Geodesic
Lyapunov functions and asymptotics at infinity of solutions of equations that are close to Hamiltonian equations
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations, Tome 163 (2019), pp. 96-107

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We consider a nonlinear nonautonomous system of two ordinary differential equations with a stable fixed point and assume that the non-Hamiltonian part of the system tends to zero at infinity. We examine the asymptotic behavior of a two-parameter family of solutions that start from a neighborhood of the stable equilibrium. The proposed construction of asymptotic solutions is based on the averaging method and the transition in the original system to new dependent variables, one of which is the angle of the limit Hamiltonian system, and the other is the Lyapunov function for the complete system.
Keywords: nonlinear differential equation, asymptotics, averaging, Lyapunov function. 
@article{INTO_2019_163_a6,
     author = {O. A. Sultanov},
     title = {Lyapunov functions and asymptotics at infinity of solutions of equations that are close to {Hamiltonian} equations},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {96--107},
     publisher = {mathdoc},
     volume = {163},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2019_163_a6/}
}
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O. A. Sultanov. Lyapunov functions and asymptotics at infinity of solutions of equations that are close to Hamiltonian equations. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations, Tome 163 (2019), pp. 96-107. http://geodesic.mathdoc.fr/item/INTO_2019_163_a6/