Kybernetika 57 no. 5, 750-775, 2021

Stability analysis of the five-dimensional energy demand-supply system

Kun Yi Yang and Chun Xia AnDOI: 10.14736/kyb-2021-5-0750

Abstract:

In this paper, a five-dimensional energy demand-supply system has been considered. On the one hand, we analyze the stability for all of the equilibrium points of the system. For each of equilibrium point, by analyzing the characteristic equation, we show the conditions for the stability or instability using Routh-Hurwitz criterion. Then numerical simulations have been given to illustrate all of cases for the theoretical results. On the other hand, by introducing the phenomenon of time delay, we establish the five-dimensional energy demand-supply model with time delay. Then we analyze the stability of the equilibrium points for the delayed system by the stability switching theory. Especially, Hopf bifurcation has been considered by showing the explicit formulae using the central manifold theorem and Poincare normalization method. For each cases of the theorems including the Hopf bifurcation, numerical simulations have been given to illustrate the effectiveness of the main results.

Keywords:

stability, energy demand-supply, equilibrium points, hopf bifurcation

Classification:

93C15, 34K20

paper.pdf

References:

  1. I. I. Delice and R. Sipahi: Delay-independent stability test for systems with multiple time-delays. IEEE Trans. Automat. Control 57 (2012), 4, 963-972.   DOI:10.1109/TAC.2011.2168992
  2. K. Q. Gu and M. Naghnaeian: Stability crossing set for systems with three delays. IEEE Trans. Automat. Control 56 (2011), 1, 11-26.   DOI:10.1109/TAC.2010.2050162
  3. K. Q. Gu, S.-I. Niculescu and J. Chen: On stability crossing curves for general systems with two delays. J. Math. Analysis Appl. 311 (2005), 231-253.   DOI:10.1016/j.jmaa.2005.02.034
  4. M. H. Koh and R. Sipahi: Effects of edge elimination on the delay margin of a class of LTI consensus dynamics. IEEE Trans. Automat. Control 63 (2018), 12, 4397-4404.   DOI:10.1109/TAC.2018.2825884
  5. Z. G. Li, J. X. Chen, S.-I. Niculescu and A. Cela: New insights in stability analysis of delayed Lotka-Volterra systems. . Franklin Inst. 355 (2018), 8683-8697.   CrossRef
  6. Y. M. Li, K. Q. Gu, J. P. Zhou and S. Y. Xu: Estimating stable delay intervals with a discretized Lyapunov-Krasovskii functional formulation. Automatica 50 (2014), 1691-1697.   DOI:10.1016/j.jfranklin.2018.09.011
  7. J. Louisell: Imaginary axis eigenvalues of matrix delay equations with a certain alternating coefficient structure. Systems Control Lett. 110 (2017), 49-54.   DOI:10.1016/j.sysconle.2017.10.007
  8. G. W. Luo and X. X. Zhang: Hopf Bifurcation of HR and FHN Neuron Systems with Time Delayed. Master Degree Thesis of Lanzhou Jiaotong University, 2018.   CrossRef
  9. M. Naghnaeian and K. Q. Gu: Stability crossing set for systems with two scalar-delay channels. Automatica 49 (2013), 2098-2106.   DOI:10.1016/j.automatica.2013.04.025
  10. N. Olgac and R. Sipahi: An exact method for the stability analysis of time-delayed linear time-invariant(LTI) systems. IEEE Trans. Automat. Control 47 (2002), 5, 793-797.   DOI:10.1109/TAC.2002.1000275
  11. T. Qi, J. Zhu and J. Chen: Fundamental limits on uncertain delays: when is a delay system stabilizable by LTI controllers? IEEE Trans. Automat. Control 62(2017), 3, 1314-1328.   DOI:10.1109/TAC.2016.2584007
  12. T. Qi, J. Zhu and J. Chen: On delay radii and bounds of MIMO systems. Automatica 77 (2017), 214-218.   DOI:10.1016/j.automatica.2016.11.038
  13. S. G. Ruan and J. J. Wei: On the zeros of transcendental functions with application to stability of delay differential equations with two delays. Dynamics Continuous Discrete Impulsive Systems Series A: Math. Analysis 10 (2003), 863-874.   CrossRef
  14. R. Sipahi and I. I. Delice: Extraction of 3D stability switching hypersurfaces of a time delay system with multiple fixed delays. Automatica 45 (2009), 1449-1454.   DOI:10.1016/j.automatica.2009.01.017
  15. M. Sun, Q. Jia and L. X. Tian: A new four-dimensional energy resourses systems and its linear feedback control. Chaos Solitons Fractals 39 (2009), 101-108.   DOI:10.1016/j.chaos.2007.01.125
  16. M. Sun and L. X. Tian: An energy resources demand-supply system and its dynamical analysis. Chaos Solitons Fractals 32 (2007), 168-180.   DOI:10.1016/j.chaos.2005.10.085
  17. M. Sun and L. X. Tian: The chaos control for a new four-dimensional energy demand-supply system. J. Jiangsu University 5 (2007), 25-30.   CrossRef
  18. Z. Wang and H. Y. Hu: Stability switches of time-delayed dynamic systems with unknown parameters. J. Sound Vibration 233 (2000), 215-233.   DOI:10.1006/jsvi.1999.2817
  19. X. Wang, F. Q. Zhang and Y. J. Zhang: Hopf bifurcation of three species system with time delays. J. Systems Sci. Math. Sci. 30 (2010), 530-540.   CrossRef
  20. G. X. Wang, Z. M. Zhou, S. M. Zhu and S. S. Wang: Oridinary Differential Equations. Higher Education Press, Beijing 2006.   CrossRef
  21. J. J. Wei, H. B. Wang and W. H. Jiang: Theory and Application of Delay Differential Equations. Sciences Press, Beijing 2012.   CrossRef
  22. Y. H. Yang and G. H. Cao: A hyperchaotic system of five-dimensional energy supply and demand under new energy constraints. J. Systems Engrg. 34 (2019), 159-169.   CrossRef
  23. K. Y. Yang, L. L. Zhang and J. Zhang: Stability analysis of a three-dimensional energy demand-supply system under delayed feedback control. Kybernetika 51 (2015), 1084-1100.   DOI:10.14736/kyb-2015-6-1084