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

Figure 9

Phase portrait of system (1.1) near the equilibrium point 𝐸₀ for the parameter values 𝜈=3, 𝜇=−0.25, 𝛼=2, and (a) 𝛽=𝛽_1𝑐+0.5=2.5625; (b) 𝛽=𝛽_1𝑐+0.05=2.1125. Time of integration is [5,350] for (a) [5,450] for (b). Initial condition is (0.001,−0.001,0.001). Stepsize = 0.1. In both cases the point 𝐸₀ is a stable focus, but the convergence in the case (b) is slower than in (a) since the parameter 𝛽 is closer to the critical value 𝛽_1𝑐=2.0625. Eigenvalues at 𝐸₀ are (a) 𝜆₁=−2.624, 𝜆_2,3=−0.062±0.853𝑖; (b) 𝜆₁=−2.737, 𝜆_2,3=−0.006±0.760𝑖.

(a)

(b)