Figure 8

Hopf bifurcation from the origin with initial values \(\pmb{(u_{1}(0),u_{2}(0),u_{3}(0),u_{4}(0))=(0.00053, 0.0004, 0.0002, 0.0004)}\) (blue curve), \(\pmb{(-0.153, -0.04, -0.02, -0.04)}\) (green curve), and parameters \(\pmb{(\alpha_{1},\alpha_{2})=(-0.001, 0.005)}\) . (a) The phase portrait in the plane \((u_{4},u_{1})\). (b) The time series in the plane \(({t},u_{1})\).