Abstract
Bounded trajectories of an autonomous system with an isolated positively homogeneous nonlinearity that is the gradient of a smooth function are studied. We prove the existence of nonstationary bounded trajectories lying in connected components of the set of points where the positively homogeneous function is negative and nonzero stationary points in those connected components whose closure has nonzero Euler characteristic. The existence of nonstationary bounded trajectories is substantiated using the Waűewski method; and the existence of stationary points, using methods for calculating the winding number of finite-dimensional vector fields.
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Funding
This work was financially supported by the Russian Science Foundation, project 23-21-00032, https://rscf.ru/en/project/23-21-00032/.
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Translated by V. Potapchouck
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Mukhamadiev, E., Naimov, A.N. & Bystretsky, M.V. On Bounded Trajectories of an Autonomous System with an Isolated Positively Homogeneous Nonlinearity. Diff Equat 59, 1003–1007 (2023). https://doi.org/10.1134/S0012266123070133
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DOI: https://doi.org/10.1134/S0012266123070133