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Yu.S. Bogdanov, S.A. Mazanik, Yu.B. Syroid (1996)
Kurs differentsial’nykh uravnenii
V.V. Nemytskii, V.V. Stepanov (1949)
Kachestvennaya teoriya differentsial’nykh uravnenii
ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 4, pp. 460–468. c Pleiades Publishing, Ltd., 2007. Original Russian Text c O.A. Lobanova, B.N. Sadovskii, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 4, pp. 449–456. ORDINARY DIFFERENTIAL EQUATIONS On Two-Dimensional Dynamical Systems with a Constraint O. A. Lobanova and B. N. Sadovskii Voronezh State University, Voronezh, Russia Received July 3, 2006 DOI: 10.1134/S0012266107040039 INTRODUCTION In the present paper, we consider problems related to the well-known Poincar´ e–Bendixson the- orem on the existence of a closed trajectory in the two-dimensional autonomous system x ˙ = f (x). We consider the case in which the values of the variable x are subjected to the constraint x ∈ Q, where Q is a convex closed set with nonempty interior in R . We assume that once the trajectory reaches the boundary of this set, its velocity is projected on the cone tangent to Q at the exit point. Such a behavior of trajectories is specific, for example, for systems governing electric chains with diode transformers [3, 4]. For a formal description of a system with a constraint, we recall that the normal and tangent cones N and T of
Differential Equations – Springer Journals
Published: May 24, 2007
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