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

Figure 14

(a) Period doubling bifurcation for the limit cycle created in the Hopf bifurcation at the point 𝐸_±. The parameter values are 𝜈=10, 𝜇=0.1, 𝛼=2, and 𝛽=𝛽_2𝑐−1.13=10.87. (b) Double symmetric homoclinic loop to the origin. Parameter values 𝜈=10, 𝜇=0.1, 𝛼=2, and 𝛽=𝛽_2𝑐−1.214322=10.785678. Initial conditions are (0.0001,0,0.0001) and (−0.0001,0,−0.0001). Eigenvalues at 𝐸₀ are (a) 𝜆₁=2.073, 𝜆_2,3=−1.536±1.697𝑖; (b) 𝜆₁=2.079, 𝜆_2,3=−1.539±1.678𝑖. Eigenvalues at 𝐸_± are (a) 𝜆₁=−4.104, 𝜆_2,3=0.052±2.300𝑖; (b) 𝜆₁=−4.111, 𝜆_2,3=0.055±2.289𝑖.

(a)

(b)