Practical computation of higher codimension Bogdanov–Takens bifurcations in energy supply–demand system

The analysis of energy crises is significant research topic in both theory and engineering. In the current paper, a nonlinear, real-life based model of energy resources transportation between two cities is considered to study codimension-1, codimension-2, and codimension-3 bifurcations using the first Lyapunov coefficient and generalized eigenvectors. The first Lyapunov coefficient (LC) provides sufficient information about the type of Hopf bifurcation and the emergence of a Bautin bifurcation at the critical points. Moreover, it is proved that there exists codimension-two Bogdanov–Takens (BT) bifurcation exhibiting Hopf, saddle–node and homoclinic local bifurcation curves at their specific parameters. Additionally, under certain conditions, the normal form of the codimension-three Bogdanov–Takens bifurcation is derived and analyzed.

Although the generalized eigenvectors in Hopf and BT bifurcations are essential for analysis, the orthogonality condition is not always satisfied, requiring tedious calculations. To facilitate this process, MATLAB codes are provided to help readers bypass these complex steps. Moreover, bifurcation diagrams for single and double parameters of the model are plotted to analyze the system’s behavior near critical points. Finally, to validate the analytical results, detailed numerical simulations of the bifurcation analysis are presented.