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Calculating the Poincaré map for two-dimensional periodic systems and Riccati equations. (English. Russian original) Zbl 1492.34040

Differ. Equ. 57, No. 10, 1313-1319 (2021); translation from Differ. Uravn. 57, No. 10, 1339-1345 (2021).
For the study of periodic solutions of non-autonomous periodic systems \(dx/dt=X(t,x)\), A.I. Mironov introduced in 1984 the so-called reflecting function which can be used to represent the corresponding Poincaré map.
In the present paper, the authors consider two-dimensional linear non-autonomous systems \[ \frac{dx}{dt} = P(t)x. \tag{1} \] They derive two groups of conditions on the entries of the matrix \(P\) such that the reflecting function can be determined explicitely. In case that \(P\) is periodic, the corresponding Poincaré map can also be determined explicitely, which yields analytic conditions for the existence of periodic solutions of (1).
Additionally, the authors show that the same approach can be applied to the Riccati equation \[ \frac{dx}{dt} = a(t)+b(t)x + c(t)x^2. \]

MSC:

34C25 Periodic solutions to ordinary differential equations
34A30 Linear ordinary differential equations and systems
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
37C60 Nonautonomous smooth dynamical systems
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[2] Arnol’d, V. I., Obyknovennye differentsial’nye uravneniya (Ordinary Differential Equations) (1984), Moscow: Nauka, Moscow · Zbl 0577.34001
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