scholarly journals Limit Cycles in Planar Piecewise Linear Hamiltonian Systems with Three Zones Without Equilibrium Points

2020 ◽  
Vol 30 (11) ◽  
pp. 2050157
Author(s):  
Alexander Fernandes da Fonseca ◽  
Jaume Llibre ◽  
Luis Fernando Mello
Keyword(s):  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Linear Hamiltonian Systems

We study the existence of limit cycles in planar piecewise linear Hamiltonian systems with three zones without equilibrium points. In this scenario, we have shown that such systems have at most one crossing limit cycle.

scholarly journals Crossing Limit Cycles of Planar Piecewise Linear Hamiltonian Systems without Equilibrium Points

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 755
Author(s):  
Rebiha Benterki ◽  
Jaume LLibre
Keyword(s):  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Upper Bounds ◽  
Linear Hamiltonian Systems ◽  
Straight Lines

In this paper, we study the existence of limit cycles of planar piecewise linear Hamiltonian systems without equilibrium points. Firstly, we prove that if these systems are separated by a parabola, they can have at most two crossing limit cycles, and if they are separated by a hyperbola or an ellipse, they can have at most three crossing limit cycles. Additionally, we prove that these upper bounds are reached. Secondly, we show that there is an example of two crossing limit cycles when these systems have four zones separated by three straight lines.

The Maximal Number of Limit Cycles in Perturbations of Piecewise Linear Hamiltonian Systems with Two Saddles

2017 ◽  
Vol 27 (08) ◽  
pp. 1750126 ◽  
Author(s):  
Yanqin Xiong ◽  
Maoan Han ◽  
Valery G. Romanovski
Keyword(s):  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Unperturbed System ◽  
Linear Hamiltonian Systems

We study the maximal number of limit cycles of piecewise linear near-Hamiltonian systems, whose unperturbed system has two saddles. Using the method developed in [Xiong, 2015], we prove that two limit cycles can be bifurcated from the periodic orbits.

scholarly journals Limit cycles of planar discontinuous piecewise linear Hamiltonian systems without equilibria separated by reducible cubics

2021 ◽  
pp. 1-38
Author(s):  
Rebiha Benterki ◽  
Jeidy Jimenez ◽  
Jaume Llibre
Keyword(s):  
Hamiltonian Systems ◽  
Differential System ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Main Role ◽  
Differential Systems ◽  
Cubic Curve ◽  
Straight Line ◽  
Linear Hamiltonian Systems ◽  
Linear Differential

Due to their applications to many physical phenomena during these last decades the interest for studying the discontinuous piecewise differential systems has increased strongly. The limit cycles play a main role in the study of any planar differential system, but to determine the maximum number of limits cycles that a class of planar differential systems can have is one of the main problems in the qualitative theory of the planar differential systems. Thus in general to provide a sharp upper bound for the number of crossing limit cycles that a given class of piecewise linear differential system can have is a very difficult problem. In this paper we characterize the existence and the number of limit cycles for the piecewise linear differential systems formed by linear Hamiltonian systems without equilibria and separated by a reducible cubic curve, formed either by an ellipse and a straight line, or by a parabola and a straight line parallel to the tangent at the vertex of the parabola. Hence we have solved the extended 16th Hilbert problem to this class of piecewise differential systems.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

ASYMPTOTIC EXPANSIONS OF MELNIKOV FUNCTIONS AND LIMIT CYCLE BIFURCATIONS

2012 ◽  
Vol 22 (12) ◽  
pp. 1250296 ◽  
Author(s):  
MAOAN HAN
Keyword(s):  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Asymptotic Expansions ◽  
Melnikov Function ◽  
Heteroclinic Loop ◽  
First Order ◽  
Melnikov Functions ◽  
Limit Cycle Bifurcation ◽  
Nilpotent Center

In the study of the perturbation of Hamiltonian systems, the first order Melnikov functions play an important role. By finding its zeros, we can find limit cycles. By analyzing its analytical property, we can find its zeros. The main purpose of this article is to summarize some methods to find its zeros near a Hamiltonian value corresponding to an elementary center, nilpotent center or a homoclinic or heteroclinic loop with hyperbolic saddles or nilpotent critical points through the asymptotic expansions of the Melnikov function at these values. We present a series of results on the limit cycle bifurcation by using the first coefficients of the asymptotic expansions.

