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Bifurcation analysis for energy transport system and its optimal control using parameter self-tuning law

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Abstract

A dynamical system for optimal path of energy and resources between two cities in China is considered in this paper. We have discussed dynamics of variables and parameters involved in mentioned system. Bifurcation analysis around non-hyperbolic equilibria is also explained for codimension 1 and 2 bifurcations. Furthermore, double-zero eigenvalue condition is calculated for the proposed model. We have adopted methodology of the generalized vectors for existence of Bogdanov–Takens bifurcation critical point and used analytical computations instead of center manifold theorem for Bogdanov–Takens bifurcation around zero equilibria. Further, with the aid of bifurcation diagram, phase portraits and time history, we discussed occurrence of period doubling, Hopf bifurcation and chaotic region of our proposed model. Based on Lyapunov function and robust control, optimal controllers are designed using Hamilton-Jacobi theorem for the stability of disturbance and aperiodic solution in optimal transportation system (3) due to energy imports from city A to city B.

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References

  • Algaba A, Dominguez-Moreno MC, Merino M, Rodriguez-Luis A (1998) Analysis of Hopf and Takens–Bogdanov bifurcations in a modified van der pol–duffing oscillator. Nonlinear Dyn 16:369–404

    MathSciNet  MATH  Google Scholar 

  • Algaba A, Dominguez-Moreno MC, Merino M, Rodriguez-Luis A (1999) On the Takens–Bogdanov bifurcation in the Chua’s equation. IEICE Trans Fundam Electron Commun Comput Sci 82(9):1722–1728

    Google Scholar 

  • Algaba A, Dominguez-Moreno MC, Merino M, Rodriguez-Luis A (2015) Takens–Bogdanov bifurcations of equilibria and periodic orbits in the Lorenz system. Commun Nonlinear Sci Numer Simul. https://doi.org/10.1016/j.cnsns.2015.06.034

    Article  MATH  Google Scholar 

  • Bi QS, Yu P (1999) Symbolic software development for computing the normal form of double Hopf bifurcation. Math Comput Model 29:49–70

    MathSciNet  MATH  Google Scholar 

  • Chen P, Islam SM (2005) Optimal control models in finance. Springer, Berlin

    MATH  Google Scholar 

  • Ding X, Sinha A (2016) Hydropower plant frequency control via feedback linearization and sliding mode control. J Dyn Sys Meas Control 138(7):0745011–0745015

    Google Scholar 

  • Dong Y, Li S, Zhang S (2017) Hopf bifurcation in a reaction–diffusion model with Degn–Harrison reaction scheme. Nonlinear Anal Real World Appl 33:284–297

    MathSciNet  MATH  Google Scholar 

  • Franke JE, Yakubu AA (2006) Multiple attractors via Cusp bifurcation in periodically varying environments. J Differ Equ Appl 11:365–377

    MathSciNet  MATH  Google Scholar 

  • Garfinkel A, Spano ML, Ditto WL, Weiss JN (1992) Controlling cardiac chaos. Science 257:1230–1235

    Google Scholar 

  • Huang J, Gong y (2013) Bifurcation analysis in a predator-prey model with constant-yield predator harvesting. Discrete Contin Dyn Syst Ser B 18(8):2101–2121

    MathSciNet  MATH  Google Scholar 

  • Kuznetsov YA (1998) Elements of applied theory. Springer, New York

    MATH  Google Scholar 

  • Kuznetsov YA (2005) Practicle computation of normal forms on center manifolds at degenerate Bogdanov–Takens bifurcation. Int J Bif Chaos 15(11):3535–3546

    MATH  Google Scholar 

  • Lam J, Tam HK (2004) Robust eigenstructure assignment with minimum subspace seperation. Int J Robust Nonlinear Control 14(15):1227–1253

    MATH  Google Scholar 

  • Li H, Wang M (2013) Hopf bifurcation analysis in a Lorenz-type system. Nonlinear Dyn 71:235–240

    MathSciNet  MATH  Google Scholar 

  • Liu X, Liu S (2012) Codimension two bifurcation analysis in two-dimensional Hindmarsh–Rose model. Nonlinear Dyn 67:847–857

    MathSciNet  MATH  Google Scholar 

  • Liu Z, Magal P, Xiao D (2016) Bogdanov–Takens bifurcation in a predator–prey model. Z Angew Math Phys 67(137):1–29

    MathSciNet  MATH  Google Scholar 

  • Marsden JE, McCracken M (1976) The Hopf bifurcation and its applications. Springer, Berlin

    MATH  Google Scholar 

  • Marwan M, Ahmad S, Aqeel M, Sabir M (2019) Control analysis of rucklidge chaotic system. J Dyn Syst Meas Control 141(4):0410101–0410107

    Google Scholar 

  • Marwan M, Mehboob M, Ahmad S, Aqeel M (2020) Hopf bifurcation of forced chen system and its stability via adaptive control with arbitrary parameters. Soft Comput 24(6):4333–4341

    MATH  Google Scholar 

  • Sang B, Huang B (2017) Bautin bifurcations of a financial system. Electron J Qual Theory Differ Equ 95:1–22

    MathSciNet  MATH  Google Scholar 

  • Shahzad M (2016) Chaos control in three dimensional cancer model by state space exact linearization based on Lie Algebra. Mathematics 33:111–112

    MATH  Google Scholar 

  • Song Z, Xu J (2012) Codimension-two bursting analysis in the delayed neural system with external stimulations. Nonlinear Dyn 67(1):309–328

