November  2019, 2(4): 333-346. doi: 10.3934/mfc.2019021

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

1. 

Springfield, MO 65801-2604, USA

2. 

Springfield, MO 65810, USA

* Corresponding author: sxryctc@163.com

Published  December 2019

Fund Project: The first author is supported by NSF 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.

Citation: Juanjuan Huang, Yan Zhou, Xuerong Shi, Zuolei Wang. A single finite-time synchronization scheme of time-delay chaotic system with external periodic disturbance. Mathematical Foundations of Computing, 2019, 2 (4) : 333-346. doi: 10.3934/mfc.2019021
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E. N. Lorenz, Deterministic non periodic flow, J Atmos Sci, 20 (1963), 130-141. 

[2]

O. E. Rossler, An equation for hyperchaos, Phys. Lett. A, 71 (1979), 155-157. 

[3]

F. L. Zhu, Observer-based synchronization of uncertain chaotic system and its application to secure communications, Chaos Soliton. Fract, 40 (2009), 2384-2391.  doi: 10.1016/j.chaos.2007.10.052.

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P. ArenaS. BaglioL. Fortuna and G. Managaro, Hyperchaos from cellular neural networks. Electron Lett, Electron Let, 31 (1995), 250-251. 

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R. VicenteJ. DaudenP. Colet and R. Toral, Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop, IEEE J Quantum Elect, 41 (2005), 541-548. 

[6]

J. MaF. Q. WuG. D. Ren and J. Tang, A class of initials-dependent dynamical systems, Appl. Math. Comput, 298 (2017), 65-76.  doi: 10.1016/j.amc.2016.11.004.

[7]

Z. AramaS. Jafaria and J. Ma, Using chaotic artificial neural networks to model memory in the brain, Commun. Nonlinear Sci, 44 (2017), 449-459.  doi: 10.1016/j.cnsns.2016.08.025.

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M. S. AzzazC. Tanougast and S. Sadoudi, A new auto-switched chaotic system and its FPGA implementation, Commun Nonlinear Sci, 18 (2013), 1792-1804.  doi: 10.1016/j.cnsns.2012.11.025.

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L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett, 64 (1990), 821-823.  doi: 10.1103/PhysRevLett.64.821.

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C. D. LiX. F. Liao and K. W. Wong, Lag synchronization of hyperchaos with application to secure communications, Chaos Soliton. Fract, 23 (2005), 183-193.  doi: 10.1016/j.chaos.2004.04.025.

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G. N. TangK. S. Xu and L. L. Jiang, Synchronization in a chaotic neural network with time delay depending on the spatial distance between neurons, Phys. Rev. E, 84 (2011), 046207-046211. 

[12]

J. MaL. HuangZ. B. Xie and C. N. Wang, Simulated test of electric activity of neurons by using Josephson junction based on synchronization scheme, Commun. Nonlinear Sci, 17 (2012), 2659-2669.  doi: 10.1016/j.cnsns.2011.10.029.

[13]

X. WangX. Z. LiuK. She and S. M. Zhong, Pinning impulsive synchronization of complex dynamical networks with various time-varying delay sizes, Nonlinear Anal-Hybri, 26 (2017), 307-318.  doi: 10.1016/j.nahs.2017.06.005.

[14]

S. LiangR. Wu and L. Chen, Adaptive pinning synchronization in fractional-order uncertain complex dynamical networks with delay, Physica A, 391 (2012), 5746-5758.  doi: 10.1016/j.physa.2012.06.050.

[15]

Y. ToopchiJ. MahdiJ. Sadati and J. Wang, Fractional PI pinning synchronization of fractional complex dynamical networks, J. Comput. Appl. Math, 347 (2019), 357-368.  doi: 10.1016/j.cam.2018.08.016.

[16]

C. F. Feng, Projective synchronization between two different time-delayed chaotic systems using active control approach, Nonlinear Dynam, 62 (2010), 453-459. 

[17]

J. MaL. MiP. ZhouY. Xu and T. Hayat, Phase synchronization between two neurons induced by coupling of electromagnetic field, Appl. Math. Comput, 307 (2017), 321-328.  doi: 10.1016/j.amc.2017.03.002.

