Bifurcations of multiple relaxation oscillations in polynomial Liénard equations

Abstract

Abstract. In this paper, we prove the presence of limit cycles of given multiplicity, together with a complete unfolding, in families of (singularly perturbed) polynomial Liénard equations. The obtained limit cycles are relaxation oscillations. Both classical Liénard equations and generalized Liénard equations are treated.

Slow–fast Bogdanov–Takens bifurcations

“…This explains (and proves) the phase portraits represented in Fig. 8, near the situations (4) and (8). Near the limiting situation (8) there are no bifurcations, implying that in reality there are no topological differences between (7), (8) and (9) in Fig.…”

“…8, near the situations (4) and (8). Near the limiting situation (8) there are no bifurcations, implying that in reality there are no topological differences between (7), (8) and (9) in Fig. 8 (they are also equal to the cases (6) and (5)).…”

“…In [8] we show that there also does not exist a uniform bound on the cyclicity of (1) with n = 2m + 1, for m 1, and a minus sign in front of x 2m+1 . This paper can hence be seen as a natural continuation in an ambitious programme that aims at studying the slow-fast bifurcations (1) for arbitrary n. Not having specific names for these bifurcations, we will call them "standard slow-fast C ± (n, 1) bifurcations".…”

“…8. To study the phase portraits near the limiting situations (4) and (8) it suffices to study the zeros of (35) near (S, E,Ỹ ) = (0, 0, 0), for B 0 = +1 (in case (4)), B 0 = −1 (in case (8)) and B 1 ∼ 0.…”

Conditions for polynomial Liénard centers

“…There are lots of work (see [2,10,18,19] and the monographs [29,31]) on the existence, uniqueness and the number of limit cycles for System (1.1). Attention was also paid to its homoclinic (heteroclinic) bifurcations and chaos (see [7,14,15,17,33]). Another interesting question is aimed to centers of Liénard system.…”