## 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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### References:

 [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. [3] Arnold, V.I., Dopolnitel’nye glavy teorii obyknovennykh differentsial’nykh uravnenii (Additional Chapters of Ordinary Differential Equations), Moscow, 1978.
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