    MathSciNet  MATH  Google Scholar 

  • Song Z, Xu J, Li Q (2009) Local and global bifurcations in an sirs epidemic model. Appl Math Comput 214(2):534–547

    MathSciNet  MATH  Google Scholar 

  • Song Z, Xu J, Zhen B (2019) Mixed-coexistence of periodic orbits and chaotic attractors in an inertial neural system with a nonmonotonic activation function. Math Biosci Eng MBE 16(6):6406–6425

    MathSciNet  Google Scholar 

  • Song Z, Zhen B, Hu D (2020) Multiple bifurcations and coexistence in an inertial two-neuron system with multiple delays. Cognit Neurodyn 14:1–16

    Google Scholar 

  • Soon LT, Salleh Z (2013) Hopf bifurcation analysis of a modified Lorenz system. AIP Conf Proc 1522:158–168

    MATH  Google Scholar 

  • Storac M, Linaro D, de lange E (2008) The Hindmarsh–Rose neuron model: bifurcation analysis and piecewise-linear approximations. Chaos 18:033128

    MathSciNet  Google Scholar 

  • Sui S, Yang J, Zhao L (2019) On the number of limit cycles for generic Lotka–Volterra system and Bogdanov–Takens system under perturbations of piecewise smooth polynomials. Nonlinear Anal Real World Appl 49:137–158

    MathSciNet  MATH  Google Scholar 

  • Sun T, Pan Y (2019) Robust adaptive control for prescribed performance tracking of constrained uncertain nonlinear system. J Frankl Inst 356(1):18–30

    MathSciNet  MATH  Google Scholar 

  • Sun M, Tian L, Xu J (2006a) Hopf bifurcation analysis of the energy resource chaotic system. Int J Nonlinear Sci 1(1):49–53

    MathSciNet  MATH  Google Scholar 

  • Sun M, Tian L, Xu J (2006b) Time-delayed feedback control of the energy resource chaotic system. Int J Nonlinear Sci 1(3):172–177

    MathSciNet  MATH  Google Scholar 

  • Sun M, Tian L, Jiang S, Xu J (2007a) Feedback control and adaptive control of the energy resource chaotic system. Chaos Solitons Fractals 32:1725–1734

    MathSciNet  MATH  Google Scholar 

  • Sun M, Tian L, Xu J (2007b) An energy resources demand–supply system and its dynamical analysis. Chaos Solitons Fractals 32:168–180

    MathSciNet  MATH  Google Scholar 

  • Tam HK, Lam J (2000) Robust deadbeat assignment with gain constraints: an LMI optimization approach. Optim Control Appl Methods 21(6):243–256

    MathSciNet  MATH  Google Scholar 

  • Tang Y, Zhang W (2004) Bogdanov–Takens bifurcation of a polynomial differential system in biochemical reaction. Comput Math Appl 48:869–883

    MathSciNet  MATH  Google Scholar 

  • Vaidyanathan S (2015a) Adaptive backstepping control of enzymes-substrates system with ferroelectric behaviour in brain-waves. Int J Pharm Tech Res 8:256–261

    Google Scholar 

  • Vaidyanathan S (2015b) Dynamics and control of Tokamak system with symmetric and magnetically confined plasma. Int J Chem Tech Res 8:795–803

    Google Scholar 

  • Wang M, Xu H, Sheng Z (2013) A novel three-dimensional energy resources chaotic system with government regulation and empirical research. J Inf Comput Sci 10(15):4879–4892

    Google Scholar 

  • Wei W, Wang M, Li D, Zuo M, Wang X (2016) Disturbance observer based active and adaptive synchronization of energy resource chaotic system. ISA Trans 65:164–173

    Google Scholar 

  • Wiggins S (2003) Introduction to applied nonlinear dynamical system and chaos. Springer, New York

    MATH  Google Scholar 

  • Wu X, Chi M (2015) Parameters study of Hopf bifurcation in railway vehicle system. J Comput Nonlinear Dyn 10(3):031012

    Google Scholar 

  • Wu Y, Hu K, Sun X-M (2018) Modeling and control design for quadrotors: a controlled hamiltonian systems approach. IEEE Trans Veh Technol 67(12):11365–11376

    Google Scholar 

  • Yang P (2019) Hopf bifurcation of an age-structured prey-predator model with Holling type II functional response incorporating a prey refuge. Nonlinear Anal Real World Appl 49:368–385

    MathSciNet  MATH  Google Scholar 

  • Yang SJ, Li CD, Huang TW (2014) Impulsive control and synchronization of memristor-based chaotic circuits. Int J Bifurc Chaos 24(12):145–162

    MATH  Google Scholar 

  • Yui D, Guan ZH, Li J, Liu I, Xiao JW, Ling G (2019) Stability and bifurcation of delay coupled genetic regularity networks with hub-structure. J Frankl Inst 356(5):2857–2869

    Google Scholar 

Download references

Acknowledgements

Authors are grateful to Higher Education Commission to support this research under project 5863/Federal/NRPU/R&D/HEC/2016.

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Institute of Space Technology, Islamabad, 44000, Pakistan

    Muhammad Marwan & Salman Ahmad

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  1. Muhammad Marwan
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  2. Salman Ahmad
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Correspondence to Muhammad Marwan.

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Marwan, M., Ahmad, S. Bifurcation analysis for energy transport system and its optimal control using parameter self-tuning law. Soft Comput 24, 17221–17231 (2020). https://doi.org/10.1007/s00500-020-05014-3

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