Limit Cycles in a Class of Quartic Kolmogorov Model with Three Positive Equilibrium Points

2015 ◽  
Vol 25 (06) ◽  
pp. 1550080 ◽  
Author(s):  
Chaoxiong Du ◽  
Yirong Liu ◽  
Qi Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Equilibrium Points ◽  
Positive Equilibrium ◽  
Bifurcation Problem ◽  
Kolmogorov Model ◽  
Limit Cycle Bifurcation ◽  
Perturbed Model

Limit cycle bifurcation problem of Kolmogorov model is interesting and significant both in theory and applications. In this paper, we will focus on investigating limit cycles for a class of quartic Kolmogorov model with three positive equilibrium points. Perturbed model can bifurcate three small limit cycles near (1, 2) or (2, 1) under a certain condition and can bifurcate one limit cycle near (1, 1). In addition, we have given some examples of simultaneous Hopf bifurcation and the structure of limit cycles bifurcated from three positive equilibrium points. The limit cycle bifurcation problem for Kolmogorov model with several positive equilibrium points are less seen in published references. Our result is good and interesting.

Limit Cycle Bifurcations from an Order-3 Nilpotent Center of Cubic Hamiltonian Systems Perturbed by Cubic Polynomials

2020 ◽  
Vol 30 (09) ◽  
pp. 2050126
Author(s):  
Li Zhang ◽  
Chenchen Wang ◽  
Zhaoping Hu
Keyword(s):  
Hamiltonian System ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Melnikov Function ◽  
First Order ◽  
Cubic Polynomials ◽  
Nilpotent Center

From [Han et al., 2009a] we know that the highest order of the nilpotent center of cubic Hamiltonian system is [Formula: see text]. In this paper, perturbing the Hamiltonian system which has a nilpotent center of order [Formula: see text] at the origin by cubic polynomials, we study the number of limit cycles of the corresponding cubic near-Hamiltonian systems near the origin. We prove that we can find seven and at most seven limit cycles near the origin by the first-order Melnikov function.

THE SAME DISTRIBUTION OF LIMIT CYCLES IN FIVE PERTURBED CUBIC HAMILTONIAN SYSTEMS

2003 ◽  
Vol 13 (01) ◽  
pp. 243-249 ◽  
Author(s):  
ZHENGRONG LIU ◽  
ZHIYAN YANG ◽  
TAO JIANG
Keyword(s):  
Qualitative Analysis ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Numerical Exploration ◽  
Perturbed Systems

Using the method of qualitative analysis we show that five perturbed cubic Hamiltonian systems have the same distribution of limit cycles and have 11 limit cycles for some parameters. The accurate location of each limit cycle is given by numerical exploration. In other words, we demonstrate the existence of 11 limit cycles and their distribution in five perturbed systems in two ways, the results obtained from both ways are the same.

Limit Cycle Bifurcations by Perturbing a Hamiltonian System with a 3-Polycycle Having a Cusp of Order One or Two

2018 ◽  
Vol 28 (03) ◽  
pp. 1850038
Author(s):  
Marzieh Mousavi ◽  
Hamid R. Z. Zangeneh
Keyword(s):  
Asymptotic Expansion ◽  
Hamiltonian System ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Perturbed Systems

In this paper, we study the asymptotic expansion of the first order Melnikov function near a 3-polycycle connecting a cusp (of order one or two) to two hyperbolic saddles for a near-Hamiltonian system in the plane. The formulas for the first coefficients of the expansion are given as well as the method of bifurcation of limit cycles. Then we use the results to study two Hamiltonian systems with this 3-polycycle and determine the number and distribution of limit cycles that can bifurcate from the perturbed systems. Moreover, a sharp upper bound for the number of limit cycles bifurcated from the whole periodic annulus is found when there is a cusp of order one.

A BIPARAMETRIC BIFURCATION IN 3D CONTINUOUS PIECEWISE LINEAR SYSTEMS WITH TWO ZONES: APPLICATION TO CHUA'S CIRCUIT

2007 ◽  
Vol 17 (02) ◽  
pp. 445-457 ◽  
Author(s):  
E. FREIRE ◽  
E. PONCE ◽  
J. ROS
Keyword(s):  
Hopf Bifurcation ◽  
Linear Systems ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Continuous Systems ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Piecewise Linear Systems ◽  
Limit Cycle Bifurcation

In this paper, a possible degeneration of the focus-center-limit cycle bifurcation for piecewise smooth continuous systems is analyzed. The case of continuous piecewise linear systems with two zones is considered, and the coexistence of two limit cycles for certain values of parameters is justified. Finally, the Chua's circuit is shown to exhibit the analyzed bifurcation. The obtained bifurcation set in the parameter plane is similar to the degenerate Hopf bifurcation for differentiable systems.

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