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 Nelin. Dinam., 2016, Volume 12, Number 1, Pages 31–52 (Mi nd511)

Original papers

On the investigation of the bifurcation and chaotic phenomena in the system with a homoclinic “figure-eight”

O. S. Kostromina

N.I.Lobachevsky State University of Nizhny Novgorod, Prospekt Gagarina, 23, Nizhny Novgorod, 603950, Russia

Abstract: Small time-periodic perturbations of an asymmetric Duffing – Van der Pol equation with a homoclinic “figure-eight” of a saddle are considered. Using the Melnikov analytical method and numerical simulations, basic bifurcations associated with the presence of a non-rough homoclinic curve in this equation are studied. In the main parameter plane the bifurcation diagram for the Poincaré map is constructed. Depending on the parameters, the boundaries of attraction basins of stable fixed (periodic) points of the direct (inverse) Poincaré map are investigated. It is ascertained that the transition moment of the fractal dimension of attraction basin boundaries of attractors through the unit may be preceded by the moment of occurrence of the first homoclinic tangency of the invariant curves of the saddle fixed point.

Keywords: bifurcations, homoclinic Poincaré structures, attraction basins, fractal dimension, sensitive dependence on initial conditions

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 1410 Russian Foundation for Basic Research 13-01-0058914-01-00344 Russian Science Foundation 14-41-00044

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UDC: 517.9
MSC: 37G25
Revised: 29.12.2015

Citation: O. S. Kostromina, “On the investigation of the bifurcation and chaotic phenomena in the system with a homoclinic “figure-eight””, Nelin. Dinam., 12:1 (2016), 31–52

Citation in format AMSBIB
\Bibitem{Kos16} \by O.~S.~Kostromina \paper On the investigation of the bifurcation and chaotic phenomena in the system with a homoclinic “figure-eight” \jour Nelin. Dinam. \yr 2016 \vol 12 \issue 1 \pages 31--52 \mathnet{http://mi.mathnet.ru/nd511}