April  2021, 14(4): 1375-1393. doi: 10.3934/dcdss.2020280

Input-to-state stability and no-inputs stabilization of delayed feedback chaotic financial system involved in open and closed economy

1. 

Department of Mathematics, Chengdu Normal University, Chengdu, 611130, China

2. 

College of mathematics, University of Electronic Science and Technology of China, Chengdu 611731, China

* Corresponding author: Ruofeng Rao

Received  August 2019 Revised  December 2019 Published  April 2021 Early access  February 2020

Fund Project: The work is supported by the Application basic research project of science and Technology Department of Sichuan Province and the Major scientific research projects of Chengdu Normal University in 2019 (CS19ZDZ01)

The stochastic inflow or withdrawal of funds in the international financial market are both positive or negative external inputs to the domestic financial system. So, in this paper, input-to-state stability criterion of delayed feedback chaotic financial system is investigated, and derived by counterevidence method, Lyapunov functional method, variational method and regional control technique, which was involved to equilibrium solution with the positive interest rate. On the other hand, if these inputs are too small to be ignored, impulse control can be applied to stability analysis of the delayed feedback system, in which the delayed impulse allows the pulse effect to lag for a period of time. The obtained stability criteria show that no matter how complex and chaos the financial system is, high-frequency effective macro-control is conducive to the global asymptotical stability of the economic system, including the open economic model with foreign investment fund inputs. Finally, numerical examples illustrate the effectiveness of all the proposed methods.

Citation: Ruofeng Rao, Shouming Zhong. Input-to-state stability and no-inputs stabilization of delayed feedback chaotic financial system involved in open and closed economy. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1375-1393. doi: 10.3934/dcdss.2020280
References:
[1]

A. Y. AziR. RaoF. ZhaoH. HuangX. Wang and H. Liu, Impulse control of financial system with probabilistic delay feedback, Appl. Math. Mech., 40 (2019), 1409-1416.  doi: 10.21656/1000-0887.400059.

[2]

S. Bhalekar and V. Daftardar-Gejji, Synchronization of different fractional order chaotic systems using active control, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 3536-3546.  doi: 10.1016/j.cnsns.2009.12.016.

[3]

W. Chen, Dynamics and control of a financial system with time-delayed feedbacks, Chaos Soli. Frac., 37 (2008), 1198-1207.  doi: 10.1016/j.chaos.2006.10.016.

[4]

S. Chen and J. Lü, Synchronization of an uncertain unified chaotic system via adaptive control, Chaos, Soli. Frac., 14 (2002), 643-647.  doi: 10.1016/S0960-0779(02)00006-1.

[5]

S. Cheng, Complicated Science and Management, J. Nanchang Univ.(Human. Soc. Sci.), 3 (2000), 1-6. 

[6]

C. HuangL. Cai and J. Cao, Linear control for synchronization of a fractional-order time-delayed chaotic financial system, Chaos, Soli. Frac., 113 (2018), 326-332.  doi: 10.1016/j.chaos.2018.05.022.

[7] D. Huang and H. Li, Theory and Method of Nonlinear Economics, Sichuan University Press, Chengdu, China, 1993. 
[8]

T. Huang, C. Li and X. Liu, Synchronization of chaotic systems with delay using intermittent linear state feedback, Chaos, 18 (2008), 033122.

[9]

C. Huang, H. Zhang, J. Cao and H. Hu, Stability and hopf bifurcation of a delayed prey-predator model with disease in the predator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950091, 23 Pages. doi: 10.1142/S0218127419500913.

[10]

M. KrichmanE. D. Sontag and Y. Wang, Input-to-state stability, SIAM J. Control Optim., 39 (2000), 1874-1928.  doi: 10.1137/S0363012999365352.

[11]

X. Li and M. Bohner, An impulsive delay differential inequality and applications, Comp. Math. Appl., 64 (2012), 1875-1881.  doi: 10.1016/j.camwa.2012.03.013.

[12]

P. Li and X. Li, Input-to-state stability of nonlinear impulsive systems via Lyapunov method involving indefnite derivative, Math. Comput. Simu., 155 (2019), 314-323.  doi: 10.1016/j.matcom.2018.06.010.

[13]

X. LiJ. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Appl. Math. Comput., 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.

[14]

X. Li and X. Yang, Lyapunov stability analysis for nonlinear systems with state-dependent state delay, Automatica, 112 (2020), 108674. doi: 10.1016/j.automatica.2019.108674.

[15]

X. LiX. Yang and T. Huang, Persistence of delayed cooperative models: Impulsive control method, Appl. Math. Comput., 342 (2019), 130-146.  doi: 10.1016/j.amc.2018.09.003.

