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.

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.

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.

Analysis of the Dynamics of Piecewise Linear Memristors

2016 ◽  
Vol 26 (13) ◽  
pp. 1650217 ◽  
Author(s):  
Fangfang Jiang ◽  
Zhicheng Ji ◽  
Qing-Guo Wang ◽  
Jitao Sun
Keyword(s):  
Piecewise Linear ◽  
Equilibrium Points ◽  
Characteristic Curve ◽  
Piecewise Linear Function ◽  
Excited Oscillation ◽  
Exciting Source ◽  
Memristive Circuits ◽  
The Stability ◽  
Straight Lines ◽  
The Mathematical Model

In this paper, we consider a class of flux controlled memristive circuits with a piecewise linear memristor (i.e. the characteristic curve of the memristor is given by a piecewise linear function). The mathematical model is described by a discontinuous planar piecewise smooth differential system, which is defined on three zones separated by two parallel straight lines [Formula: see text] (called as discontinuity lines in discontinuous differential systems). We first investigate the stability of equilibrium points and the existence and uniqueness of a crossing limit cycle for the memristor-based circuit under self-excited oscillation. We then analyze the existence of periodic orbits of forced nonlinear oscillation for the memristive circuit with an external exciting source. Finally, we give numerical simulations to show good matches between our theoretical and simulation results.

The Cyclicity of Period Annulus of Degenerate Quadratic Hamiltonian Systems with Polycycles S(2) or S(3) Under Perturbations of Piecewise Smooth Polynomials

2020 ◽  
Vol 30 (15) ◽  
pp. 2050230
Author(s):  
Jiaxin Wang ◽  
Liqin Zhao
Keyword(s):  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bounds ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Period Annulus ◽  
Quadratic Hamiltonian

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the bifurcation of limit cycles for degenerate quadratic Hamilton systems with polycycles [Formula: see text] or [Formula: see text] under the perturbations of piecewise smooth polynomials with degree [Formula: see text]. Roughly speaking, for [Formula: see text], a polycycle [Formula: see text] is cyclically ordered collection of [Formula: see text] saddles together with orbits connecting them in specified order. The discontinuity is on the line [Formula: see text]. If the first order Melnikov function is not equal to zero identically, it is proved that the upper bounds of the number of limit cycles bifurcating from each of the period annuli with the boundary [Formula: see text] and [Formula: see text] are respectively [Formula: see text] and [Formula: see text] (taking into account the multiplicity).

scholarly journals LOWER BOUNDS FOR THE MAXIMUM NUMBER OF LIMIT CYCLES OF DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH A STRAIGHT LINE OF SEPARATION

2013 ◽  
Vol 23 (04) ◽  
pp. 1350066 ◽  
Author(s):  
J. LLIBRE ◽  
M. A. TEIXEIRA ◽  
J. TORREGROSA
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Unique Equilibrium ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Straight Line ◽  
Linear Differential

In this paper, we provide a lower bound for the maximum number of limit cycles of planar discontinuous piecewise linear differential systems defined in two half-planes separated by a straight line. Here, we only consider nonsliding limit cycles. For those systems, the interior of any limit cycle only contains a unique equilibrium point or a unique sliding segment. Moreover, the linear differential systems that we consider in every half-plane can have either a focus (F), or a node (N), or a saddle (S), these equilibrium points can be real or virtual. Then, we can consider six kinds of planar discontinuous piecewise linear differential systems: FF, FN, FS, NN, NS, SS. We provide for each of these types of discontinuous differential systems examples with two limit cycles.

The Extended 16th Hilbert Problem for Discontinuous Piecewise Linear Centers Separated by a Nonregular Line

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Marina Esteban ◽  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Limit Cycles ◽  
Hilbert Problem ◽  
Piecewise Linear ◽  
Upper Bounds ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Straight Line ◽  
Linear Differential ◽  
16Th Hilbert Problem ◽  
To Receive

The study of the piecewise linear differential systems goes back to Andronov, Vitt and Khaikin in 1920’s, and nowadays such systems still continue to receive the attention of many researchers mainly due to their applications. We study the discontinuous piecewise differential systems formed by two linear centers separated by a nonregular straight line. We provide upper bounds for the maximum number of limit cycles that these discontinuous piecewise differential systems can exhibit and we show that these upper bounds are reached. Hence, we solve the extended 16th Hilbert problem for this class of piecewise differential systems.

scholarly journals Limit cycles from a monodromic infinity in planar piecewise linear systems

2021 ◽  
Vol 496 (2) ◽  
pp. 124818
Author(s):  
Emilio Freire ◽  
Enrique Ponce ◽  
Joan Torregrosa ◽  
Francisco Torres
Keyword(s):  
Linear Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Piecewise Linear Systems
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