Figure 4

Detailed bifurcation analysis for traditional rivalry. (A) Bifurcation diagram of the Wilson model (1) with fixed inputs varying adaptation strength h, \(g=1.5\). In addition to WTA, RIV, and SIM, two other regions with different dynamical behavior are revealed: (B) Mixed-mode oscillations (MMOs) emerging from high amplitude relaxation oscillations (RIV) with discontinuous transitions between segments. Each period of these MMOs has one high and one or more low-amplitude oscillations (see Fig. 6 for time histories). On MMO branches n:m defines the n high to m low-amplitude oscillations ratio. The number of low-amplitude oscillations starts from one and is increased by one as we move down in the bifurcation parameter. (C) Low amplitude winner-take-all (LAWTA) oscillations emerge from supercritical Hopf bifurcation on the WTA branch and by further increasing the bifurcation parameter, a cascade of period-doubling bifurcations emerges. Panels B and C show the maximum of \(E_{1} \) & \(E_{2} \) on the limit cycle branches. The minimum of MMOs is close to zero. (D) Boundaries of different dynamical behaviors are shown in parameter space \((h,g)\). The region with the periodic solution (RIV) is confined by Hopf bifurcation (red solid line) from beneath and by fold bifurcation (L, green dashed line) from above. Other parameters: \(J_{\mathrm{HL}}=J_{\mathrm{VR}}=10\)