Figure 4

The bifurcation diagram by fixing \(k=2.5\) and varying m in \((0.16,0.2)\). In (a), (b) the x and y components of equilibria Ê, \(E_{2}\), \(E_{3}\) are plotted respectively, the blue line represents Ê and it is always stable, the red line represents \(E_{3}\) and it is unstable, the red dotted line represents an unstable branch of \(E_{2}\), and the black line represents a stable branch of \(E_{2}\). The complex behavior of system (2.4a)–(2.4b) by separating critical values of m by black dotted lines into the regions namely A, B, C, D, and E. In A, Ê only exists and in B, C, D the Ê, \(E_{2}\), \(E_{3}\) exist. In C, the unstable \(E_{2}\) is surrounded by stable limit cycle (green curve), and when \(m=c=0.2\) the Ê, \(E_{3}\) coincide with the saddle equilibrium \(E_{0}\)