Differential Equations Defined by the Sum of two Quasi-Homogeneous Vector Fields

1997 ◽  
Vol 49 (2) ◽  
pp. 212-231 ◽  
Author(s):  
B. Coll ◽  
A. Gasull ◽  
R. Prohens
Keyword(s):  
Differential Equation ◽  
Critical Points ◽  
Limit Cycles ◽  
Vector Fields ◽  
Polynomial Systems ◽  
Polar Coordinates ◽  
The Stability ◽  
Derivatives Of ◽  
Homogeneous Vector Fields ◽  
Homogeneous Vector

AbstractIn this paper we prove, that under certain hypotheses, the planar differential equation: ˙x = X1(x, y) + X2(x, y), ˙y = Y1(x, y) + Y2(x, y), where (Xi, Yi), i = 1, 2, are quasi-homogeneous vector fields, has at most two limit cycles. The main tools used in the proof are the generalized polar coordinates, introduced by Lyapunov to study the stability of degenerate critical points, and the analysis of the derivatives of the Poincar´e return map. Our results generalize those obtained for polynomial systems with homogeneous non-linearities.

Structural stability of planar quasi-homogeneous vector fields

2018 ◽  
Vol 468 (1) ◽  
pp. 212-226 ◽  
Author(s):  
A. Algaba ◽  
N. Fuentes ◽  
E. Gamero ◽  
C. Garcia
Keyword(s):  
Structural Stability ◽  
Vector Fields ◽  
Homogeneous Vector Fields ◽  
Homogeneous Vector

ON INTEGRABILITY OF DIFFERENTIAL EQUATIONS DEFINED BY THE SUM OF HOMOGENEOUS VECTOR FIELDS WITH DEGENERATE INFINITY

2001 ◽  
Vol 11 (03) ◽  
pp. 711-722 ◽  
Author(s):  
J. CHAVARRIGA ◽  
I. A. GARCÍA ◽  
J. GINÉ
Keyword(s):  
Vector Fields ◽  
Short Proof ◽  
Angular Speed ◽  
Darboux Integrability ◽  
Arbitrary Degree ◽  
Projective Techniques ◽  
Points Of View ◽  
Homogeneous Vector Fields ◽  
Integrating Factors ◽  
Homogeneous Vector

The paper deals with polynomials systems with degenerate infinity from different points of view. We show the utility of the projective techniques for such systems, and a more detailed study in the quadratic and cubic cases is carried out. On the other hand, some results on Darboux integrability in the affine plane for a class of systems are given. In short we show the explicit form of generalized Darboux inverse integrating factors for the above kind of systems. Finally, a short proof of the center cases for arbitrary degree homogeneous systems with degenerate infinity is given, and moreover we solve the center problem for quartic systems with degenerate infinity and constant angular speed.

Limit Cycles of Planar Piecewise Smooth Quadratic Systems with Focus-Parabolic Type Critical Points

2021 ◽  
Vol 31 (06) ◽  
pp. 2150090
Author(s):  
Liping Sun ◽  
Zhengdong Du
Keyword(s):  
Critical Points ◽  
Limit Cycles ◽  
Parabolic Type ◽  
Piecewise Smooth ◽  
Polar Coordinates ◽  
Quadratic Systems ◽  
Focus Type ◽  
Almost All ◽  
Lyapunov Constants

It is very important to determine the maximum number of limit cycles of planar piecewise smooth quadratic systems and it has become a focal subject in recent years. Almost all of the previous studies on this problem focused on systems with focus–focus type critical points. In this paper, we consider planar piecewise smooth quadratic systems with focus-parabolic type critical points. By using the generalized polar coordinates to compute the corresponding Lyapunov constants, we construct a class of planar piecewise smooth quadratic systems with focus-parabolic type critical points having six limit cycles. Our results improve the results obtained by Coll, Gasull and Prohens in 2001, who constructed a class of such systems with four limit cycles.

scholarly journals Perturbations of a Hamiltonian family of cubic vector fields

1991 ◽  
Vol 44 (1) ◽  
pp. 139-147
Author(s):  
A.M. Urbina ◽  
M. Cañas ◽  
G. León de la Barra ◽  
M. León de la Barra
Keyword(s):  
Vector Field ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Vector Fields ◽  
Global Analysis ◽  
Hamiltonian Vector Field ◽  
Hamiltonian Graph ◽  
Polynomial Vector Fields ◽  
The Stability ◽  
Type Of Stability

This paper is related with the configurations of limit cycles for cubic polynomial vector fields in two variables (χ3).It is an open question to decide whether every limit cycle configuration in χ3 can be obtained by perturbation of a corresponding Hamiltonian configuration of centres and graphs.In this work, by considering perturbations of the Hamiltonian vector field XH = (Hy, − Hx), where H(x, y) = [a(x + h)2 + by2 − 1] [a(x − h)2 + by2 − 1], we make a global analysis of the possible cases.The vector field XH has three centres (C−, C+ and the origin) and two saddles. By means of quadratic perturbations the centres become fine foci where C−and C+ have the same type of stability but opposed to that one of the origin and infinity. Further introducing cubic perturbations changes the stability of C−, C+ and the cycle at infinity and generates limit cycles. Lastly extra linear terms change the stability of the origin and generate another limit cycle.Finally, we analyse the rupture of saddle connection of the Hamiltonian field under perturbation, via Melnikov's integral, in order to complete the study of the global phase portrait and to consider the possibility of new limit cycles emerging from the Hamiltonian graph.

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