, resulting in the appearance or disappearance of some components of the basin. This study focuses on the properties of a discrete dynamical system consisting of two symmetrically coupled non-invertible maps, specifically those with an invariant one-dimensional submanifold (or one-dimensional maps). These maps exhibit a complex structure of basins with the coexistence of symmetric chaotic attractors, riddled basins, blow-out, on-off intermittency, and, most significantly, the appearance of chaotic synchronization with a correlation between all the characteristics. The numerical method of critical curves can be used to demonstrate a wide range of dynamic scenarios and explain the bifurcations that lead to their occurrence. These curves play a crucial role in a system of two symmetrically coupled maps, and their significance will be discussed. [ABSTRACT FROM AUTHOR]

This article addresses the challenging problem of fixed-time output-constrained synchronization for master–slave chaotic financial systems with unknown parameters and perturbations. A fixed-time neural adaptive control approach is originally proposed with the aid of the barrier Lyapunov function (BLF) and neural network (NN) identification. The BLF is introduced to preserve the synchronization errors always within the predefined output constraints. The NN is adopted to identify the compound unknown item in the synchronization error system. Unlike the conventional NN identification, the concept of indirect NN identification is employed, and only a single adaptive learning parameter is required to be adjusted online. According to the stability argument, the proposed controller can ensure that all error variables in the closed-loop system regulate to the minor residual sets around zero in fixed time. Finally, simulations and comparisons are conducted to verify the efficiency and benefits of the proposed control strategy. It can be concluded from the simulation results that the proposed fixed-time neural adaptive controller is capable of achieving better synchronization performance than the compared linear feedback controller. [ABSTRACT FROM AUTHOR]

This paper introduces a fractional-order financial risk system for the first time. Employing well-known tools and analyses such as bifurcations diagrams and spectral entropy, the dynamical behaviors of the system associated with fractional derivative are investigated. The impacts of the fractional derivative on the system's behavior and its dynamical feature are shown. Then, tracking control and stabilization of the systems are studied. As it is obvious, the existence of faults and failures in the process of control of financial and economic systems is undeniable — this issue necessitates applying proper control techniques for the systems. So as to achieve appropriate results in the control of fractional financial risk system, two finite-time fault-tolerant controllers are proposed, namely, finite-time active fault-tolerant control and finite-time passive fault-tolerant control. Not only do these techniques force the system to reach desired values in finite time, but also, the proposed techniques are robust against uncertainties, faults, and failures in actuators. Through finite-time observers, the effects of all uncertainties are taken to account in the active controller. Finally, numerical simulations of tracking control and stabilization are presented. For numerical simulations, the fractional financial risk system is considered to be in the presence of unknown disturbance as well as faults and failures in actuators. It is assumed that the system is in the presence of various types of actuator faults. Numerical results affirm the ability of the offered control techniques for pushing the states of the fractional-order risk system to the desired value in a short period of time. [ABSTRACT FROM AUTHOR]

• A new neural network-based terminal sliding mode technique is proposed to stabilize and synchronize fractional-order chaotic ecological systems in finite time. • The Chebyshev neural network is implemented to estimate unknown functions of the system. • Through the proposed Chebyshev neural network observer, the effects of external disturbances are fully taken into account. • The finite-time convergence of the closed-loop system, which is a new concept for ecological systems, is proven In the present study, a new neural network-based terminal sliding mode technique is proposed to stabilize and synchronize fractional-order chaotic ecological systems in finite-time. The Chebyshev neural network is implemented to estimate unknown functions of the system. Moreover, through the proposed Chebyshev neural network observer, the effects of external disturbances are fully taken into account. The weights of the Chebyshev neural network observer are adjusted based on adaptive laws. The finite-time convergence of the closed-loop system, which is a new concept for ecological systems, is proven. Then, the dependency of the system on the value of the fractional time derivatives is investigated. Lastly, the proposed control scheme is applied to the fractional-order ecological system. Through numerical simulations, the performance of the developed technique for synchronization and stabilization are assessed and compared with a conventional method. The numerical simulations strongly corroborate the effective performance of the proposed control technique in terms of accuracy, robustness, and convergence time for the unknown nonlinear system in the presence of external disturbances. [ABSTRACT FROM AUTHOR]

