Switching and Sliding Control of Limit Cycles in Planar Systems – Nonsmooth Bifurcation Techniques

2006 ◽  
pp. 67-87
Author(s):  
Fabiola Angulo ◽  
Mario Bernardo ◽  
Gerard Olivar
Keyword(s):  
Limit Cycles ◽  
Sliding Control ◽  
Planar Systems ◽  
Nonsmooth Bifurcation

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.

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