Advertisement

A Novel Stability Criterion of Time-varying delay Fractional-order Financial Systems Based a New Functional Transformation Lemma

  • Zhe ZhangEmail author
  • Jing Zhang
  • Fanyong Cheng
  • Feng Liu
Regular Papers Control Theory and Applications
  • 33 Downloads

Abstract

This paper puts forward a novel stability criterion of all cases of the time-delay fractional-order financial systems(FFS) including FFS without time delay, FFS with constant time delay and FFS with time-varying delay. This novel stability criterion is mainly based on a new stability judgment method which contains the deduction of Wirtinger inequality, Integral mean value theorem, fractional-order Lyapunov method, and a new functional transformation lemma which we deduced. This new functional transformation lemma simplifies the structure of the novel stability criterion with fewer constraints. Thus, compared with the previous stability criterion of FFS, the novel stability criterion of FFS has clearer structure and lower conservatism. Moreover, the novel stability criterion of FFS can also satisfy all fractional-order operators from 0 to 1. Last but not least, some numerical simulation examples are provided to verify the effectiveness and the benefit of the proposed novel stability criterion of FFS.

Keywords

Financial system stability analysis stability criterion time-varying delay 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Liang, R. Wu, and L. Chen, “BIBO stability of fractional-order controlled nonlinear systems,” International Journal of Systems Science, vol. 48, no. 7, pp.1507-1514, December 2017.Google Scholar
  2. [2]
    H. Yu, G. Cai, and Y. Li, “Dynamic analysis and control of a new hyperchaotic finance system,” Nonlinear Dynamics, vol. 67, no. 3, pp. 2171–2182, September 2011.MathSciNetzbMATHGoogle Scholar
  3. [3]
    L. Zhang, G. Cai, and X. Fang, “Stability and hybrid synchronization of a time-delay financial hyperchaotic system,” Journal of Information and Computing Science, vol. 10, no. 3, pp. 189–198, May 2015.Google Scholar
  4. [4]
    J. Ding, W. Yang, and H. Yao, “A new modified hyperchaotic finance system and its control,” Journal of Information and Computing Science, vol. 8, no. 1, pp. 59–66, April 2009.MathSciNetzbMATHGoogle Scholar
  5. [5]
    U. E. Kocamaz, A. Göksu, Y. Uyaroğlu, and H. Takn, “Controlling hyperchaotic finance system with combining passive and feedback controllers,” Information Technology & Control, vol. 47, no. 1, pp. 45–55, May 2018.Google Scholar
  6. [6]
    C. Lin, “A four-dimensional hyperchaotic finance system and its control problems,” Journal of Control Science & Engineering, no. 6, pp. 1–12, February 2018.Google Scholar
  7. [7]
    M. J. Hai and C. Y. Shu, “Study for the bifurcation topological structure and the global complicated character of a kind of non-linear finance system,” Applied Mathematics and Mechanics, vol. 22, no. 11, pp. 1375–1382, November 2001.Google Scholar
  8. [8]
    L. He and X. Wang, “Parameters estimation and stability analysis of nonlinear fractional-order economic system based on empirical data,” Abstract and Applied Analysis, vol. 2014, no. 2, pp. 1–11, April 2014.MathSciNetGoogle Scholar
  9. [9]
    Y. G. Yang, W. Xu, Y. H. Sun, and Y. W. Xiao, “Stochastic bifurcations in the nonlinear vibroimpact system with fractional derivative under random excitation,” Communications in Nonlinear Science & Numerical Simulation, no. 42, pp. 62–72, January 2017.MathSciNetGoogle Scholar
  10. [10]
    C. J. Wu, S. Lv, J. C. Long, J. H. Yang, and A. F. S. Miguel, “Self-similarity and adaptive aperiodic stochastic resonance in a fractional-order system,” Nonlinear Dynamics, vol. 91, no. 3, pp. 1697–1711, February 2018.Google Scholar
  11. [11]
    B. B. He, H. C. Zhou, Y. Q. Chen, and C. H. Kou, “Asymptotical stability of fractional order systems with time delay via an integral inequality,” Iet Control Theory & Applications, vol. 12, no. 12, pp. 1748–1754, April 2018.MathSciNetGoogle Scholar
  12. [12]
    M. Ardashir and G. Sehraneh, “Robust synchronization of uncertain fractional-order chaotic systems with timevarying delay,” Nonlinear Dynamics, vol. 93, no. 4, pp. 1809–1821, September 2018.zbMATHGoogle Scholar
