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 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.

BIFURCATION OF LIMIT CYCLES BY PERTURBING PIECEWISE HAMILTONIAN SYSTEMS

2010 ◽  
Vol 20 (05) ◽  
pp. 1379-1390 ◽  
Author(s):  
XIA LIU ◽  
MAOAN HAN
Keyword(s):  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Melnikov Function ◽  
Unperturbed System ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
General Perturbation ◽  
Smooth System

In this paper, the general perturbation of piecewise Hamiltonian systems on the plane is considered. When the unperturbed system has a family of periodic orbits, similar to the perturbations of smooth system, an expression of the first order Melnikov function is derived, which can be used to study the number of limit cycles bifurcated from the periodic orbits. As applications, the number of bifurcated limit cycles of several concrete piecewise systems are presented.

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 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.

Bifurcation of Periodic Orbits Crossing Switching Manifolds Multiple Times in Planar Piecewise Smooth Systems

2019 ◽  
Vol 29 (12) ◽  
pp. 1950160
Author(s):  
Zhihui Fan ◽  
Zhengdong Du
Keyword(s):  
Limit Cycle ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Sufficient Conditions ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Piecewise Smooth Systems ◽  
Smooth Curves ◽  
Switching Curve ◽  
Smooth System

In this paper, we discuss the bifurcation of periodic orbits in planar piecewise smooth systems with discontinuities on finitely many smooth curves intersecting at the origin. We assume that the unperturbed system has either a limit cycle or a periodic annulus such that the limit cycle or each periodic orbit in the periodic annulus crosses every switching curve transversally multiple times. When the unperturbed system has a limit cycle, we give the conditions for its stability and persistence. When the unperturbed system has a periodic annulus, we obtain the expression of the first order Melnikov function and establish sufficient conditions under which limit cycles can bifurcate from the annulus. As an example, we construct a concrete nonlinear planar piecewise smooth system with three zones with 11 limit cycles bifurcated from the periodic annulus.

scholarly journals Limit Cycles Bifurcating from the Periodic Orbits of a Discontinuous Piecewise Linear Differentiable Center with Two Zones

2015 ◽  
Vol 25 (11) ◽  
pp. 1550144 ◽  
Author(s):  
Jaume Llibre ◽  
Douglas D. Novaes ◽  
Marco A. Teixeira
Keyword(s):  
Equilibrium Point ◽  
Periodic Solutions ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Piecewise Linear ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Straight Line ◽  
Linear Differential

We study a class of discontinuous piecewise linear differential systems with two zones separated by the straight line x = 0. In x > 0, we have a linear saddle with its equilibrium point living in x > 0, and in x < 0 we have a linear differential center. Let p be the equilibrium point of this linear center, when p lives in x < 0, we say that it is real, and when p lives in x > 0 we say that it is virtual. We assume that this discontinuous piecewise linear differential system formed by the center and the saddle has a center q surrounded by periodic orbits ending in a homoclinic orbit of the saddle, independent if p is real, virtual or p is in x = 0. Note that q = p if p is real or p is in x = 0. We perturb these three classes of systems, according to the position of p, inside the class of all discontinuous piecewise linear differential systems with two zones separated by x = 0. Let N be the maximum number of limit cycles which can bifurcate from the periodic solutions of the center q with these perturbations. Our main results show that N = 2 when p is on x = 0, and N ≥ 2 when p is a real or virtual center. Furthermore, when p is a real center we found an example satisfying N ≥ 3.

scholarly journals Bifurcation of Limit Cycles of a Class of Piecewise Linear Differential Systems in with Three Zones

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Yanyan Cheng
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Averaging Method ◽  
Piecewise Linear ◽  
Dimensional System ◽  
Bifurcation Of Limit Cycles ◽  
Linear Differential Systems ◽  
Main Technique ◽  
Perturbed Systems ◽  
Linear Differential

We study the bifurcation of limit cycles from periodic orbits of a four-dimensional system when the perturbation is piecewise linear with two switching boundaries. Our main result shows that when the parameter is sufficiently small at most, six limit cycles can bifurcate from periodic orbits in a class of asymmetric piecewise linear perturbed systems, and, at most, three limit cycles can bifurcate from periodic orbits in another class of asymmetric piecewise linear perturbed systems. Moreover, there are perturbed systems having six limit cycles. The main technique is the averaging method.

scholarly journals Bifurcation of Limit Cycles by Perturbing a Piecewise Linear Hamiltonian System

2013 ◽  
Vol 2013 ◽  
pp. 1-19 ◽  
Author(s):  
Yanqin Xiong ◽  
Maoan Han
Keyword(s):  
Hamiltonian System ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Melnikov Function ◽  
Phase Portraits ◽  
Unperturbed System ◽  
Linear Hamiltonian System ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Global Center

This paper concerns limit cycle bifurcations by perturbing a piecewise linear Hamiltonian system. We first obtain all phase portraits of the unperturbed system having at least one family of periodic orbits. By using the first-order Melnikov function of the piecewise near-Hamiltonian system, we investigate the maximal number of limit cycles that bifurcate from a global center up to first order ofε.

Subharmonic Orbits of Rectangular Thin Plate With Parametrically and Externally Excitations

2012 ◽  
Author(s):  
Wei Zhang ◽  
Min Sun ◽  
Qian Wang ◽  
Jianen Chen
Keyword(s):  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Thin Plate ◽  
Plate Theory ◽  
Nonlinear Dynamical System ◽  
Sufficient Conditions ◽  
Type Equation ◽  
Unperturbed System ◽  
Rectangular Thin Plate ◽  
Subharmonic Orbits

In this paper, the bifurcations of subharmonic orbits for a four-dimensional rectangular thin plate with parametrically and externally excitations is considered for the first time. The formulas of the rectangular thin plate are derived by using the von Karman-type equation, the Reddy’s third-order shear deformation plate theory and the Galerkin’s approach. The unperturbed system is composed of two independent planar Hamiltonian systems such that the unperturbed system has a family of periodic orbits. The problem addressed here is the determination of sufficient conditions for some of the periodic orbits to generate subharmonic orbits after periodic perturbations. Thus, based on periodic transformations and Poincaré map the subharmonic Melnikov method is improved to enable us to analyze directly the non-autonomous nonlinear dynamical system, which is applied to the non-autonomous governing equations of motion for the parametrically and externally excited rectangular thin plate. The results obtained here indicate that subharmonic motions can occur in the rectangular thin plate. The method succeeds in establishing the existence of subharmonics in perturbed Hamiltonian systems as well as in discussing their bifurcations. Numerical simulation is also employed to find the subharmonic motions of the parametrically and externally excited rectangular thin plate.

LIMIT CYCLES FOR A CLASS OF QUINTIC NEAR-HAMILTONIAN SYSTEMS NEAR A NILPOTENT CENTER

2009 ◽  
Vol 19 (06) ◽  
pp. 2107-2113 ◽  
Author(s):  
JIAO JIANG ◽  
JIZHOU ZHANG ◽  
MAOAN HAN
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Homoclinic Bifurcation ◽  
Unperturbed System ◽  
Bifurcation Method ◽  
Simple Cubic ◽  
Stability Change ◽  
Centrally Symmetric ◽  
Nilpotent Center

In this paper we deal with a centrally symmetric quintic near-Hamiltonian system whose unperturbed system is a simple cubic Hamiltonian system having a nilpotent center. We prove that the system can have ten limit cycles by using a homoclinic bifurcation method based on stability-change.

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