In this paper, dynamical behavior and synchronization of a non-equilibrium four-dimensional chaotic system are studied. The system only includes one constant term and has hidden attractors. Some dynamical features of the governing system, such as invariance and symmetry, the existence of attractors and dissipativity, chaotic flow with a plane of equilibria, and offset boosting of the chaotic attractor, are stated and discussed and a new disturbance-observer-based adaptive terminal sliding mode control (ATSMC) method with input saturation is proposed for the control and synchronization of the chaotic system. To deal with unexpected noises, an extended Kalman filter (EKF) is implemented along with the designed controller. Through the concept of Lyapunov stability, the proposed control technique guarantees the finite time convergence of the uncertain system in the presence of disturbances and control input limits. Furthermore, to decrease the chattering phenomena, a genetic algorithm is used to optimize the controller parameters. Finally, numerical simulations are presented to demonstrate the performance of the designed control scheme in the presence of noise, disturbances, and control input saturation. [ABSTRACT FROM AUTHOR]

Four-dimensional chaotic systems are a very interesting topic for researchers, given their special features. This paper presents a novel fractional-order four-dimensional chaotic system with self-excited and hidden attractors, which includes only one constant term. The proposed system presents the phenomenon of multi-stability, which means that two or more different dynamics are generated from different initial conditions. It is one of few published works in the last five years belonging to the aforementioned category. Using Lyapunov exponents, the chaotic behavior of the dynamical system is characterized, and the sensitivity of the system to initial conditions is determined. Also, systematic studies of the hidden chaotic behavior in the proposed system are performed using phase portraits and bifurcation transition diagrams. Moreover, a design technique of a new fuzzy adaptive sliding mode control (FASMC) for synchronization of the fractional-order systems has been offered. This control technique combines an adaptive regulation scheme and a fuzzy logic controller with conventional sliding mode control for the synchronization of fractional-order systems. Applying Lyapunov stability theorem, the proposed control technique ensures that the master and slave chaotic systems are synchronized in the presence of dynamic uncertainties and external disturbances. The proposed control technique not only provides high performance in the presence of the dynamic uncertainties and external disturbances, but also avoids the phenomenon of chattering. Simulation results have been presented to illustrate the effectiveness of the presented control scheme. • A novel fractional-order four-dimensional chaotic system with self-excited and hidden attractors is introduced. • We have characterized the chaos of the dynamical system and determined the sensitivity of the system. • Systematic studies of the hidden chaotic behavior in the proposed system are performed. • We have designed a fuzzy adaptive sliding mode control to synchronize the proposed system. [ABSTRACT FROM AUTHOR]

In this paper, by considering the Caputo-like delta difference definition, a fractional difference order map with chaotic dynamics and with no equilibria is proposed. The complex dynamical behaviors associated with fractional difference order maps are analyzed employing the phase portraits, bifurcations diagrams, and Lyapunov exponents. The complexity of the sequence generated by the chaotic difference map is studied using the permutation entropy approach. Afterwards, projective synchronization of the systems is investigated. Fuzzy logic engines as intelligent schemes are strong tools for control of various systems. However, studies that apply fuzzy logic engines for control of fractional-order discrete-time systems are rare. Hence, in the current study, by taking advantages of fuzzy systems, a new controller is proposed for the fractional-order discrete-time map. The fuzzy logic engine is implemented in order to enhance the performance and agility of the proposed control technique. The stability of the closed-loop systems and asymptotic convergence of the projective synchronization error based on the proposed control scheme are proven. Finally, numerical simulations which clearly confirm that the offered control technique is able to push the states of the fractional-order discrete-time system to the desired value in a short period of time are presented. [ABSTRACT FROM AUTHOR]