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

Figure 16

Phase portraits of system (1.1) before the Hopf bifurcation at the points 𝐸_±. The parameter values are near the point 𝑄₁ of Theorem 4.6 in the parameter space: 𝜇=𝜇_𝑄1=0.015289525, 𝜈=𝜈_𝑄1=4.036345241, and 𝛽=𝛽_𝑄+0.5=4.523353871 in (a); 𝛽=𝛽_𝑄+0.01=4.033353871 in (b). Time of integration is [−4,50] (a); [−4,1380] (b). Initial conditions are (0.001,0,0.001) and (−0.001,0,−0.001). Stepsize = 0.01. Eigenvalues at 𝐸₀ are (a) 𝜆₁=0.280, 𝜆₂=−1.602, 𝜆₃=−0.617; (b) 𝜆₁=0.357, 𝜆₂=−1.937, 𝜆₃=−0.358. Eigenvalues at 𝐸_± are (a) 𝜆₁=−1.891, 𝜆_2,3=−0.115±0.530𝑖; (b) 𝜆₁=−2.119, 𝜆_2,3=−0.002±0.484𝑖.

(a)

(b)