scholarly journals Darboux integrability of a cubic differential system with two parallel invariant straight lines

2021 ◽  
Vol 38 (1) ◽  
pp. 129-137
Author(s):  
DUMITRU COZMA ◽  
Keyword(s):  
Singular Point ◽  
Differential System ◽  
Darboux Integrability ◽  
Straight Lines

In this paper we prove the Darboux integrability of a cubic differential system with a singular point of a center typer having at least two parallel invariant straight lines.

scholarly journals Cubic differential systems with 2-invariant straight lines

2013 ◽  
Vol 22 (2) ◽  
pp. 185-192
Author(s):  
DUMITRU COZMA ◽  
Keyword(s):  
Singular Point ◽  
Differential System ◽  
Differential Systems ◽  
Focus Type ◽  
Straight Lines

For cubic differential system with a singular point O(0, 0) of a center or a focus type having three invariant straight lines of which one is 2-invariant it is proved that the origin is a center if and only if the first five Lyapunov quantities vanish.

SOME BIFURCATION DIAGRAMS FOR LIMIT CYCLES OF QUADRATIC DIFFERENTIAL SYSTEMS

2001 ◽  
Vol 11 (01) ◽  
pp. 197-206 ◽  
Author(s):  
H. S. Y. CHAN ◽  
K. W. CHUNG ◽  
DONGWEN QI
Keyword(s):  
Singular Point ◽  
Differential System ◽  
Limit Cycles ◽  
Quadratic Differential ◽  
Bifurcation Diagrams ◽  
Differential Systems ◽  
Numerical Examples ◽  
Quadratic Differential System ◽  
Realistic Parameter ◽  
Parameter Values

Concrete numerical examples of quadratic differential systems having three limit cycles surrounding one singular point are shown. In case another finite singular point also exists, a (3, 1) distribution of limit cycles is also obtained. This is the highest number of limit cycles known to occur in a quadratic differential system so far. Representative bifurcation diagrams are drawn for realistic parameter values.

Darboux integrability of the simple chaotic flow with a line equilibria differential system

2020 ◽  
Vol 135 ◽  
pp. 109712
Author(s):  
Adnan A. Jalal ◽  
Azad I. Amen ◽  
Nejmaddin A. Sulaiman
Keyword(s):  
Differential System ◽  
Darboux Integrability ◽  
Chaotic Flow

CENTER CONDITIONS FOR A CUBIC DIFFERENTIAL SYSTEM HAVING AN INTEGRATING FACTOR

2020 ◽  
Vol 8 (2) ◽  
pp. 6-13
Author(s):  
D. Cozma ◽  
A. Matei
Keyword(s):  
Singular Point ◽  
Differential System ◽  
Integrating Factor ◽  
Focus Type ◽  
Center Conditions ◽  
Integrating Factors

We find conditions for a singular point O(0, 0) of a center or a focus type to be a center, in a cubic differential system with one irreducible invariant cubic. The presence of a center at O(0, 0) is proved by constructing integrating factors.

scholarly journals Lyapunov, Perron and upper-limit stability properties of autonomous differential systems

2020 ◽  
Vol 56 ◽  
pp. 63-78
Author(s):  
I.N. Sergeev
Keyword(s):  
Singular Point ◽  
Global Stability ◽  
Differential System ◽  
Lyapunov Stability ◽  
Phase Region ◽  
Differential Systems ◽  
Stability Properties ◽  
Upper Limit

For a singular point of an autonomous differential system, the natural concepts of its Perron and upper-limit stability are defined, reminiscent of Lyapunov stability. Numerous varieties of them are introduced: from asymptotic and global stability to complete and total instability. Their logical connections with each other are investigated: cases of their coincidence are revealed and examples of their possible differences are given. The invariance of most of these properties with respect to the narrowing of the phase region of the system is established.

Characterizations of linear differential systems with a regular singular point

1972 ◽  
Vol 18 (2) ◽  
pp. 93-98 ◽  
Author(s):  
W. A. Harris
Keyword(s):  
Singular Point ◽  
Differential System ◽  
Fundamental Matrix ◽  
Linear Differential System ◽  
Regular Singular Point ◽  
Constant Matrix ◽  
Linear Differential Systems ◽  
Significant Difference ◽  
Linear Differential ◽  
Image Position

The linear differential systemwhere w is a vector with n components and A is an n by n matrix is said to have z = 0 as a regular singular point if there exists a fundamental matrix of the formsuch that S is holomorphic at z = 0 and R is a constant matrix ((1), p. 111; (2), p. 73). For such systems A will have at most a pole at z = 0 and we may writewhere p is an integer, Ã is holomorphic at z = 0, and Ã(0) ≠ 0. However, the converse is not true. When p ≦ − 1, A is holomorphic at z = 0, and every fundamental matrix is holomorphic at z = 0. If p ≧ 1, the non-negative integer p is called (after Poincaré) the rank of the singularity and there is a significant difference between the cases p = 0 and p ≧ 1. If p = 0 the linear differential system (1) is known to have z = 0 as a regular singular point ((1), p. 111) ; whereas, if p ≧ 1, z = 0 may or may not be a regular singular point.

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