Global Analysis on the Discontinuous Limit Case of a Smooth Oscillator

2016 ◽  
Vol 26 (04) ◽  
pp. 1650061 ◽  
Author(s):  
Hebai Chen
Keyword(s):  
Limit Cycles ◽  
Global Analysis ◽  
Sufficient Conditions ◽  
Global Bifurcation ◽  
Phase Portraits ◽  
Special Method ◽  
Global Dynamics ◽  
Limit Case ◽  
Grazing Bifurcation ◽  
Necessary And Sufficient

Global dynamics of a class of planar Filippov systems with symmetry, which is a discontinuous limit case of a smooth oscillator, is studied. Necessary and sufficient conditions for the existence and the number of limit cycles are given. It is shown that at most two limit cycles or a pair of grazing loops exist. A special method is introduced to study grazing bifurcation. The monotonicity and the [Formula: see text] smoothness of the grazing bifurcation curve are proved. All global phase portraits and a complete global bifurcation diagram are described. Finally, some numerical examples are demonstrated.

scholarly journals The existence of positive solutions for Kirchhoff-type problems via the sub-supersolution method

2018 ◽  
Vol 26 (1) ◽  
pp. 5-41 ◽  
Author(s):  
Baoqiang Yan ◽  
Donal O’Regan ◽  
Ravi P. Agarwal
Keyword(s):  
Positive Solutions ◽  
Bifurcation Theory ◽  
Sufficient Conditions ◽  
Global Bifurcation ◽  
Existence Of A Solution ◽  
Kirchhoff Type ◽  
Existence Of Positive Solutions ◽  
Global Bifurcation Theory ◽  
Necessary And Sufficient ◽  
Kirchhoff Type Problems

Abstract In this paper we discuss the existence of a solution between wellordered subsolution and supersolution of the Kirchhoff equation. Using the sub-supersolution method together with a Rabinowitz-type global bifurcation theory, we establish the existence of positive solutions for Kirchhoff-type problems when the nonlinearity is singular or sign-changing. Moreover, we obtain some necessary and sufficient conditions for the existence of positive solutions for the problem when N = 1.

Global Dynamics in the Leslie–Gower Model with the Allee Effect

2018 ◽  
Vol 28 (12) ◽  
pp. 1850151 ◽  
Author(s):  
Valery A. Gaiko ◽  
Cornelis Vuik
Keyword(s):  
Singular Point ◽  
Bifurcation Analysis ◽  
Limit Cycles ◽  
Allee Effect ◽  
Global Bifurcation ◽  
Global Dynamics ◽  
Global Bifurcations ◽  
Biomedical System

We complete the global bifurcation analysis of the Leslie–Gower system with the Allee effect which models the dynamics of the populations of predators and their prey in a given ecological or biomedical system. In particular, studying global bifurcations of limit cycles, we prove that such a system can have at most two limit cycles surrounding one singular point.

scholarly journals Global Analysis of a Liénard System with Quadratic Damping

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Feng Guo
Keyword(s):  
Necessary Conditions ◽  
Global Analysis ◽  
Global Bifurcation ◽  
Phase Portraits ◽  
Liénard System ◽  
Closed Orbits ◽  
Lienard System ◽  
Sufficient And Necessary Conditions ◽  
Global Phase Portrait ◽  
Quadratic Damping

In this paper, the global analysis of a Liénard equation with quadratic damping is studied. There are 22 different global phase portraits in the Poincaré disc. Every global phase portrait is given as well as the complete global bifurcation diagram. Firstly, the equilibria at finite and infinite of the Liénard system are discussed. The properties of the equilibria are studied. Then, the sufficient and necessary conditions of the system with closed orbits are obtained. The degenerate Bogdanov-Takens bifurcation is studied and the bifurcation diagrams of the system are given.

Global Dynamics of Memristor Oscillator

2016 ◽  
Vol 26 (12) ◽  
pp. 1650198 ◽  
Author(s):  
Hebai Chen
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Pitchfork Bifurcation ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Generalized Hopf Bifurcation ◽  
Limit Cycle Bifurcation ◽  
Sector Method ◽  
Double Limit ◽  
The Continuum

In this paper, we investigate the global dynamics of a memristor oscillator [Formula: see text] which comes from [Corinto et al., 2011], where [Formula: see text], and [Formula: see text]. Clearly, the case [Formula: see text] is trivial. So far, all results of this oscillator were given only for the case [Formula: see text], where the set of equilibria may change among a singleton, three points and a singular continuum and at most one limit cycle can arise and no limit cycles arise from the continuum. Compared with the case [Formula: see text], this oscillator displays more complicated dynamics for the case when [Formula: see text]. More clearly, one limit cycle may arise from the continuum and at most three limit cycles appear in the case of three equilibria, where generalized pitchfork bifurcation, saddle-node bifurcation, generalized Hopf bifurcation, double limit cycle bifurcation and homoclinic bifurcation may occur. Finally all global phase portraits are given for [Formula: see text] cases on the Poincaré disc, where a generalized normal sector method is applied. Moreover, our partial analytical results are demonstrated by numerical examples.

scholarly journals Multiple cycles and the Bautin bifurcation in the Goodwin model of a class struggle

2013 ◽  
Vol 18 (3) ◽  
pp. 265-274 ◽  
Author(s):  
Giovanni Bella
Keyword(s):  
Steady State ◽  
Limit Cycles ◽  
Sufficient Conditions ◽  
Class Struggle ◽  
Necessary And Sufficient Conditions ◽  
Goodwin Model ◽  
Multiple Cycles ◽  
Bautin Bifurcation ◽  
Necessary And Sufficient

The aim of this paper is to present the necessary and sufficient conditions for the emergence of a generalized Hopf (i.e., Bautin) bifurcation in the Goodwin’s model of a class struggle, and determine the parameter regions where multiple attracting and repelling limit cycles around the steady state may coexist.

LIMIT CYCLES AND CENTERS IN A CUBIC PLANAR SYSTEM

2010 ◽  
Vol 20 (12) ◽  
pp. 4127-4135 ◽  
Author(s):  
LEONID CHERKAS ◽  
VALERY G. ROMANOVSKI ◽  
YEPENG XING
Keyword(s):  
Limit Cycles ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Planar System ◽  
Large Size ◽  
Necessary And Sufficient

In this paper, we obtain the necessary and sufficient conditions for centers for a cubic planar system with Z2-symmetry studied by Yu and Han [2004]. We also give an example of such a system with 12 limit cycles of "real" (relatively large) size.

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.

scholarly journals A classification of explosions in dimension one

2009 ◽  
Vol 29 (2) ◽  
pp. 715-731 ◽  
Author(s):  
E. SANDER ◽  
J. A. YORKE
Keyword(s):  
Sufficient Conditions ◽  
Global Bifurcation ◽  
Necessary And Sufficient Conditions ◽  
Homoclinic Tangency ◽  
Discontinuous Change ◽  
One Dimensional ◽  
One Dimensional Maps ◽  
Dimension One ◽  
Necessary And Sufficient

AbstractA discontinuous change in the size of an attractor is the most easily observed type of global bifurcation. More generally, anexplosionis a discontinuous change in the set of recurrent points. An explosion often results from heteroclinic and homoclinic tangency bifurcations. We prove that, for one-dimensional maps, explosions are generically the result of either tangency or saddle-node bifurcations. Furthermore, we give necessary and sufficient conditions for generic tangency bifurcations to lead to explosions.

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