ON THE SLIDING BIFURCATION OF A CLASS OF PLANAR FILIPPOV SYSTEMS

2013 ◽  
Vol 23 (03) ◽  
pp. 1350040 ◽  
Author(s):  
DINGHENG PI ◽  
JIANG YU ◽  
XIANG ZHANG
Keyword(s):  
Limit Cycles ◽  
Homoclinic Bifurcation ◽  
Qualitative Theory ◽  
Piecewise Smooth ◽  
The Other ◽  
Heteroclinic Cycle ◽  
Differential Systems ◽  
Sliding Bifurcation ◽  
Bifurcation Phenomena ◽  
Limit Cycle Bifurcation

In this paper, we study the sliding bifurcation phenomena of a class of planar piecewise smooth differential systems consisting of linear and quadratic subsystems. Using the differential inclusion and the qualitative theory of ordinary differential equations, we find some new interesting phenomena appearing in the piecewise smooth differential systems. In brief, we prove that the system may have sliding homoclinic bifurcation, sliding cycle bifurcation, semistable limit cycle bifurcation and heteroclinic cycle bifurcation. In addition, the mentioned systems can have at most two limit cycles, and the maximal number of limit cycles can be realized and central nested with one bifurcated from the sliding–crossing bifurcation of a sliding cycle and the other from the saddle homoclinic bifurcation. These two limit cycles collide and then both disappear. This novel scenario is verified by our systems.

Bifurcation Analysis of Planar Piecewise Linear System with Different Dynamics

2016 ◽  
Vol 26 (11) ◽  
pp. 1650185
Author(s):  
Xiaoshi Guo ◽  
Dingheng Pi ◽  
Zhensheng Gao
Keyword(s):  
Linear System ◽  
Periodic Orbit ◽  
Necessary Conditions ◽  
Piecewise Linear ◽  
Homoclinic Bifurcation ◽  
Sliding Bifurcation ◽  
Piecewise Linear System ◽  
Bifurcation Phenomena ◽  
Sufficient And Necessary Conditions ◽  
Limit Cycle Bifurcation

In this paper, we investigate the bifurcation phenomena of a planar piecewise linear system. This piecewise linear system comprises two linear subsystems. The two linear subsystems have different types of dynamics. One subsystem has node or saddle dynamic and the other has focus dynamic. Some sufficient and necessary conditions for the existence of periodic orbit are given by studying the properties of Poincaré maps. Our results show that two crossing periodic orbits can bifurcate from this piecewise linear system. Moreover, we establish some sufficient and necessary conditions for the existence of sliding periodic orbit, crossing–sliding periodic orbit and sliding homoclinic orbit passing through a pseudo saddle and so on. We find that this piecewise system can appear multiply as two limit cycle bifurcation, buckling bifurcation, critical crossing cycle bifurcation, sliding homoclinic bifurcation, pseudo homoclinic bifurcation and so on. To our knowledge, sliding bifurcation phenomena are usually ignored when people study piecewise linear systems.

scholarly journals The acyclicity of a quadratic differential system

2021 ◽  
pp. 32-41
Author(s):  
Адам Дамирович Ушхо ◽  
Вячеслав Бесланович Тлячев ◽  
Дамир Салихович Ушхо
Keyword(s):  
Differential Equations ◽  
Differential System ◽  
Limit Cycles ◽  
Quadratic Differential ◽  
Qualitative Theory ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Quadratic Differential System ◽  
Straight Line

Дан краткий обзор некоторых основных публикаций, посвященных исследованию вопроса о предельных циклах и сепаратрисах квадратичных дифференциальных систем. Рассмотрено наличие замкнутых траекторий для определенного класса автономных квадратичных систем на плоскости. Доказательство основано на применении теории прямых изоклин, признаков Дюлака и Бендиксона качественной теории дифференциальных уравнений. Предложенное доказательство покрывает результаты известной работы Л.А. Черкаса и Л.С. Жилевич. We now give a brief overview of some of the main publications devoted to the study of the question of limit cycles and separatrices of quadratic differential systems. In this paper, we consider the existence of closed trajectories for a certain class of autonomous quadratic systems on the plane. The proof is based on the application of the theory of straight line isoclines, Dulac and Bendixon criteria of the qualitative theory of differential equations. The proposed proof covers the results of the well-known work of L.A. Cherkas and L.S. Zhilevich.

Limit Cycles Appearing From Piecewise Smooth Perturbations to a Reversible Nonlinear Center

2021 ◽  
Vol 31 (16) ◽  
Author(s):  
Dan Sun ◽  
Linping Peng
Keyword(s):  
Limit Cycles ◽  
Upper Bounds ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Complex Method ◽  
Discontinuous Systems ◽  
Lower And Upper Bounds ◽  
Argument Principle ◽  
Period Annulus ◽  
Limit Cycle Bifurcation

This paper deals with the limit cycle bifurcation from a reversible differential center of degree [Formula: see text] due to small piecewise smooth homogeneous polynomial perturbations. By using the averaging theory for discontinuous systems and the complex method based on the Argument Principle, we obtain lower and upper bounds for the maximum number of limit cycles bifurcating from the period annulus around the center of the unperturbed system.

