Periodic solutions of a generalized Van der Pol–Mathieu differential equation

Bautin bifurgation of a modified generalized Van der Pol-Mathieu equation

“…The following lemma will be useful for deriving the averaged equation. The proof of this lemma can be found in [2]. Now we shall state Theorems 2 and 3 which show the conditions for Generalized Andronov-Hopf (Bautin) bifurcation and the topological normal form for Bautin bifurcation.…”

“…In this section we derive the autonomous system from the differential equation 3and we use the same method as in [2]. We carry out the substitution τ = (ω 0 + d o ε)t and get the equation…”

“…4β2 + λ. This inequality implies that the equation (42) has the discriminant D = β 2 1 − 4β 2 (α − λ sin(2ϕ)) > 0. The set of points ρ + is empty for β 1 < 0.…”

“…4β2 − λ implies that the equation (42) has discriminant D = β 2 1 − 4β 2 (α − λ sin(2ϕ)) < 0 and therefore the set of points, for which it is true that h(ρ, ϕ) = 0, does not exist. The derivation ρ is always negative for a positive real number ρ according to the condition for α.…”

“…The Van der Pol-Mathieu equation 1d 2 x dt 2 − ε(α 0 − β 0 x 2 ) dx dt + ω 2 0 (1 + εh 0 cos γt)x = 0 with a small detuning parameter ε describes the dynamics of dusty plasmas and it has recently been studied near 1 : 2 resonance in [5]. In [7], using the averaging method, the existence of stable periodic and stable quasi-periodic solutions near the parametric frequency is proved.…”