  13. [13]
    B. Du, Y. H. Wei, S. Liang and Y. Wang, “Rational approximation of fractional order systems by vector fitting method,” International Journal of Control Automation & Systems, vol. 15, no. 1, pp. 186–195, February 2017.Google Scholar
  14. [14]
    S. Song, X. N. Song, N. Pathak, and T. B. Ines, “Multiswitching adaptive synchronization of two fractional-order chaotic systems with different structure and different order,” International Journal of Control Automation & Systems, vol. 15, no.43, pp. 1524–1535, August 2017.Google Scholar
  15. [15]
    Q. Wang and D. L. Qi, “Synchronization for fractional order chaotic systems with uncertain parameters,” International Journal of Control Automation & Systems, vol. 14, no. 1, pp. 211–216, February 2016.Google Scholar
  16. [16]
    M. Saliha, C. Mohammed, and B. Djillali, “A novel approach of admissibility for singular linear continuous-time fractional-order systems,” International Journal of Control Automation & Systems, vol. 15, no. 2, pp. 959–6964, April 2017.zbMATHGoogle Scholar
  17. [17]
    S. Wang and R. C. Wu, “Dynamic analysis of a 5D fractional-order hyperchaotic system,” International Journal of Control Automation & Systems, vol. 15, no. 3, pp. 1003–1010, June 2017.Google Scholar
  18. [18]
    B. S. Vadivoo, R. Raja, J. D. Cao, H. Zhang, and X. D. Li, “Controllability analysis of nonlinear neutral-type fractional-order differential systems with state delay and impulsive effects,” International Journal of Control Automation & Systems, vol. 16, no. 2, pp. 659–669, April 2018.Google Scholar
  19. [19]
    Z. B. Wu, Y. Z. Zou, and N. J. Huang, “A system of fractional-order interval projection neural networks,” Journal of Computational & Applied Mathematics, vol. 294, no. 1, pp. 389–402, March 2016.MathSciNetzbMATHGoogle Scholar
  20. [20]
    I. Matychyn and V. Onyshchenko, “On time-optimal control of fractional-order systems,” Journal of Computational & Applied Mathematics, vol. 339, pp. 245–257, September 2018.MathSciNetzbMATHGoogle Scholar
  21. [21]
    B. Bayour and F. M. T. Delfim, “Existence of solution to a local fractional nonlinear differential equation,” Journal of Computational & Applied Mathematics, vol. 312, no. 1, pp. 127–133, March 2017.MathSciNetzbMATHGoogle Scholar
  22. [22]
    V. Gafiychuk, D. Bohdan, and V. M. Vitalii, “Mathematical modeling of time fractional reaction-diffusion systems,” Journal of Computational & Applied Mathematics, vol. 220, no. 1, pp. 215–225, October 2008.MathSciNetGoogle Scholar
  23. [23]
    G. A. Javier and D. A. Manuel, “Boundedness and convergence on fractional order systems,” Journal of Computational& Applied Mathematics, vol. 296, pp. 815–826, April 2016.MathSciNetzbMATHGoogle Scholar
  24. [24]
    Y. J. Fan, X. Huang, Z. Wang, and Y. X. Li, “Nonlinear dynamics and chaos in a simplified memristor-based fractional-order neural network with discontinuous memductance function,” Nonlinear Dynamics, vol. 93, no. 2, pp. 611–627, July 2018.zbMATHGoogle Scholar
  25. [25]
    R. Z. Luo and Y. H. Zeng, “The control and synchronization of fractional-order Genesio-Tesi system,” Nonlinear Dynamics, vol. 88, no. 3, pp. 2111–2121, May 2017.MathSciNetzbMATHGoogle Scholar
  26. [26]
    B. K. Lenka and S. Banerjee, “Sufficient conditions for asymptotic stability and stabilization of autonomous fractional order systems,” Communications in Nonlinear Science & Numerical Simulation, vol. 56, pp. 365–379, March 2018.MathSciNetGoogle Scholar
  27. [27]
    Y. Q. Chen, Y. H. Wei, X. Zhou, and Y. Wang, “Stability for nonlinear fractional order systems: an indirect approach,” Nonlinear Dynamics, vol. 89, no. 2, pp. 1011–1018, July 2017.zbMATHGoogle Scholar
  28. [28]
    S. Rathinasamy and Y. Ren, “Approximate controllability of fractional differential equations with state-dependent delay,” Results in Mathematics, vol. 63, no. 3–4, pp. 949–963, June 2013.MathSciNetzbMATHGoogle Scholar
  29. [29]
