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

Figure 12

Phase portraits of system (1.1) before the Hopf bifurcation at the points 𝐸_±. The parameter values are 𝜈=10, 𝜇=0.1, 𝛼=2, and 𝛽=𝛽_2𝑐+0.8=12.8 in (a); 𝛽=𝛽_2𝑐+0.2=12.2 in (b). Time of integration is [−4.5,50] for (a); [−4.5,270] for (b). Initial conditions are (0.0001,0,0.0001) and (−0.0001,0,−0.0001). Stepsize = 0.05. Eigenvalues at 𝐸₀ are (a) 𝜆₁=1.951, 𝜆_2,3=−1.475±2.093𝑖; (b) 𝜆₁=1.987, 𝜆_2,3=−1.493±1.976𝑖. Eigenvalues at 𝐸_± are (a) 𝜆₁=−3.928, 𝜆_2,3=−0.035±2.552𝑖; (b) 𝜆₁=−3.981, 𝜆_2,3=−0.009±2.475𝑖.

(a)

(b)