Corrigendum to “Chaos in a Financial System with Fractional Order and Its Control via Sliding Mode”

In this paper, the dynamical behaviors and chaos control of a fractional-order financial system are discussed. The lowest fractional order found from which the system generates chaos is 2.49 for the commensurate order case and 2.57 for the incommensurate order case. Also, the period-doubling route to chaos was found in this system. The results of this study were validated by the existence of a positive Lyapunov exponent. Besides, in order to control chaos in this fractional-order financial system with uncertain dynamics, a sliding mode controller is derived. The proposed controller stabilizes the commensurate and incommensurate fractional-order systems. Numerical simulations are carried out to verify the analytical results.

Theory of chaos synchronization and quasi-period synchronization of an all optic 2n-D LAN

Theory of chaos synchronization and quasi-period synchronization of an all optics local area network (O-LAN) is deeply studied and discussed, where two coupled-lasers are used as network’s double-star and the other single-lasers are used as network nodes. The LAN operates double-star lasers to drive node lasers in two links to perform a 2n−D (n is a positive integer, dimensions (D)) laser network. The O-LAN has the characteristics of an all optics LAN with double-center and two link nodes. Our theoretical and numerical results prove that the double-center lasers can obtain their synchronizations with each laser in two link nodes. A route to chaos after a quasi-period bifurcation is analyzed to illustrate dynamics distribution region of O-LAN. We find five quasi-period regions, four chaos regions, where there is a region where instability mixes with the first chaos, and a stable region. We find also that O-LAN can obtain its parallel multi-dynamics synchronizations, such as cycle-one synchronization, cycle-2 synchronization, cycle-3 synchronization, cycle-4 synchronization, cycle-5 synchronization, other quasi-period synchronization and chaos synchronization, shown in two links of O-LAN by shifting the currents of the lasers in one link. The theory of all optics LAN and its obtained results are useful to study on complex dynamic system, optics network, artificial intelligence, chaos and its synchronization.

Chaos in a Financial System with Fractional Order and Its Control via Sliding Mode

In this paper, the dynamical behaviors and chaos control of a fractional-order financial system are discussed. The lowest fractional order found from which the system generates chaos is 2.49 for the commensurate order case and 2.13 for the incommensurate order case. Also, period-doubling route to chaos was found in this system. The results of this study were validated by the existence of a positive Lyapunov exponent. Besides, in order to control chaos in this fractional-order financial system with uncertain dynamics, a sliding mode controller is derived. The proposed controller stabilizes the commensurate and incommensurate fractional-order systems. Numerical simulations are carried out to verify the analytical results.

Circuit Implementation of a Modified Chaotic System with Hyperbolic Sine Nonlinearities Using Bi-Color LED

In this paper, a chaotic three dimansional dynamical system is proposed, that is a modification of the system in Volos et al. (2017). The new system has two hyperbolic sine nonlinear terms, as opposed to the original system that only included one, in order to optimize system’s chaotic behavior, which is confirmed by the calculation of the maximal Lyapunov exponents and Kaplan-Yorke dimension. The system is experimentally realized, using Bi-color LEDs to emulate the hyperbolic sine functions. An extended dynamical analysis is then performed, by computing numerically the system’s bifurcation and continuation diagrams, Lyapunov exponents and phase portraits, and comparing the numerical simulations with the circuit simulations. A series of interesting phenomena are unmasked, like period doubling route to chaos, coexisting attractors and antimonotonicity, which are all verified from the circuit realization of the system. Hence, the circuit setup accurately emulates the chaotic dynamics of the proposed system.

Antimonotonicity, Crisis, and Route to Chaos in a Tumor Growth Model

In this chapter, a new model of a tumor growth is dynamically investigated. The model is presented in a form of a system of three ordinary differential equations, which describe the avascular, vascular, and metastasis tumor growth, respectively. For the investigation of system's dynamics and especially of the population of the immune cells in system's behavior, some of the most well-known tools from nonlinear theory, such as the phase portrait, the Poincaré map, the bifurcation diagram the Kaplan-Yorke dimension, and the Lyapunov exponents have been used. Interesting phenomena related with chaos theory, such as a period-doubling route to chaos, crisis phenomena, and antimonotonicity, have been revealed for the first time in this model.