Phase portraits of Abel quadratic differential systems of second kind with symmetries

Antoni Ferragut, Johanna D. García-Saldaña and Claudia Valls

Dyn. Syst., vol. 34(2), 301333, 2019.

Abstract

We provide normal forms and the global phase portraits on the Poincaré disk of the Abel quadratic differential equations of the second kind having a symmetry with respect to an axis or to the origin. Moreover, we also provide the bifurcation diagrams for these global phase portraits.

Phase portraits (80)

• \(QS1^{(0)}_{1}\).
• \(QS2_{3}\).
• \(QS3^{(1)}_{2}\).
• \(QS4_{1}\).
• \(QS5^{(0)}_{3}\).
• \(QS6_{2}\).
• \(QS8^{(0)}_{4}\). \(QS8^{(1)}_{2}\). \(QS8^{(1)}_{3}\).
• \(QS9_{1}\).
• \(QS10^{(0)}_{16}\). \(QS10^{(1)}_{5}\).
• \(QS11^{(1)}_{20}\). \(QS11^{(2)}_{6}\).
• \(QS12^{(0)}_{1}\).
• \(QS13^{(0)}_{1}\). \(QS13^{(1)}_{1}\).
• \(QS14^{(1)}_{1}\). \(QS14^{(2)}_{1}\).
• \(QS15^{(0)}_{1}\). \(QS15^{(1)}_{1}\).
• \(QS16^{(0)}_{1}\). \(QS16^{(1)}_{1}\).
• \(QS18_{1}\).
• \(QS19^{(0)}_{1}\). \(QS19^{(0)}_{2}\). \(QS19^{(0)}_{3}\). \(QS19^{(1)}_{1}\).
• \(QS21_{1}\).
• \(QS22^{(1)}_{4}\). \(QS22^{(2)}_{1}\). \(QS22^{(2)}_{2}\).
• \(QS23^{(0)}_{1}\).
• \(QS24_{1}\).
• \(QS25^{(1)}_{3}\).
• \(QS26_{1}\).
• \(QS28^{(1)}_{1}\).
• \(QS29^{(3)}_{1}\).
• \(QS31^{(1)}_{7}\).
• \(QS34^{(2)}_{1}\).
• \(QS35^{(2)}_{2}\).
• \(QS44^{(1)}_{1}\).
• \(QS45^{(2)}_{1}\).
• \(QS47^{(2)}_{1}\).
• \(QS48^{(3)}_{1}\).
• \(QS60_{2}\).
• \(QS62_{1}\).
• \(QS64^{(3)}_{1}\).
• \(QS65^{(3)}_{2}\).
• \(QS67^{(4)}_{1}\).
• \(QS68^{(4)}_{1}\).
• \(QS69^{(4)}_{1}\).
• \(QS70^{(5)}_{1}\).
• \(QS71^{(4)}_{1}\).
• \(QS72^{(5)}_{1}\).
• \(QS73^{(2)}_{1}\).
• \(QS78_{1}\).
• \(QS79^{(2)}_{3}\).
• \(QS81^{(2)}_{3}\).
• \(QS83^{(4)}_{1}\).
• \(QS88^{(2)}_{1}\).
• \(QS90_{1}\).
• \(QS94_{1}\).
• \(QS97_{1}\).
• \(QS119^{(6)}_{1}\).
• \(QS121^{(5)}_{1}\).
• \(QS122^{(6)}_{1}\).
• \(QS123^{(4)}_{1}\).
• \(QS124^{(3)}_{1}\).
• \(QS125^{(3)}_{1}\).
• \(QS126^{(3)}_{1}\).
• \(QS128^{(5)}_{1}\).
• \(QS129^{(4)}_{1}\).
• \(QS131^{(5)}_{1}\).
• \(QS137^{(3)}_{1}\).
• \(QS138^{(4)}_{1}\).
• \(QS139^{(6)}_{1}\).
• \(QS144^{(4)}_{1}\).
• \(QS145^{(5)}_{1}\).
• \(QS147^{(6)}_{1}\).

Comments

• Type of paper: Bifurcation diagram.

• This paper produces the complete set of phase portraits for Abel quadratic differential systems of second kind with symmetries.

• Even though some of the pictures provided by the paper are not complete, one can easily grasp the complete set of phase portraits. For example, their phase portraits 17, 19, 23, 24, 39, 40, 48, 59, 60, 63, 65, 67, 77, 78, 81 and 83 need of an elliptic orbit (or even two in some cases).

• There are some cusps which are drawn having a non-null angle between their separatrices as in 41, 42, 45 and 46. Even thought the phase portrait is topologically right, it is strange to see.

• The paper shows 84 phase portraits and Theorem 3 states that they are all topologically different, but there are 4 couples which are equivalent. Concretely, 35 with 54 (\(QS19^{(0)}_{3}\)), 55 with 58 (\(QS13^{(0)}_{1}\)), 63 with 67 (\(QS88^{(2)}_{1}\)) and 77 with 84 (\(QS123^{(4)}_{1}\)). Maybe there is some geometric property that makes them different, but not topologically.