Bifurcation analysis for energy transport system and its optimal control using parameter self-tuning law
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METHODOLOGIES AND APPLICATION
Bifurcation analysis for energy transport system and its optimal control using parameter self-tuning law Muhammad Marwan1
· Salman Ahmad1
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 nonhyperbolic 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. Keywords Nonlinear dynamical system · Bifurcation theory · Normal form theory · Optimal control · Stability
1 Introduction Energy is one of the basic needs of every country exists on this globe. There are several important resources from which energy can be produced; coal, natural gas, petroleum, nuclear power and water power. Energy can be classified into two types; renewable and non-renewable energy. The issue of energy resources supply and demand has been valued with the advancement in modern technology. “Life on earth is no more a single bulb, which enlighten our lives with modern facilities”. In (2007b), Sun et al. discussed transportation problem of energy from eastern region of china to western region and other resources from western to eastern china, which was impossible in the absence of quantitative technique. Government of China has put forward some important suggestions and make a cooperation between the east and the west to bring some positive changes on both ends and proCommunicated by V. Loia.
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Muhammad Marwan [email protected] Salman Ahmad [email protected]
1
Department of Applied Mathematics and Statistics, Institute of Space Technology, Islamabad 44000, Pakistan
vided a win–win policy for both big cities. Wei et al. (2016) suggested productions on the energy consumption and distribution in the domain of uncertainties. Sun et al. (2007a) worked on economic and energy transport system under the assumption that both of them are chaotic. The control of chaotic systems through different techniques has gained popularity in the fields of engineering (Vaidyanathan 2015b; Marwan et al. 2019), physics (Yang et al. 2014; Ding and Sinha 2016), biology (Shahzad 2016; Vaidyanathan 2015a; Garfinkel et al. 1992),
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