[18]

X. R. Shi and Z. L. Wang, The alternating between complete synchronization and hybrid synchronization of hyperchaotic Lorenz system with time delay, Nonlinear Dynam, 69 (2012), 1177-1190.  doi: 10.1007/s11071-012-0339-9.

[19]

J. MaF. LiL. Huang and W. Y. Jin, Complete synchronization, phase synchronization and parameters estimation in a realistic chaotic system, Commun. Nonlinear Sci, 16 (2011), 3770-3785. 

[20]

W. W. Zhang and J. D. Cao, Lag projective synchronization of fractional-order delayed chaotic systems, J. Franklin I, 356 (2019), 1522-1534.  doi: 10.1016/j.jfranklin.2018.10.024.

[21]

Z. L. Wang and X. R. Shi, Chaotic bursting lag synchronization of Hindmarsh-Rose system via a single controller, Appl. Math. Comput., 215 (2009), 1091-1097.  doi: 10.1016/j.amc.2009.06.039.

[22]

X. P. ZhangD. Li and X. H. Zhang, Adaptive fuzzy impulsive synchronization of chaotic systems with random parameters, Chaos, Soliton, Fract, 104 (2017), 77-83.  doi: 10.1016/j.chaos.2017.08.006.

[23]

M. M. Al-Sawalha and M. S. Noorani, Adaptive anti-synchronization of two identical and different hyperchaotic systems with uncertain parameters, Commun. Nonlinear Sci, 15 (2010), 1036-1047.  doi: 10.1016/j.cnsns.2009.05.037.

[24]

Z. M. Odibat, Adaptive feedback control and synchronization of non-identical chaotic fractional order systems, Nonlinear Dynam, 60 (2010), 479-487. 

[25]

E. K. Ugur and C. Bar, Control and synchronization of chaos with sliding mode control based on cubic reaching rule, Chaos, Soliton, Fract, 105 (2017), 92-98.  doi: 10.1016/j.chaos.2017.10.008.

[26]

J. MeiM. H. JiangW. M. Xu and B. Wang, Finite-time synchronization control of complex dynamical networks with time delay, Commun. Nonlinear Sci, 18 (2013), 2462-2478.  doi: 10.1016/j.cnsns.2012.11.009.

[27]

H. WangZ. Z. HanO. Y. Xie and W. Zhang, Finite-time chaos control of unified chaotic systems with uncertain parameters, Nonlinear Dynam, 55 (2009), 323-328.  doi: 10.1007/s11071-008-9364-0.

[28]

G. HeJ. A. Fang and Z. Li, Finite-time synchronization of cyclic switched complex networks under feedback control, J. Franklin I, 354 (2017), 3780-3796.  doi: 10.1016/j.jfranklin.2016.10.016.

[29]

M. C. Pai, Chaotic sliding mode controllers for uncertain time-delay chaotic systems with input nonlinearity, Appl. Math. Comput, 271 (2015), 757-767.  doi: 10.1016/j.amc.2015.09.058.

[30]

X. R. ShiZ. L. Wang and L. X. Han, Finite-time stochastic synchronization of time-delay neural networks with noise disturbance, Nonlinear Dynam, 88 (2017), 2747-2755.  doi: 10.1007/s11071-017-3408-2.

[31]

Z. W. CaiL. H. Huang and L. L. Zhang, Finite-time synchronization of master-slave neural networks with time-delays and discontinuous activations, Appl. Math. Model., 47 (2017), 208-226.  doi: 10.1016/j.apm.2017.03.024.

[32]

H. WangZ. Z. HanQ. Y. Xie and W. Zhang, Finite-time chaos synchronization of unified chaotic system with uncertain parameters, Commun. Nonlinear Sci, 14 (2009), 2239-2247. 

[33]

H. WangZ. Z. HanQ. Y. Xie and W. Zhang, Finite-time chaos control via nonsingular terminal sliding mode control, Commun. Nonlinear Sci, 14 (2009), 2728-2733.  doi: 10.1016/j.cnsns.2008.08.013.

[34]

R. Z. Luo and H. P. Su, Finite-time control and synchronization of a class of systems via the twisting controller, Chinese J. Phys, 55 (2017), 2199-2207.  doi: 10.1016/j.cjph.2017.09.003.

