Abstract
We propose a three-dimensional autonomous nonlinear system, called the generalTsystem, which has potential applications in secure communications and the electronic circuit. For the generalTsystem with delayed feedback, regarding the delay as bifurcation parameter, we investigate the effect of the time delay on its dynamics. We determine conditions for the existence of the Hopf bifurcations and analyze their direction and stability. Also, the fractional order generalT-system is proposed and analyzed. We provide some numerical simulations, where the chaos attractor and the dynamics of the Lyapunov coefficients are taken into consideration. The effectiveness of the chaotic control and synchronization on schemes for the new fractional order chaotic system are verified by numerical simulations.
Highlights
Lorenz found the first canonical chaotic attractor [1]
Tigan and Opris [2] proposed and analyzed a new chaotic three-dimensional nonlinear system, called T system, which is similar to the Lorenz system
Based on (1), (2), and (3), we propose a general T system described by the following differential equations: ẋ (t) = a (y (t) − x (t)), ẏ (t) = b1x (t) − b2x (t) z (t), (4)
Summary
Lorenz found the first canonical chaotic attractor [1]. During the time, it has been proved that chaos can occur in simple three-dimensional autonomous systems with one, two, and three nonlinearities. Tigan and Opris [2] proposed and analyzed a new chaotic three-dimensional nonlinear system, called T system, which is similar to the Lorenz system. Li et al [15] have proposed a new Lorenz-like chaotic system derived from (1). Based on (1), (2), and (3), we propose a general T system described by the following differential equations:. As in [13], in this paper we use the Caputo definition for the fractional derivative. The aim is to provide a new investigation of the Hopf bifurcation and chaos control on the general T system given by (5) and an analysis of the fractional general T-system as well. The chaotic dynamics in the general T-system with fractional derivative is taken into account.
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