[16]

X. Li, Q. Zhu and D. O'Regan, $p$th Moment exponential stability of impulsive stochastic functional differential equations and application to control problems of NNs, J. Franklin Instit. EAM, 351 (2014), 4435–4456. doi: 10.1016/j.jfranklin.2014.04.008.

[17]

B. Liu, Asymptotic behavior of solutions to a class of non-autonomous delay differential equations, J. Math. Anal. Appl., 446 (2017), 580-590.  doi: 10.1016/j.jmaa.2016.09.001.

[18]

J. Ma and Y. Chen, Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (Ⅰ), Appl. Math. Mech., 11 (2001), 1240-1251.  doi: 10.1023/A:1016313804297.

[19]

J. Ma and Y. Chen, Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (Ⅱ), Appl. Math. Mech., 12 (2001), 1375-1382.  doi: 10.1023/A:1022806003937.

[20]

C. NingY. HeM. WuQ. Liu and J. She, Input-to-state stability of nonlinear systems based on an indefnite Lyapunov function, Syst. Control Lett., 61 (2012), 1254-1259.  doi: 10.1016/j.sysconle.2012.08.009.

[21]

C. NingY. HeM. Wu and S. Zhou, Indefinite derivative Lyapunov-Krasovskii functional method for input to state stability of nonlinear systems with time-delay, Appl. Math. Comput., 270 (2015), 534-542.  doi: 10.1016/j.amc.2015.08.063.

[22]

Z. Pu and R. Rao, Delay-dependent LMI-based robust stability criterion for discrete and distributed time-delays Markovian jumping reaction-diffusion CGNNs under Neumann boundary value, Neurocomputing, 171 (2016), 1367-1374.  doi: 10.1016/j.neucom.2015.07.063.

[23]

R. Rao, Global Stability of a Markovian Jumping Chaotic Financial System with Partially Unknown Transition Rates under Impulsive Control Involved in the Positive Interest Rate, Mathematics, 7 (2019), 579. doi: 10.3390/math7070579.

[24]

R. Rao, Delay-Dependent exponential stability for nonlinear reaction-diffusion uncertain Cohen-Grossberg neural networks with partially known transition rates via Hardy-Poincare inequality, Chin. Ann. Math. Ser.B, 35 (2014), 575-598.  doi: 10.1007/s11401-014-0839-7.

[25]

R. RaoJ. Hang and S. Zhong, Global exponential stability of reaction-diffusion BAM neural networks, J. Jilin Univ. (Sci. Ed.), 50 (2012), 1086-1090. 

[26]

R. RaoS. Zhong and Z. Pu, Fixed point and p-stability of T-S fuzzy impulsive reaction-diffusion dynamic neural networks with distributed delay via Laplacian semigroup, Neurocomputing, 335 (2019), 170-184.  doi: 10.1016/j.neucom.2019.01.051.

[27]

R. RaoS. Zhong and X. Wang, Stochastic stability criteria with LMI conditions for Markovian jumping impulsive BAM neural networks with mode-dependent time-varying delays and nonlinear reaction-diffusion, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 258-273.  doi: 10.1016/j.cnsns.2013.05.024.

[28]

E. D. Sontag, Comments on integral variants of ISS, Syst. Control Lett., 34 (1998), 93-100.  doi: 10.1016/S0167-6911(98)00003-6.

[29]

J. WangX. Chen and L. Huang, The number and stability of limit cycles for planar piecewise linear systems of node-saddle type, J. Math. Anal. Appl., 469 (2019), 405-427.  doi: 10.1016/j.jmaa.2018.09.024.

[30]

J. WangC. Huang and L. Huang, Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type, Nonlinear Anal. HS, 33 (2019), 162-178.  doi: 10.1016/j.nahs.2019.03.004.

[31]

D. YangX. Li and J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Anal. HS, 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.

[32]

X. YangX. LiQ. Xi and P. Duan, Review of stability and stabilization for impulsive delayed systems, Math. Biosci. Engin., 15 (2018), 1495-1515.  doi: 10.3934/mbe.2018069.

[33]

R. Zhang, Bifurcation analysis for a kind of nonlinear finance system with delayed feedback and its application to control of chaos, J. Appl. Math., 2012 (2012), Article ID 316390, 1–18. doi: 10.1155/2012/316390.

[34]

Y. Zhang and Q. Wang, Comment on "Synchronization of chaotic systems with delay using intermittent linear state feedback", Chaos, 18 (2008), 048102, 1p. doi: 10.1063/1.3046536.

[35]

X. ZhaoZ. Li and S. Li, Synchronization of a chaotic finance system, Appl. Math. Comput., 217 (2011), 6031-6039.  doi: 10.1016/j.amc.2010.07.017.

[36]

M. Zhao and J. Wang, $H_\infty$ control of a chaotic finance system in the presence of external disturbance and input time-delay, Appl. Math. Comput., 233 (2014), 320-327.  doi: 10.1016/j.amc.2013.12.085.

