Figure 3

The two-parameter bifurcation diagram for system (2.4a)–(2.4b), k varying in \((0,4)\) and m varying in \((0,0.4)\). From Fig. 2(c) the black dark curve (\(E_{1}\) exists) represents \(\operatorname{Det}(E_{1})=0\) which separates Region I (no interior equilibria) and Region II (\(E_{2}\) and \(E_{3}\) exist), the black dashed line (\(E_{4}\) exists) represents \(m=c\) which separates Region II and Region III (\(E_{5}\) exists), and the blue line (Hopf bifurcation curve) represents \(\operatorname{Tr}(E_{2,4,5})=0\) which separates the stable and unstable region of equilibria \(E_{2,4,5}\). Red dot represents the Bogdanov–Takens point, which is the intersecting point of \(\operatorname{Det}(E_{1})=0\) and \(\operatorname{Tr}(E_{2,4,5})=0\)