scholarly journals Bifurcation of limit cycles from a non-smooth perturbation of a two-dimensional isochronous cylinder

2016 ◽  
Vol 140 (5) ◽  
pp. 519-540
Author(s):  
C.A. Buzzi ◽  
R.D. Euzébio ◽  
A.C. Mereu
Keyword(s):  
Limit Cycles ◽  
Two Dimensional ◽  
Bifurcation Of Limit Cycles ◽  
Smooth Perturbation

scholarly journals LIMIT CYCLES OF DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS

2011 ◽  
Vol 21 (11) ◽  
pp. 3181-3194 ◽  
Author(s):  
PEDRO TONIOL CARDIN ◽  
TIAGO DE CARVALHO ◽  
JAUME LLIBRE
Keyword(s):  
Limit Cycles ◽  
Piecewise Linear ◽  
Displacement Function ◽  
Two Dimensional ◽  
Bifurcation Of Limit Cycles ◽  
Differential Systems ◽  
First Order ◽  
Linear Differential Systems ◽  
Order Expansion ◽  
Linear Differential

We study the bifurcation of limit cycles from the periodic orbits of a two-dimensional (resp. four-dimensional) linear center in ℝn perturbed inside a class of discontinuous piecewise linear differential systems. Our main result shows that at most 1 (resp. 3) limit cycle can bifurcate up to first-order expansion of the displacement function with respect to the small parameter. This upper bound is reached. For proving these results, we use the averaging theory in a form where the differentiability of the system is not needed.

scholarly journals Bifurcation of limit cycles from a two-dimensional center inside

2010 ◽  
Vol 72 (3-4) ◽  
pp. 1387-1392 ◽  
Author(s):  
Jaume Llibre ◽  
Amar Makhlouf
Keyword(s):  
Limit Cycles ◽  
Two Dimensional ◽  
Bifurcation Of Limit Cycles

scholarly journals Bifurcation of Limit Cycles and Center Conditions for Two Families of Kukles-Like Systems with Nilpotent Singularities

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Peiluan Li ◽  
Yusen Wu ◽  
Xiaoquan Ding
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Singular Points ◽  
Center Problem ◽  
Bifurcation Of Limit Cycles ◽  
Center Conditions

We solve theoretically the center problem and the cyclicity of the Hopf bifurcation for two families of Kukles-like systems with their origins being nilpotent and monodromic isolated singular points.

BIFURCATION OF LIMIT CYCLES AND ISOCHRONOUS CENTERS FOR A QUARTIC SYSTEM

2013 ◽  
Vol 23 (10) ◽  
pp. 1350172 ◽  
Author(s):  
WENTAO HUANG ◽  
AIYONG CHEN ◽  
QIUJIN XU
Keyword(s):  
Limit Cycles ◽  
Necessary Conditions ◽  
Polynomial System ◽  
Isochronous Center ◽  
Eighth Order ◽  
Bifurcation Of Limit Cycles ◽  
Quartic Polynomial ◽  
Fine Focus ◽  
Sufficient And Necessary Conditions ◽  
Isochronous Centers

For a quartic polynomial system we investigate bifurcations of limit cycles and obtain conditions for the origin to be a center. Computing the singular point values we find also the conditions for the origin to be the eighth order fine focus. It is proven that the system can have eight small amplitude limit cycles in a neighborhood of the origin. To the best of our knowledge, this is the first example of a quartic system with eight limit cycles bifurcated from a fine focus. We also give the sufficient and necessary conditions for the origin to be an isochronous center.

scholarly journals Hopf Bifurcation of Limit Cycles in Discontinuous Liénard Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Yanqin Xiong ◽  
Maoan Han
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Special Form ◽  
Bifurcation Of Limit Cycles ◽  
Liénard Systems

We consider a class of discontinuous Liénard systems and study the number of limit cycles bifurcated from the origin when parameters vary. We establish a method of studying cyclicity of the system at the origin. As an application, we discuss some discontinuous Liénard systems of special form and study the cyclicity near the origin.

scholarly journals Tiling Efflorescence of Expanding Kernels in a Fixed Periodic Array

2021 ◽  
Vol 3 (1) ◽  
pp. 13-34
Author(s):  
Robert J Marks II
Keyword(s):  
Fourier Analysis ◽  
Fixed Points ◽  
Limit Cycles ◽  
Circular Cone ◽  
Two Dimensional ◽  
Periodic Functions ◽  
Fundamental Properties ◽  
Art And Architecture ◽  
The Trinity ◽  
Func Tion

Continually expanding periodically translated kernels on the two dimensional grid can yield interesting, beau- tiful and even familiar patterns. For example, expand- ing circular pillbox shaped kernels on a hexagonal grid, adding when there is overlap, yields patterns includ- ing maximally packed circles and a triquetra-type three petal structure used to represent the trinity in Chris- tianity. Continued expansion yields the flower-of-life used extensively in art and architecture. Additional expansion yields an even more interesting emerging ef- florescence of periodic functions. Example images are given for the case of circular pillbox and circular cone shaped kernels. Using Fourier analysis, fundamental properties of these patterns are analyzed. As a func- tion of expansion, some effloresced functions asymp- totically approach fixed points or limit cycles. Most interesting is the case where the efflorescence never repeats. Video links are provided for viewing efflores- cence in real time.

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