Local Integrability and Linearizability of a $$(1:-1:-1)$$ ( 1 : - 1 : - 1 ) Resonant Quadratic System

2015 ◽  
Vol 29 (2) ◽  
pp. 597-613 ◽  
Author(s):  
Maša Dukarić ◽  
Regilene Oliveira ◽  
Valery G. Romanovski
Keyword(s):  
Quadratic System ◽  
Local Integrability

Local Integrability of Poincaré–Dulac Normal Forms

2018 ◽  
Vol 23 (7-8) ◽  
pp. 933-947
Author(s):  
Shogo Yamanaka
Keyword(s):  
Normal Forms ◽  
Local Integrability

scholarly journals On acyclicity of quadratic system with two non-rough foci

2021 ◽  
pp. 41-46
Author(s):  
Адам Дамирович Ушхо
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Phase Plane ◽  
Second Order ◽  
Quadratic System ◽  
Equilibrium States ◽  
System Of Differential Equations ◽  
Right Hand ◽  
The Right

Доказывается, что система дифференциальных уравнений, правые части которой представляют собой полиномы второй степени, не имеет предельных циклов, если в ограниченной части фазовой плоскости она имеет только два состояния равновесия и при этом они являются состояниями равновесия второй группы. It is proved that a system of differential equations, the right-hand sides of which are second-order polynomials, has no limit cycles if it has only two equilibrium states in the bounded part of the phase plane, and they are the equilibrium states of the second group.

Study of a simple 3D quadratic system with homoclinic flip bifurcations of inward twist case C

2019 ◽  
Vol 77 ◽  
pp. 324-337 ◽  
Author(s):  
A. Algaba ◽  
M.C. Domínguez-Moreno ◽  
M. Merino ◽  
A.J. Rodríguez-Luis
Keyword(s):  
Quadratic System

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.

A general time-dependent invariant for and integrability of the quadratic system

1994 ◽  
Vol 444 (1922) ◽  
pp. 643-650 ◽  
Keyword(s):  
Direct Method ◽  
First Integral ◽  
Quadratic System ◽  
Time Dependent ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Linear Polynomial ◽  
A Direct Method ◽  
Special Case ◽  
General Time

We derive a general time-dependent invariant (first integral) for the quadratic system (QS) that requires only one condition on the coefficients of the QS. The general invariant could yield asymptotic behaviour of phase-space trajectories. With more conditions imposed on the coefficients, the general invariant reduces to polynomial form and is equivalent to polynomial invariants found using a direct method. For the special case of a linear polynomial invariant where one of the variables is analytically invertible, the solution of the QS is reduced to a quadrature.

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