Bifurcation Analysis in Planar Quadratic Differential Systems with Boundary

2020 ◽  
Vol 30 (07) ◽  
pp. 2030017
Author(s):  
Jocelyn A. Castro ◽  
Fernando Verduzco
Keyword(s):  
Quadratic Differential ◽  
Sufficient Conditions ◽  
The Other ◽  
Tangency Point ◽  
Differential Systems ◽  
Quadratic Differential System ◽  
Bifurcation Phenomena ◽  
Straight Line ◽  
Boundary Equilibrium ◽  
Parameterized Family

Given a planar quadratic differential system delimited by a straight line, we are interested in studying the bifurcation phenomena that can arise when the position on the boundary of two tangency points are considered as parameters of bifurcation. First, under generic conditions, we find a two-parametric family of quadratic differential systems with at least one tangency point. After that, we find a normal form for this parameterized family. Next, we study two subfamilies, one of them characterized by the existence of two fold points of different nature, and the other one, characterized by the existence of one fold point and one boundary equilibrium point. For the first family, we find sufficient conditions for the existence of stationary bifurcations: saddle-node, transcritical and pitchfork, while for the second family, the existence of the called transcritical Bogdanov–Takens bifurcation is proved. Finally, the results are illustrated with two examples.

Stability and Limit Cycle Bifurcation for Two Kinds of Generalized Double Homoclinic Loops in Planar Piecewise Smooth Systems

2014 ◽  
Vol 24 (12) ◽  
pp. 1450153
Author(s):  
Feng Liang ◽  
Maoan Han
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Upper Bound ◽  
Piecewise Smooth ◽  
Piecewise Smooth Systems ◽  
Homoclinic Loop ◽  
Limit Cycle Bifurcation

In this paper, we present two kinds of generalized double homoclinic loops in planar piecewise smooth systems. For their stability a criterion is provided. Under nondegenerate conditions, we prove that for each case there are at most five limit cycles which can be bifurcated from the generalized double homoclinic loop. Especially, we construct two concrete systems to show that the upper bound can be achieved in both cases.

scholarly journals Limit Cycles of a Class of Piecewise Smooth Liénard Systems

2016 ◽  
Vol 26 (01) ◽  
pp. 1650009 ◽  
Author(s):  
Lijuan Sheng
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Function Theory ◽  
Polynomial Systems ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Piecewise Polynomial ◽  
Multiple Parameters ◽  
Liénard Systems ◽  
Limit Cycle Bifurcation

In this paper, we study the problem of limit cycle bifurcation in two piecewise polynomial systems of Liénard type with multiple parameters. Based on the developed Melnikov function theory, we obtain the maximum number of limit cycles of these two systems.

On the Limit Cycles Bifurcating from Piecewise Quasi-Homogeneous Differential Center

2016 ◽  
Vol 26 (07) ◽  
pp. 1650116 ◽  
Author(s):  
Shimin Li ◽  
Kuilin Wu
Keyword(s):  
Lower Bound ◽  
Limit Cycles ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Differential Systems ◽  
First Order

In this paper, a class of piecewise smooth quasi-homogeneous differential systems are considered. Using the first order Melnikov function derived in [Liu & Han, 2010], we obtain a lower bound of the maximum number of limit cycles which bifurcate from the periodic annulus of the center under polynomial perturbation. The results reveal that piecewise smooth quasi-homogeneous differential systems can bifurcate more limit cycles than the smooth systems.

Sliding Bifurcation in Planar Piecewise Smooth Systems with Two Zones Separated by an Ellipse

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Fang Wu ◽  
Lihong Huang ◽  
Jiafu Wang
Keyword(s):  
Bifurcation Diagram ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Sufficient Conditions ◽  
Piecewise Smooth ◽  
Piecewise Smooth System ◽  
Sliding Bifurcation ◽  
Switching Curve ◽  
Necessary And Sufficient ◽  
Smooth System

The objective of this paper is to study the sliding bifurcation in a planar piecewise smooth system with an elliptic switching curve. Some new phenomena are observed, such as a crossing limit cycle containing four intersections with the switching curve, sliding cycles having four sliding segments, and sliding cycles consisting of the entire switching curve. Firstly, we investigate the bifurcation of sliding cycle from a sliding heteroclinic connection to two cusps and show the appearance of one sliding cycle with two folds. To plot the bifurcation diagram, a planar piecewise linear system with two zones separated by an ellipse are considered. Moreover, we study in more detail the unfolding of a sliding cycle connecting four cusps by exhibiting its complete bifurcation diagram. More precisely, we explore the necessary and sufficient conditions for the existence of limit cycles and derive the concrete bifurcation curves. Additionally, a simple piecewise smooth system with nonlinear subsystems is studied, which shows the possibility of the existence of two nested limit cycles. Finally, numerical simulations are given to confirm the theoretical analysis.

Limit Cycles for a Class of Piecewise Smooth Quadratic Differential Systems with Multiple Parameters

2016 ◽  
Vol 26 (10) ◽  
pp. 1650171 ◽  
Author(s):  
Xuekang Bo ◽  
Yun Tian
Keyword(s):  
Limit Cycles ◽  
Quadratic Differential ◽  
Small Amplitude ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Isochronous Center ◽  
Differential Systems ◽  
First Order ◽  
Multiple Parameters ◽  
Perturbation Parameters

This paper considers a class of quadratic differential systems with an isochronous center under small piecewise smooth perturbations. Two perturbation parameters at different scales are included in the system. By using the first order Melnikov function, we obtain some new results on the number of small-amplitude limit cycles bifurcating around an isochronous center.

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