On periodic solutions for forced Van der Pol type equations

Sufficient conditions for the existence of periodic solutions of the extended Duffing–Van der Pol oscillator

“…We remark that we cannot prove analytically the non-existence of 5 periodic solutions for system (8) considering m = 0, n 1 = n 3 = n 4 = 1 and n 2 > 1. Actually, using the algebraic manipulator Mathematica we obtained evidences that this number of periodic solutions cannot happen, but an analytical treatment is not trivial.…”

“…Proof of Theorem 5: Now we take m = 0, n 2 = n 3 = n 4 = 1 and n 1 > 1. The expression of the vector field of system (8) in this case is (y, −x + ε n 1 ry − ε n 1 rx 2 y − εax 3 − εℓx 5 + εd cos t) T , and we get F 1 (t, x) = (0, −ax 3 − ℓx 5 + d cos t). Hence function f = (f 1 , f 2 ) writes…”

“…We start studying the possible values to the powers of ε in system (8). This is important because these powers play an important role in averaging theory described in section 3 because they determine the terms of order 1 that we are interested in.…”

“…Observing Table 1 we see that in fact we have only 18 cases. Moreover in order to apply averaging theory, for which one of the 18 cases we must integrate the equations of the non-perturberd part of vector field of system (8). This is not a simple task because the major part of these equations are nonlinear and non-autonomous.…”

“…In [2] and [11] it is studied many aspects of system (4) for the case where α and λ are zero and ρ is large. Furthermore in [8] the authors provide sufficient conditions to the existence of one periodic solution under some analytical hypotheses for a general forced Van der Pol system considering ρ equal to zero. Periodic solutions for system (4) were found in [17], where the authors investigate chaos control.…”