    K. M. ALi, T. Hamed, and B. Oscar, “On dynamic sliding mode control of nonlinear fractional-order systems using sliding observer,” Nonlinear Dynamics, vol. 92, no. 3, pp. 1379–1393, May 2018.zbMATHGoogle Scholar
  30. [30]
    F. D. Marius, F. Michal, V. K. Nikolay, and G. R. Chen, “Complex dynamics, hidden attractors and continuous approximation of a fractional-order hyperchaotic PWC system,” Nonlinear Dynamics, vol. 91, no. 4, pp. 2523–25, March 2018.zbMATHGoogle Scholar
  31. [31]
    O. Zaid, “A note on phase synchronization in coupled chaotic fractional order systems,” Nonlinear Analysis Real World Applications, vol. 13, no. 2, pp. 779–789, April 2012.MathSciNetzbMATHGoogle Scholar
  32. [32]
    M. P. Aghababa, “Stabilization of a class of fractionalorder chaotic systems using a non-smooth control methodology,” Nonlinear Dynamics, vol. 89, no. 2, pp. 1357–1370, July 2017.MathSciNetzbMATHGoogle Scholar
  33. [33]
    M. V. Fidel and M. G. Rafael, “A reduced-order fractional integral observer for synchronization and antisynchronization of fractional-order chaotic systems,” IET Control Theory & Applications, vol. 12, no. 12, pp. 1755–1762, August 2018.MathSciNetGoogle Scholar
  34. [34]
    Y. G. Tang, N. Li, M. M. Liu, Y. Lu, and W. W. Wang, “Identification of fractional-order systems with time delays using block pulse functions,” Mechanical Systems& Signal Processing, vol. 91, pp. 382–394, July, 2017.Google Scholar
  35. [35]
    M. Saliha, C. Mohammed, and B. Djillali, “New admissibility conditions for singular linear continuous-time fractional-order systems,” Journal of The Franklin Institute, vol. 354, no. 2, pp. 752–766, January 2017.MathSciNetzbMATHGoogle Scholar
  36. [36]
    A. G. Mohammed, B. Djillali, and C. Mohammed, “Influence of discretization step on positivity of a certain class of two-dimensional continuous-discrete fractional linear systems,” IMA Journal of Mathematical Control and Information, vol. 35, no. 3, pp. 845860, September 2018.MathSciNetzbMATHGoogle Scholar
  37. [37]
    M. W. Zheng, L. X. Li, H. P. Peng, J. H. Xiao, Y. X. Yang, Y. P. Zhang, and H. Zhao, “Finite-time stability and synchronization of memristor-based fractional-order fuzzy cellular neural networks,” Communications in Nonlinear Science & Numerical Simulation, vol. 59, pp. 272–291, June 2018.MathSciNetGoogle Scholar
  38. [38]
    A. Ricardo, “A Caputo fractional derivative of a function with respect to another function,” Communications in Nonlinear Science & Numerical Simulation, vol. 44, pp. 460–481, March 2017.MathSciNetGoogle Scholar
  39. [39]
    D. Baleanu, G. C. Wu, and S. D. Zeng, “Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations,” Chaos Solitons & Fractals, vol. 102, pp. 99–105, September 2017.MathSciNetzbMATHGoogle Scholar
  40. [40]
    D. Y. Chen, R. F. Zhang, X. Z Liu, and X. Y. Ma, “Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks,” Communications in Nonlinear Science & Numerical Simulation, vol. 19, no. 12, pp. 4105–4121, December 2014.MathSciNetGoogle Scholar
  41. [41]
    F. Jarad, T. Abdeljawad, and D. Baleanu, “Stability of q-fractional non-autonomous systems,” Nonlinear Analysis Real World Applications, vol. 14, no. 1, pp. 780–784, February 2013.MathSciNetzbMATHGoogle Scholar
  42. [42]
    G. F. Anaya, G. N. Antonio, J. J. Galante, R. M. Vega, and E. G. H. Martínez, “Lyapunov functions for a class of nonlinear systems using Caputo derivative,” Communications in Nonlinear Science & Numerical Simulation, vol. 43, pp. 91–99, February 2017.MathSciNetGoogle Scholar
  43. [43]
    Z. Zhang, J. Zhang, and Z. Y. Ai, “A novel stability criterion of the time-lag fractional-order gene regulatory network system for stability analysis,” Communications in Nonlinear Science & Numerical Simulation, vol. 66, pp. 96–108, January 2019.MathSciNetGoogle Scholar

Copyright information

© ICROS, KIEE and Springer 2019

Authors and Affiliations

  1. 1.College of Electrical and Information EngineeringHunan UniversityChangshaChina
  2. 2.Fujian Provincial Key Laboratory of Information Processing and Intelligent ControlMinjiang UniversityFuzhouChina
  3. 3.School of AutomationChina University of Geosciences (Wuhan)WuhanChina

Personalised recommendations