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Time symmetry preserving perturbations of differential systems. (English. Russian original) Zbl 1087.34521
Differ. Equ. 40, No. 10, 1395-1403 (2004); translation from Differ. Uravn. 40, No. 10, 1325-1332 (2004).
Two differential systems, the original one \[ \frac {dx}{dt} = X(t,x) , \quad t \in\mathbb{R}, x \in D \subset \mathbb{R}^n, \qquad \tag{1} \] and the perturbed system \[ \frac {dx}{dt} = X(t,x) + \alpha (t) \Delta (t,x) , \quad t \in \mathbb{R}<, x \in D \subset \mathbb{R}^n, \qquad \tag{2} \] where \(\alpha (t)\) is a continuous scalar odd function and \(\Delta(t,x)\) is an arbitrary continuously differentiable vector function, are considered. An equivalence of the differential systems (1) and (2) in the sense of the coincidence of the so-called reflection functions is studied.. It is proved that if the reflection functions of the two systems coincide, then their shift operators also coincide on a symmetric interval of the form \([-\tau, \tau]\) and therefore the mappings for the period \([-\omega, \omega]\) coincide for periodic systems. This result simplifies a qualitative analysis of solution sets of differential systems.

MSC:
34C41 Equivalence and asymptotic equivalence of ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations
34D10 Perturbations of ordinary differential equations
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[1] Mironenko, V.I., Otrazhayushchaya funktsiya i periodicheskie resheniya differentsial’nykh uravnenii (Reflection Function and Periodic Solutions of Differential Equations), Minsk, 1986. · Zbl 0607.34038
[2] Krasnosel’skii, M.A., Operator sdviga po traektoriyam differentsial’nykh uravnenii (The Operator of Shift Along Trajectories of Differential Equations), Moscow, 1966.
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