scholarly journals LOCAL BIFURCATION OF CRITICAL PERIODS IN QUADRATIC-LIKE CUBIC SYSTEMS

2019 ◽  
Vol 9 (5) ◽  
pp. 1901-1926
Author(s):  
Zhiheng Yu ◽  
◽  
Zhaoxia Wang ◽  
Keyword(s):  
Local Bifurcation ◽  
Critical Periods ◽  
Bifurcation Of Critical Periods ◽  
Cubic Systems

Bifurcations of Critical Periods for a Class of Quintic Liénard Equation

2020 ◽  
Vol 30 (14) ◽  
pp. 2050201
Author(s):  
Zhiheng Yu ◽  
Lingling Liu
Keyword(s):  
Phase Portraits ◽  
Liénard Equation ◽  
Local Bifurcation ◽  
Exact Order ◽  
Critical Periods ◽  
Isochronous Centers ◽  
Bifurcation Of Critical Periods ◽  
Lienard 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.

Local Bifurcation of Critical Periods in Vector Fields With Homogeneous Nonlinearities of the Third Degree

1993 ◽  
Vol 36 (4) ◽  
pp. 473-484 ◽  
Author(s):  
C. Rousseau ◽  
B. Toni
Keyword(s):  
Vector Field ◽  
Finite Order ◽  
Periodic Orbits ◽  
Vector Fields ◽  
Local Bifurcation ◽  
Critical Periods ◽  
The Third ◽  
Bifurcation Of Critical Periods

AbstractIn this paper we study the local bifurcation of critical periods of periodic orbits in the neighborhood of a nondegenerate centre of a vector field with a homogeneous nonlinearity of the third degree. We show that at most three local critical periods bifurcate from a weak linear centre of finite order or from the linear isochrone and at most two local critical periods from the nonlinear isochrone. Moreover, in both cases, there are perturbations with the maximum number of critical periods.

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

2015 ◽  
Vol 25 (11) ◽  
pp. 1550143 ◽  
Author(s):  
Yusen Wu ◽  
Wentao Huang ◽  
Yongqiang Suo
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Hamiltonian Vector ◽  
Local Bifurcation ◽  
Hamiltonian Vector Field ◽  
Critical Periods ◽  
Weak Center ◽  
Hamiltonian Vector Fields ◽  
Center Conditions ◽  
Bifurcation Of Critical Periods

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.

scholarly journals Bifurcation of critical periods of polynomial systems

2015 ◽  
Vol 259 (8) ◽  
pp. 3825-3853 ◽  
Author(s):  
Brigita Ferčec ◽  
Viktor Levandovskyy ◽  
Valery G. Romanovski ◽  
Douglas S. Shafer
Keyword(s):  
Polynomial Systems ◽  
Critical Periods ◽  
Bifurcation Of Critical Periods
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