[37]

H. ZhuR. Rakkiyappan and X. Li, Delayed state-feedback control for stabilization of neural networks with leakage delay, Neural Net., 105 (2018), 249-255.  doi: 10.1016/j.neunet.2018.05.013.

show all references

References:
[1]

A. Y. AziR. RaoF. ZhaoH. HuangX. Wang and H. Liu, Impulse control of financial system with probabilistic delay feedback, Appl. Math. Mech., 40 (2019), 1409-1416.  doi: 10.21656/1000-0887.400059.

[2]

S. Bhalekar and V. Daftardar-Gejji, Synchronization of different fractional order chaotic systems using active control, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 3536-3546.  doi: 10.1016/j.cnsns.2009.12.016.

[3]

W. Chen, Dynamics and control of a financial system with time-delayed feedbacks, Chaos Soli. Frac., 37 (2008), 1198-1207.  doi: 10.1016/j.chaos.2006.10.016.

[4]

S. Chen and J. Lü, Synchronization of an uncertain unified chaotic system via adaptive control, Chaos, Soli. Frac., 14 (2002), 643-647.  doi: 10.1016/S0960-0779(02)00006-1.

[5]

S. Cheng, Complicated Science and Management, J. Nanchang Univ.(Human. Soc. Sci.), 3 (2000), 1-6. 

[6]

C. HuangL. Cai and J. Cao, Linear control for synchronization of a fractional-order time-delayed chaotic financial system, Chaos, Soli. Frac., 113 (2018), 326-332.  doi: 10.1016/j.chaos.2018.05.022.

[7] D. Huang and H. Li, Theory and Method of Nonlinear Economics, Sichuan University Press, Chengdu, China, 1993. 
[8]

T. Huang, C. Li and X. Liu, Synchronization of chaotic systems with delay using intermittent linear state feedback, Chaos, 18 (2008), 033122.

[9]

C. Huang, H. Zhang, J. Cao and H. Hu, Stability and hopf bifurcation of a delayed prey-predator model with disease in the predator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950091, 23 Pages. doi: 10.1142/S0218127419500913.

[10]

M. KrichmanE. D. Sontag and Y. Wang, Input-to-state stability, SIAM J. Control Optim., 39 (2000), 1874-1928.  doi: 10.1137/S0363012999365352.

[11]

X. Li and M. Bohner, An impulsive delay differential inequality and applications, Comp. Math. Appl., 64 (2012), 1875-1881.  doi: 10.1016/j.camwa.2012.03.013.

[12]

P. Li and X. Li, Input-to-state stability of nonlinear impulsive systems via Lyapunov method involving indefnite derivative, Math. Comput. Simu., 155 (2019), 314-323.  doi: 10.1016/j.matcom.2018.06.010.

[13]

X. LiJ. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Appl. Math. Comput., 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.

[14]

X. Li and X. Yang, Lyapunov stability analysis for nonlinear systems with state-dependent state delay, Automatica, 112 (2020), 108674. doi: 10.1016/j.automatica.2019.108674.

[15]

X. LiX. Yang and T. Huang, Persistence of delayed cooperative models: Impulsive control method, Appl. Math. Comput., 342 (2019), 130-146.  doi: 10.1016/j.amc.2018.09.003.

[16]

X. Li, Q. Zhu and D. O'Regan, $p$th Moment exponential stability of impulsive stochastic functional differential equations and application to control problems of NNs, J. Franklin Instit. EAM, 351 (2014), 4435–4456. doi: 10.1016/j.jfranklin.2014.04.008.

[17]

B. Liu, Asymptotic behavior of solutions to a class of non-autonomous delay differential equations, J. Math. Anal. Appl., 446 (2017), 580-590.  doi: 10.1016/j.jmaa.2016.09.001.

[18]

J. Ma and Y. Chen, Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (Ⅰ), Appl. Math. Mech., 11 (2001), 1240-1251.  doi: 10.1023/A:1016313804297.

[19]

J. Ma and Y. Chen, Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (Ⅱ), Appl. Math. Mech., 12 (2001), 1375-1382.  doi: 10.1023/A:1022806003937.

[20]

C. NingY. HeM. WuQ. Liu and J. She, Input-to-state stability of nonlinear systems based on an indefnite Lyapunov function, Syst. Control Lett., 61 (2012), 1254-1259.  doi: 10.1016/j.sysconle.2012.08.009.

[21]

C. NingY. HeM. Wu and S. Zhou, Indefinite derivative Lyapunov-Krasovskii functional method for input to state stability of nonlinear systems with time-delay, Appl. Math. Comput., 270 (2015), 534-542.  doi: 10.1016/j.amc.2015.08.063.