[35]

J. H. Lü and G. R. Chen, A new chaotic attractor coined, Int. J. Bifur. Chaos, 12 (2002), 659-661.  doi: 10.1142/S0218127402004620.

show all references

References:
[1]

E. N. Lorenz, Deterministic non periodic flow, J Atmos Sci, 20 (1963), 130-141. 

[2]

O. E. Rossler, An equation for hyperchaos, Phys. Lett. A, 71 (1979), 155-157. 

[3]

F. L. Zhu, Observer-based synchronization of uncertain chaotic system and its application to secure communications, Chaos Soliton. Fract, 40 (2009), 2384-2391.  doi: 10.1016/j.chaos.2007.10.052.

[4]

P. ArenaS. BaglioL. Fortuna and G. Managaro, Hyperchaos from cellular neural networks. Electron Lett, Electron Let, 31 (1995), 250-251. 

[5]

R. VicenteJ. DaudenP. Colet and R. Toral, Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop, IEEE J Quantum Elect, 41 (2005), 541-548. 

[6]

J. MaF. Q. WuG. D. Ren and J. Tang, A class of initials-dependent dynamical systems, Appl. Math. Comput, 298 (2017), 65-76.  doi: 10.1016/j.amc.2016.11.004.

[7]

Z. AramaS. Jafaria and J. Ma, Using chaotic artificial neural networks to model memory in the brain, Commun. Nonlinear Sci, 44 (2017), 449-459.  doi: 10.1016/j.cnsns.2016.08.025.

[8]

M. S. AzzazC. Tanougast and S. Sadoudi, A new auto-switched chaotic system and its FPGA implementation, Commun Nonlinear Sci, 18 (2013), 1792-1804.  doi: 10.1016/j.cnsns.2012.11.025.

[9]

L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett, 64 (1990), 821-823.  doi: 10.1103/PhysRevLett.64.821.

[10]

C. D. LiX. F. Liao and K. W. Wong, Lag synchronization of hyperchaos with application to secure communications, Chaos Soliton. Fract, 23 (2005), 183-193.  doi: 10.1016/j.chaos.2004.04.025.

[11]

G. N. TangK. S. Xu and L. L. Jiang, Synchronization in a chaotic neural network with time delay depending on the spatial distance between neurons, Phys. Rev. E, 84 (2011), 046207-046211. 

[12]

J. MaL. HuangZ. B. Xie and C. N. Wang, Simulated test of electric activity of neurons by using Josephson junction based on synchronization scheme, Commun. Nonlinear Sci, 17 (2012), 2659-2669.  doi: 10.1016/j.cnsns.2011.10.029.

[13]

X. WangX. Z. LiuK. She and S. M. Zhong, Pinning impulsive synchronization of complex dynamical networks with various time-varying delay sizes, Nonlinear Anal-Hybri, 26 (2017), 307-318.  doi: 10.1016/j.nahs.2017.06.005.

[14]

S. LiangR. Wu and L. Chen, Adaptive pinning synchronization in fractional-order uncertain complex dynamical networks with delay, Physica A, 391 (2012), 5746-5758.  doi: 10.1016/j.physa.2012.06.050.

[15]

Y. ToopchiJ. MahdiJ. Sadati and J. Wang, Fractional PI pinning synchronization of fractional complex dynamical networks, J. Comput. Appl. Math, 347 (2019), 357-368.  doi: 10.1016/j.cam.2018.08.016.

[16]

C. F. Feng, Projective synchronization between two different time-delayed chaotic systems using active control approach, Nonlinear Dynam, 62 (2010), 453-459. 

[17]

J. MaL. MiP. ZhouY. Xu and T. Hayat, Phase synchronization between two neurons induced by coupling of electromagnetic field, Appl. Math. Comput, 307 (2017), 321-328.  doi: 10.1016/j.amc.2017.03.002.

[18]

X. R. Shi and Z. L. Wang, The alternating between complete synchronization and hybrid synchronization of hyperchaotic Lorenz system with time delay, Nonlinear Dynam, 69 (2012), 1177-1190.  doi: 10.1007/s11071-012-0339-9.

[19]

J. MaF. LiL. Huang and W. Y. Jin, Complete synchronization, phase synchronization and parameters estimation in a realistic chaotic system, Commun. Nonlinear Sci, 16 (2011), 3770-3785. 

