Non-bifurcation of critical periods from semi-hyperbolic polycycles of quadratic centres

In this paper we consider the unfolding of saddle-node \[ X= \frac{1}{xU_a(x,y)}\Big(x(x^{\mu}-\varepsilon)\partial_x-V_a(x)y\partial_y\Big), \] parametrized by $(\varepsilon,\,a)$ with $\varepsilon \approx 0$ and $a$ in an open subset $A$ of $ {\mathbb {R}}^{\alpha },$ and we study the Dulac time $\mathcal {T}(s;\varepsilon,\,a)$ of one of its hyperbolic sectors. We prove (theorem 1.1) that the derivative $\partial _s\mathcal {T}(s;\varepsilon,\,a)$ tends to $-\infty$ as $(s,\,\varepsilon )\to (0^{+},\,0)$ uniformly on compact subsets of $A.$ This result is addressed to study the bifurcation of critical periods in the Loud's family of quadratic centres. In this regard we show (theorem 1.2) that no bifurcation occurs from certain semi-hyperbolic polycycles.

Bifurcations of Critical Periods for a Class of Quintic Liénard Equation

In this paper, we investigate a quintic Liénard equation which has a center at the origin. We give the conditions for the parameters for the isochronous centers and weak centers of exact order. Then, we present the global phase portraits for the system having isochronous centers. Moreover, we prove that at most four critical periods can bifurcate and show with appropriate perturbations that local bifurcation of critical periods occur from the centers.

Number of Critical Periods for Perturbed Rigidly Isochronous Centers

This paper deals with the bifurcation of critical periods from a rigidly quartic isochronous center. It shows that under any small homogeneous perturbation of degree four, up to any order in [Formula: see text], there are at most two critical periods bifurcating from the periodic orbits of the unperturbed system, and the upper bound is sharp. In addition, we further prove that under any small polynomial perturbation of degree [Formula: see text], up to the first order in [Formula: see text], there are at most [Formula: see text] critical periods bifurcating from the periodic orbits of the unperturbed quartic system.

Weak Center and Bifurcation of Critical Periods in a Cubic Z2-Equivariant Hamiltonian Vector Field

This paper focuses on the problems of weak center and local bifurcation of critical periods for a class of cubic Z2-equivariant planar Hamiltonian vector fields. By computing the period constants carefully, one can see that there are three weak centers: (±1, 0) and the origin. The corresponding weak center conditions are also derived. Meanwhile, we address the problem of the coexistence of bifurcation of critical periods that occurred from (±1, 0) and the origin.

Critical Period Bifurcation by Perturbing a Reversible Rigidly Isochronous Center with Multiple Parameters

This paper focuses on bifurcation of critical periods by perturbing a rigidly isochronous center with multiple parameters. First, we give expressions of period bifurcation functions (PBF for short) in the form of integrals, and then study the first PBF T1(ρ, λ) with a new method. Compared with the result in [Liu & Han, 2014], more critical periods can be found by our method.