TY - JOUR
T1 - Phase Portraits of the Equation x¨ + axx˙ + bx3 = 0
AU - Llibre, Jaume
AU - Valls, Clàudia
PY - 2024
Y1 - 2024
N2 - The second-order differential equation x¨ + axx˙ + bx3 = 0 with a, b ∈ R has been studied by several authors mainly due to its applications. Here, for the first time, we classify all its phase portraits in function of its parameters a and b. This classification is done in the Poincaré disc in order to control the orbits which scape or come from infinity. We prove that there are exactly six topologically different phase portraits in the Poincar'e disc of the first order differential system associated by the second-order differential equation. Additionally we show that this system is always integrable providing explicitly its first integrals.
AB - The second-order differential equation x¨ + axx˙ + bx3 = 0 with a, b ∈ R has been studied by several authors mainly due to its applications. Here, for the first time, we classify all its phase portraits in function of its parameters a and b. This classification is done in the Poincaré disc in order to control the orbits which scape or come from infinity. We prove that there are exactly six topologically different phase portraits in the Poincar'e disc of the first order differential system associated by the second-order differential equation. Additionally we show that this system is always integrable providing explicitly its first integrals.
KW - Second-order differential equation
KW - Poincaré compactification
KW - Global phase portraits
UR - https://www.scopus.com/pages/publications/85203126185
U2 - 10.1134/S1560354724560053
DO - 10.1134/S1560354724560053
M3 - Article
SN - 1560-3547
VL - 29
SP - 825
EP - 837
JO - Regular and Chaotic Dynamics
JF - Regular and Chaotic Dynamics
IS - 6
ER -