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# A single finite-time synchronization scheme of time-delay chaotic system with external periodic disturbance

The first author is supported by NSFC grant 11872327 and 51777180

• In this paper, dynamical behaviors of three-dimensional chaotic system with time-delay and external periodic disturbance are investigated. When the periodic perturbation term and time-delay are added, the system presents more abundant dynamic behaviors, which can be switched between periodic state and chaotic state. Based on Lyapunov stability theory, a sufficient condition for finite-time synchronization is given. A single controller is proposed to realize finite-time synchronization of time-delay chaotic system with external periodic disturbance. The addressed scheme is provided in the form of linear inequality which is simple and easy to be realized. At the same time, it also displays that when delay term $\tau$ takes different values, the time of synchronization shows certain difference. The feasibility and effectiveness of the finite-time synchronization method is verified by theoretical analysis and numerical simulation.

Erratum: The affiliations of all four authors have been corrected to School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, 224002, China. NSF has been corrected to NSFC under Fund Project. We apologize for any inconvenience this may cause.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

 Citation: • • Figure 1.  Phase trajectory and the time series of Eq.(9) with $a = 15,b = 3,c = 7$ (a) ($x_1,x_2$) (b) ($t ,x_1$)

Figure 2.  Phase trajectory and the time series of Eq.(9) with $a = 15,b = 0.91,c = 7$ (a) ($x_1,x_2$) (b) ($t ,x_1$)

Figure 3.  Phase trajectory and the time series of Eq.(9) with $a = 15,b = 0.5,c = 7$ (a) ($x_1,x_2$) (b) ($t ,x_1$)

Figure 4.  Phase trajectory and the time series of Eq.(9) with $a = 15,b = 3,c = 2$ (a) ($x_1,x_2$) (b) ($t ,x_1$)

Figure 5.  Phase trajectory and the time series of Eq.(10) with $a = 15,b = 0.91,c = 7,A = 10,\omega = 0.001$ (a) ($x_1,x_2$) (b) ($t ,x_1$)

Figure 6.  Phase trajectory and the time series of Eq.(10) with $a = 15,b = 3,c = 7,A = 100,\omega = 0.001$ (a) ($x_1,x_2$) (b) ($t ,x_1$)

Figure 7.  Phase trajectory and the time series of Eq.(11) with $a = 15,b = 0.91,c = 7,A = 10,\omega = 0.001,\tau = 0.3$

Figure 8.  Phase trajectory and the time series of Eq.(11) with $a = 15,b = 3,c = 7,A = 0.1,\omega = 0.001,\tau = 0.2$

Figure 9.  Phase trajectory and the time series of Eq.(11) with $a = 15,b = 3,c = 7,A = 0.1,\omega = 0.001,\tau = 0.3$

Figure 10.  2D overview chaotic attractor and the chaotic time series of Eq.(11) with $\tau = 0.005$ (a) ($x_1,x_2$) (b) ($t ,x_1$)

Figure 11.  2D overview chaotic attractor and the chaotic time series of Eq.(11) with $\tau = 0.3$ (a) ($x_1,x_2$) (b) ($t ,x_1$)

Figure 12.  The error dynamics between systems (11) and (12) with $\tau = 0.3$ (a) $e_1$ (b) $e_2$ (b) $e_3$

Figure 13.  The error states $e_1$ between systems (11) and (12) with $\tau = 0.005,\quad\tau = 0.05\quad and \quad \tau = 0.3$

Figure 14.  The error states $e_2$ between systems (11) and (12) with $\tau = 0.005,\quad\tau = 0.05\quad and \quad \tau = 0.3$

Figure 15.  The error states $e_3$ between systems (11) and (12) with $\tau = 0.005,\quad \tau = 0.05\quad and \quad \tau = 0.3$

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