[22]

Z. Pu and R. Rao, Delay-dependent LMI-based robust stability criterion for discrete and distributed time-delays Markovian jumping reaction-diffusion CGNNs under Neumann boundary value, Neurocomputing, 171 (2016), 1367-1374.  doi: 10.1016/j.neucom.2015.07.063.

[23]

R. Rao, Global Stability of a Markovian Jumping Chaotic Financial System with Partially Unknown Transition Rates under Impulsive Control Involved in the Positive Interest Rate, Mathematics, 7 (2019), 579. doi: 10.3390/math7070579.

[24]

R. Rao, Delay-Dependent exponential stability for nonlinear reaction-diffusion uncertain Cohen-Grossberg neural networks with partially known transition rates via Hardy-Poincare inequality, Chin. Ann. Math. Ser.B, 35 (2014), 575-598.  doi: 10.1007/s11401-014-0839-7.

[25]

R. RaoJ. Hang and S. Zhong, Global exponential stability of reaction-diffusion BAM neural networks, J. Jilin Univ. (Sci. Ed.), 50 (2012), 1086-1090. 

[26]

R. RaoS. Zhong and Z. Pu, Fixed point and p-stability of T-S fuzzy impulsive reaction-diffusion dynamic neural networks with distributed delay via Laplacian semigroup, Neurocomputing, 335 (2019), 170-184.  doi: 10.1016/j.neucom.2019.01.051.

[27]

R. RaoS. Zhong and X. Wang, Stochastic stability criteria with LMI conditions for Markovian jumping impulsive BAM neural networks with mode-dependent time-varying delays and nonlinear reaction-diffusion, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 258-273.  doi: 10.1016/j.cnsns.2013.05.024.

[28]

E. D. Sontag, Comments on integral variants of ISS, Syst. Control Lett., 34 (1998), 93-100.  doi: 10.1016/S0167-6911(98)00003-6.

[29]

J. WangX. Chen and L. Huang, The number and stability of limit cycles for planar piecewise linear systems of node-saddle type, J. Math. Anal. Appl., 469 (2019), 405-427.  doi: 10.1016/j.jmaa.2018.09.024.

[30]

J. WangC. Huang and L. Huang, Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type, Nonlinear Anal. HS, 33 (2019), 162-178.  doi: 10.1016/j.nahs.2019.03.004.

[31]

D. YangX. Li and J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Anal. HS, 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.

[32]

X. YangX. LiQ. Xi and P. Duan, Review of stability and stabilization for impulsive delayed systems, Math. Biosci. Engin., 15 (2018), 1495-1515.  doi: 10.3934/mbe.2018069.

[33]

R. Zhang, Bifurcation analysis for a kind of nonlinear finance system with delayed feedback and its application to control of chaos, J. Appl. Math., 2012 (2012), Article ID 316390, 1–18. doi: 10.1155/2012/316390.

[34]

Y. Zhang and Q. Wang, Comment on "Synchronization of chaotic systems with delay using intermittent linear state feedback", Chaos, 18 (2008), 048102, 1p. doi: 10.1063/1.3046536.

[35]

X. ZhaoZ. Li and S. Li, Synchronization of a chaotic finance system, Appl. Math. Comput., 217 (2011), 6031-6039.  doi: 10.1016/j.amc.2010.07.017.

[36]

M. Zhao and J. Wang, $H_\infty$ control of a chaotic finance system in the presence of external disturbance and input time-delay, Appl. Math. Comput., 233 (2014), 320-327.  doi: 10.1016/j.amc.2013.12.085.

[37]

H. ZhuR. Rakkiyappan and X. Li, Delayed state-feedback control for stabilization of neural networks with leakage delay, Neural Net., 105 (2018), 249-255.  doi: 10.1016/j.neunet.2018.05.013.

Table 1.  Comparisons of Theorem 3.1 and [23,Theorem 1]
[23,Theorem 1] Theorem 3.1
stability type exponential stability input-to-state stability
admitting external inputs No Yes
Requirements on initial function No boundedness restraint
[23,Theorem 1] Theorem 3.1
stability type exponential stability input-to-state stability
admitting external inputs No Yes
Requirements on initial function No boundedness restraint
Table 2.  Comparisons of Theorem 4.1, Corollary 2 and [23,Theorem 2]
[23,Theorem 2] Corollary 2 Theorem 4.1
stability type E-S A-S A-S
interest rate $8.93\%$ $8.93\%$ $8.93\%$
delayed feedback model No Yes Yes
time-delays on impulse No $\rho_k\equiv0$ $\rho_k\equiv0.05$
[23,Theorem 2] Corollary 2 Theorem 4.1
stability type E-S A-S A-S
interest rate $8.93\%$ $8.93\%$ $8.93\%$
delayed feedback model No Yes Yes
time-delays on impulse No $\rho_k\equiv0$ $\rho_k\equiv0.05$
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