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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


On Quasi-periodic Perturbations of Duffing Equation

Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 397--406 | DOI:10.5890/DNC.2016.12.005

A.D. Morozov; T.N. Dragunov

Institute of IT, Mathematics and Mechanics, Lobachevsky University of Nizhny Novgorod, 23 Gagarin Ave, Nizhny Novgorod, 603950, Russia

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Quasi-periodic two-frequency perturbations are studied in a system which is close to a nonlinear two-dimensional Hamiltonian one. The example of Duffing equation with a saddle and two separatix loops is considered. Several problems are studied: dynamical behavior in a neighborhood of a resonance level of the unperturbed system, conditions for the existence of resonance quasi-periodic solutions (two-dimensional resonance tori), global behavior of solutions inside domains separated from the unperturbed separatrix. In a neighborhood of the unperturbed separatrix the problem of relative position of stable an unstable separatrix manifolds is studied, conditions for the existence of doubly asymptotic solutions are found.


Our work was partially supported by the RFFR grant No 14-01-00344, RSCF, grant No 14-41-00044 and the Ministry of Education and Science of Russian Federation, Project 1410.


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