Bifurcation Analysis of a Van der Pol-Duffing Circuit with Parallel Resistor

Figure 13

Phase portraits of system (1.1) after the Hopf bifurcation at the points 𝐸_±. The parameter values are 𝜈=10, 𝜇=0.1, 𝛼=2, and 𝛽=𝛽_2𝑐−0.2=11.8. The equilibria become unstable and two symmetric limit cycles arise. Time of integration is [−4.5,270] for (a); [500,550] for (b). Initial conditions are (0.0001,0,0.0001), (−0.0001,0,−0.0001), (0.5,0.01,0.5), and (−0.5,0.01,−0.5) for (a); (0.0001,0,0.0001) and (−0.0001,0,−0.0001) for (b). Stepsize = 0.05. Eigenvalues at 𝐸₀ are 𝜆₁=2.012, 𝜆_2,3=−1.506±1.895𝑖. Eigenvalues at 𝐸_± are 𝜆₁=−4.018, 𝜆_2,3=0.009±2.423𝑖.

(a)

(b)