scholarly journals Critical Periods of Perturbations of Reversible Rigidly Isochronous Centers

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Jiamei Zhou ◽  
Na Li ◽  
Maoan Han
Keyword(s):  
Polynomial System ◽  
Polynomial Systems ◽  
New Method ◽  
Critical Periods ◽  
Period Function ◽  
Number Of Zeros ◽  
Isochronous Centers ◽  
Bifurcation Of Critical Periods

We study the problem of bifurcation of critical periods of a time-reversible polynomial system of degreen. We first present a new method to find the number of zeros of the period function. Then applying our results, we study the number of critical periods for some polynomial systems and obtain new results.

Bifurcation of Critical Periods from a Quartic Isochronous Center

2014 ◽  
Vol 24 (06) ◽  
pp. 1450089 ◽  
Author(s):  
Linping Peng ◽  
Zhaosheng Feng
Keyword(s):  
Periodic Orbits ◽  
Upper Bound ◽  
Unperturbed System ◽  
Isochronous Center ◽  
Critical Periods ◽  
Period Function ◽  
Number Of Zeros ◽  
Perturbed Systems ◽  
Bifurcation Of Critical Periods

This paper is focused on the bifurcation of critical periods from a quartic rigidly isochronous center under any small quartic homogeneous perturbations. By studying the number of zeros of the first several terms in the expansion of the period function in ε, it shows that under any small quartic homogeneous perturbations, up to orders 1 and 2 in ε, there are at most two critical periods bifurcating from the periodic orbits of the unperturbed system respectively, and the upper bound can be reached. Up to order 3 in ε, there are at most six critical periods from the periodic orbits of the unperturbed system. Moreover, we consider a family of perturbed systems of this quartic rigidly isochronous center, and obtain that up to any order in ε, there are at most two critical periods bifurcating from the periodic orbits of the unperturbed one, and the upper bound is sharp.

scholarly journals Bifurcation of critical periods of polynomial systems

2015 ◽  
Vol 259 (8) ◽  
pp. 3825-3853 ◽  
Author(s):  
Brigita Ferčec ◽  
Viktor Levandovskyy ◽  
Valery G. Romanovski ◽  
Douglas S. Shafer
Keyword(s):  
Polynomial Systems ◽  
Critical Periods ◽  
Bifurcation Of Critical Periods

A Higher-Order Period Function and Its Application

2015 ◽  
Vol 25 (10) ◽  
pp. 1550140 ◽  
Author(s):  
Linping Peng ◽  
Lianghaolong Lu ◽  
Zhaosheng Feng
Keyword(s):  
Periodic Orbits ◽  
Upper Bound ◽  
Critical Period ◽  
Higher Order ◽  
Quadratic System ◽  
Critical Periods ◽  
Explicit Formulas ◽  
Period Function ◽  
Isochronous System ◽  
Isochronous Centers

This paper derives explicit formulas of the q th period bifurcation function for any perturbed isochronous system with a center, which improve and generalize the corresponding results in the literature. Based on these formulas to the perturbed quadratic and quintic rigidly isochronous centers, we prove that under any small homogeneous perturbations, for ε in any order, at most one critical period bifurcates from the periodic orbits of the unperturbed quadratic system. For ε in order of 1, 2, 3, 4 and 5, at most three critical periods bifurcate from the periodic orbits of the unperturbed quintic system. Moreover, in each case, the upper bound is sharp. Finally, a family of perturbed quintic rigidly isochronous centers is shown, which has three, for ε in any order, as the exact upper bound of the number of critical periods.

Bifurcations of Critical Periods for a Class of Quintic Liénard Equation

2020 ◽  
Vol 30 (14) ◽  
pp. 2050201
Author(s):  
Zhiheng Yu ◽  
Lingling Liu
Keyword(s):  
Phase Portraits ◽  
Liénard Equation ◽  
Local Bifurcation ◽  
Exact Order ◽  
Critical Periods ◽  
Isochronous Centers ◽  
Bifurcation Of Critical Periods ◽  
Lienard Equation

In this paper, we investigate a quintic Liénard equation which has a center at the origin. We give the conditions for the parameters for the isochronous centers and weak centers of exact order. Then, we present the global phase portraits for the system having isochronous centers. Moreover, we prove that at most four critical periods can bifurcate and show with appropriate perturbations that local bifurcation of critical periods occur from the centers.

scholarly journals Amplitude Independent Frequency Synchroniser for a Cubic Planar Polynomial System

2012 ◽  
Vol 7 (2) ◽  
Author(s):  
Islam Boussaada
Keyword(s):  
New York ◽  
Initial Conditions ◽  
Polynomial System ◽  
Polynomial Systems ◽  
Algebraic Equations ◽  
Compact Region ◽  
Synchronization Problem ◽  
The Family ◽  
Cubic System ◽  
Isochronous Centers

