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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Authors are grateful to Higher Education Commission to support this research under project 5863/Federal/NRPU/R&D/HEC/2016.
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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 (2020). https://doi.org/10.1007/s00500-020-05014-3
- Nonlinear dynamical system
- Bifurcation theory
- Normal form theory
- Optimal control