[20]

W. W. Zhang and J. D. Cao, Lag projective synchronization of fractional-order delayed chaotic systems, J. Franklin I, 356 (2019), 1522-1534.  doi: 10.1016/j.jfranklin.2018.10.024.

[21]

Z. L. Wang and X. R. Shi, Chaotic bursting lag synchronization of Hindmarsh-Rose system via a single controller, Appl. Math. Comput., 215 (2009), 1091-1097.  doi: 10.1016/j.amc.2009.06.039.

[22]

X. P. ZhangD. Li and X. H. Zhang, Adaptive fuzzy impulsive synchronization of chaotic systems with random parameters, Chaos, Soliton, Fract, 104 (2017), 77-83.  doi: 10.1016/j.chaos.2017.08.006.

[23]

M. M. Al-Sawalha and M. S. Noorani, Adaptive anti-synchronization of two identical and different hyperchaotic systems with uncertain parameters, Commun. Nonlinear Sci, 15 (2010), 1036-1047.  doi: 10.1016/j.cnsns.2009.05.037.

[24]

Z. M. Odibat, Adaptive feedback control and synchronization of non-identical chaotic fractional order systems, Nonlinear Dynam, 60 (2010), 479-487. 

[25]

E. K. Ugur and C. Bar, Control and synchronization of chaos with sliding mode control based on cubic reaching rule, Chaos, Soliton, Fract, 105 (2017), 92-98.  doi: 10.1016/j.chaos.2017.10.008.

[26]

J. MeiM. H. JiangW. M. Xu and B. Wang, Finite-time synchronization control of complex dynamical networks with time delay, Commun. Nonlinear Sci, 18 (2013), 2462-2478.  doi: 10.1016/j.cnsns.2012.11.009.

[27]

H. WangZ. Z. HanO. Y. Xie and W. Zhang, Finite-time chaos control of unified chaotic systems with uncertain parameters, Nonlinear Dynam, 55 (2009), 323-328.  doi: 10.1007/s11071-008-9364-0.

[28]

G. HeJ. A. Fang and Z. Li, Finite-time synchronization of cyclic switched complex networks under feedback control, J. Franklin I, 354 (2017), 3780-3796.  doi: 10.1016/j.jfranklin.2016.10.016.

[29]

M. C. Pai, Chaotic sliding mode controllers for uncertain time-delay chaotic systems with input nonlinearity, Appl. Math. Comput, 271 (2015), 757-767.  doi: 10.1016/j.amc.2015.09.058.

[30]

X. R. ShiZ. L. Wang and L. X. Han, Finite-time stochastic synchronization of time-delay neural networks with noise disturbance, Nonlinear Dynam, 88 (2017), 2747-2755.  doi: 10.1007/s11071-017-3408-2.

[31]

Z. W. CaiL. H. Huang and L. L. Zhang, Finite-time synchronization of master-slave neural networks with time-delays and discontinuous activations, Appl. Math. Model., 47 (2017), 208-226.  doi: 10.1016/j.apm.2017.03.024.

[32]

H. WangZ. Z. HanQ. Y. Xie and W. Zhang, Finite-time chaos synchronization of unified chaotic system with uncertain parameters, Commun. Nonlinear Sci, 14 (2009), 2239-2247. 

[33]

H. WangZ. Z. HanQ. Y. Xie and W. Zhang, Finite-time chaos control via nonsingular terminal sliding mode control, Commun. Nonlinear Sci, 14 (2009), 2728-2733.  doi: 10.1016/j.cnsns.2008.08.013.

[34]

R. Z. Luo and H. P. Su, Finite-time control and synchronization of a class of systems via the twisting controller, Chinese J. Phys, 55 (2017), 2199-2207.  doi: 10.1016/j.cjph.2017.09.003.

[35]

J. H. Lü and G. R. Chen, A new chaotic attractor coined, Int. J. Bifur. Chaos, 12 (2002), 659-661.  doi: 10.1142/S0218127402004620.

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 $

(a) ($ x_1,x_2 $) (b) ($ t ,x_1 $)

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 $

(a) ($ x_1,x_2 $) (b) ($ t ,x_1 $)

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 $

(a) ($ x_1,x_2 $) (b) ($ t ,x_1 $)

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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