The problem of local linearizability of the planar linear center perturbed by cubic non- linearities in all generalities on the system parameters (14 parameters) is far from being solved. The synchronization problem (as noted in Pikovsky, A., Rosenblum, M., and Kurths, J., 2003, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, Cambridge University Press, UK, and Blekhman, I. I., 1988, Synchronisation in Science and Technology, ASME Press Translations, New York) consists in bringing appropriate modifications on a given system to obtain a desired dynamic. The desired phase portrait along this paper contains a compact region around a singular point at the origin in which lie periodic orbits with the same period (independently from the chosen initial conditions). In this paper, starting from a five parameters non isochronous Chouikha cubic system (Chouikha, A. R., 2007, “Isochronous Centers of Lienard Type Equations and Applications,” J. Math. Anal. Appl., 331, pp. 358–376) we identify all possible monomial perturbations of degree d ∈ {2, 3} insuring local linearizability of the perturbed system. The necessary conditions are obtained by the Normal Forms method. These conditions are real algebraic equations (multivariate polynomials) in the parameters of the studied ordinary differential system. The efficient algorithm FGb (J. C. Faugère, “FGb Salsa Software,” http://fgbrs.lip6.fr) for computing the Gröbner basis is used. For the family studied in this paper, an exhaustive list of possible parameters values insuring local linearizability is established. All the found cases are already known in the literature but the contexts are different since our object is the synchronisation rather than the classification. This paper can be seen as a direct continuation of several new works concerned with the hinting of cubic isochronous centers, (in particular Bardet, M., and Boussaada, I., 2011, “Compexity Reduction of C-algorithm,” App. Math. Comp., in press; Boussaada, I., Chouikha, A. R., and Strelcyn, J.-M., 2011, “Isochronicity Conditions for some Planar Polynomial Systems,” Bull. Sci. Math, 135(1), pp. 89–112; Bardet, M., Boussaada, I., Chouikha, A. R., and Strelcyn, J.-M., 2011, “Isochronicity Conditions for some Planar Polynomial Systems,” Bull. Sci. Math, 135(2), pp. 230–249; and furthermore, it can be considered as an adaptation of a qualitative theory method to a synchronization problem.

Critical Period Bifurcation by Perturbing a Reversible Rigidly Isochronous Center with Multiple Parameters

2015 ◽  
Vol 25 (05) ◽  
pp. 1550070 ◽  
Author(s):  
Na Li ◽  
Maoan Han
Keyword(s):  
Critical Period ◽  
New Method ◽  
Isochronous Center ◽  
Critical Periods ◽  
Multiple Parameters ◽  
Bifurcation Of Critical Periods

This paper focuses on bifurcation of critical periods by perturbing a rigidly isochronous center with multiple parameters. First, we give expressions of period bifurcation functions (PBF for short) in the form of integrals, and then study the first PBF T1(ρ, λ) with a new method. Compared with the result in [Liu & Han, 2014], more critical periods can be found by our method.

Number of Critical Periods for Perturbed Rigidly Isochronous Centers

2016 ◽  
Vol 26 (13) ◽  
pp. 1650220 ◽  
Author(s):  
Lianghaolong Lu ◽  
Linping Peng ◽  
Zhaosheng Feng
Keyword(s):  
Periodic Orbits ◽  
Upper Bound ◽  
Unperturbed System ◽  
Isochronous Center ◽  
Critical Periods ◽  
First Order ◽  
Isochronous Centers ◽  
Bifurcation Of Critical Periods

This paper deals with the bifurcation of critical periods from a rigidly quartic isochronous center. It shows that under any small homogeneous perturbation of degree four, up to any order in [Formula: see text], there are at most two critical periods bifurcating from the periodic orbits of the unperturbed system, and the upper bound is sharp. In addition, we further prove that under any small polynomial perturbation of degree [Formula: see text], up to the first order in [Formula: see text], there are at most [Formula: see text] critical periods bifurcating from the periodic orbits of the unperturbed quartic system.

Number of Limit Cycles from a Class of Perturbed Piecewise Polynomial Systems

2021 ◽  
Vol 31 (09) ◽  
pp. 2150123
Author(s):  
Xiaoyan Chen ◽  
Maoan Han
Keyword(s):  
Limit Cycles ◽  
Polynomial Systems ◽  
Melnikov Function ◽  
Unperturbed System ◽  
Piecewise Polynomial ◽  
First Order ◽  
Number Of Zeros ◽  
Period Annulus

In this paper, we study Poincaré bifurcation of a class of piecewise polynomial systems, whose unperturbed system has a period annulus together with two invariant lines. The main concerns are the number of zeros of the first order Melnikov function and the estimation of the number of limit cycles which bifurcate from the period annulus under piecewise polynomial perturbations of degree [Formula: see text].

BIFURCATION OF LIMIT CYCLES AND ISOCHRONOUS CENTERS FOR A QUARTIC SYSTEM

2013 ◽  
Vol 23 (10) ◽  
pp. 1350172 ◽  
Author(s):  
WENTAO HUANG ◽  
AIYONG CHEN ◽  
QIUJIN XU
Keyword(s):  
Limit Cycles ◽  
Necessary Conditions ◽  
Polynomial System ◽  
Isochronous Center ◽  
Eighth Order ◽  
Bifurcation Of Limit Cycles ◽  
Quartic Polynomial ◽  
Fine Focus ◽  
Sufficient And Necessary Conditions ◽  
Isochronous Centers

For a quartic polynomial system we investigate bifurcations of limit cycles and obtain conditions for the origin to be a center. Computing the singular point values we find also the conditions for the origin to be the eighth order fine focus. It is proven that the system can have eight small amplitude limit cycles in a neighborhood of the origin. To the best of our knowledge, this is the first example of a quartic system with eight limit cycles bifurcated from a fine focus. We also give the sufficient and necessary conditions for the origin to be an isochronous center.

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