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Transactions of the Moscow Mathematical Society

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Poincaré index and periodic solutions of perturbed autonomous systems

Author: O. Yu. Makarenkov
Translated by: E. Khukhro
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 70 (2009).
Journal: Trans. Moscow Math. Soc. 2009, 1-30
MSC (2000): Primary 34C25; Secondary 34A26, 34C23, 34D10, 47H11
Published electronically: December 3, 2009
MathSciNet review: 2573636
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Abstract: Classical conditions for the bifurcation of periodic solutions in perturbed auto-oscillating and conservative systems go back to Malkin and Mel'nikov, respectively. These authors' papers were based on the Lyapunov-Schmidt reduction and the implicit function theorem, which lead to the requirement that both the cycles and the zeros of the bifurcation functions be simple. In this paper a geometric approach is put forward which does not assume these requirements, but imposes a certain condition on the Poincaré index of a generating cycle with respect to some auxiliary vector field. The approach is based on calculating the topological degree of the Poincaré operator of the perturbed system with respect to interior and exterior neighbourhoods of a generating cycle, as a consequence of which the conclusion of the main theorem guarantees bifurcation of a certain number of periodic solutions towards the interior of the cycle, and of a certain number of periodic solutions towards the exterior of the cycle. Concrete examples are given, where this approach either establishes bifurcation of a greater number of periodic solutions compared with the known classical results, or provides additional information on the location of these solutions.

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Additional Information

O. Yu. Makarenkov
Affiliation: Research Institute of Mathematics, Voronezh State University, Voronezh, Russia

Keywords: Poincar\'{e} index, periodic solution, bifurcation, perturbation.
Published electronically: December 3, 2009
Additional Notes: This research was supported by the Russian Federal Agency for Science and Innovation and the Programme for Basic Research and Higher Education of the U.S. Civilian Research and Development Foundation (grant BF6M10), by the President of the Russian Federation fund for the support of young scientists (grant MK-1620.2008.1) and by the Marie Curie Foundation (grant PIIF-GA-2008-221331).
The main results of the paper were reported at the seminar of the Department of Differential Equations of the Steklov Mathematical Institute of the Russian Academy of Sciences on 22 March, 2006.
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