scholarly journals Uniqueness and hyperbolicity of limit cycles for autonomous planar systems with zero diagonal coefficient

2017 ◽  
Vol 67 ◽  
pp. 75-80 ◽  
Author(s):  
Zhang Daoxiang ◽  
Ping Yan
Keyword(s):  
Limit Cycles ◽  
Planar Systems

MONODROMY AND STABILITY FOR NILPOTENT CRITICAL POINTS

2005 ◽  
Vol 15 (04) ◽  
pp. 1253-1265 ◽  
Author(s):  
M. J. ÁLVAREZ ◽  
A. GASULL
Keyword(s):  
Critical Points ◽  
Limit Cycles ◽  
Stability Problem ◽  
Short Proof ◽  
Sufficient Conditions ◽  
Planar Systems ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Lyapunov Constants

We give a new and short proof of the characterization of monodromic nilpotent critical points. We also calculate the first generalized Lyapunov constants in order to solve the stability problem. We apply the results to several families of planar systems obtaining necessary and sufficient conditions for having a center. Our method also allows us to generate limit cycles from the origin.

On the limit cycles for a class of planar systems

1995 ◽  
Vol 24 (4) ◽  
pp. 605-614 ◽  
Author(s):  
Zheng Zuo-Huan
Keyword(s):  
Limit Cycles ◽  
Planar Systems

DEGENERATE HOPF BIFURCATION IN NONSMOOTH PLANAR SYSTEMS

2012 ◽  
Vol 22 (03) ◽  
pp. 1250057 ◽  
Author(s):  
FENG LIANG ◽  
MAOAN HAN
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Upper Bounds ◽  
Weak Focus ◽  
Liénard Systems ◽  
General Systems ◽  
Planar Systems

In this paper, we mainly discuss Hopf bifurcation for planar nonsmooth general systems and Liénard systems with foci of parabolic–parabolic (PP) or focus–parabolic (FP) type. For the bifurcation near a focus, when the focus is kept fixed under perturbations we prove that there are at most k limit cycles which can be produced from an elementary weak focus of order 2k + 2 ( resp. k + 1)(k ≥ 1) if the focus is of PP (resp. FP) type, and we present the conditions to ensure these upper bounds are achievable. For the bifurcation near a center, the Hopf cyclicicy is studied for these systems. Some interesting applications are presented.

New Double Bifurcation of Nilpotent Focus

2021 ◽  
Vol 31 (04) ◽  
pp. 2150053
Author(s):  
Feng Li ◽  
Hongwei Li ◽  
Yuanyuan Liu
Keyword(s):  
Hopf Bifurcation ◽  
Singular Point ◽  
Limit Cycles ◽  
Second Order ◽  
Singular Points ◽  
Weak Focus ◽  
Bifurcation Phenomenon ◽  
Planar Systems ◽  
Nilpotent Singular Point

In this paper, a new bifurcation phenomenon of nilpotent singular point is analyzed. A nilpotent focus or center of the planar systems with 3-multiplicity can be broken into two complex singular points and a second order elementary weak focus. Then, two more limit cycles enclosing the second order elementary weak focus can bifurcate through the multiple Hopf bifurcation.

Small Amplitude Limit Cycles for Cubic Systems

1993 ◽  
Vol 36 (1) ◽  
pp. 54-63 ◽  
Author(s):  
V. Guíñez ◽  
E. Sáez ◽  
I. Szántó
Keyword(s):  
Periodic Orbit ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Small Amplitude ◽  
The Other ◽  
Simultaneous Generation ◽  
The Family ◽  
Planar Systems ◽  
Image Position ◽  
Cubic Systems

AbstractIn this article we study the simultaneous generation of limit cycles out of singular points and infinity for the family of cubic planar systemsWith a suitable choice of parameters, the origin and four other singularities are foci and infinity is a periodic orbit. We prove that it is possible to obtain the following configuration of limit cycles: two small amplitude limit cycles out of the origin, a small amplitude limit cycle out of each of the other four foci, and a large amplitude limit cycle out of infinity. We also obtain other configurations with fewer limit cycles.

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