Application of the Melnikov Method to the Triki–Biswas Equation
https://doi.org/10.1134/S2304487X2104009X
Abstract
A second order periodically perturbed ordinary differential equation, which is a traveling wave reduction of the Triki–Biswas equation, used to describe the propagation pulses in nonlinear optics has been considered. A stability analysis of the studied system of ordinary differential equations is carried out. It has been found that, depending on the control parameters, the considered system of equations can have three or five stationary points, which affects the structure of its stable and unstable manifolds. The Melnikov function of the system of equations under study is constructed for the case of three stationary points, where the stable and unstable manifolds of the unperturbed system of equations coincide with its homoclinic orbit. It has been found that homoclinic chaos always occurs in the system under consideration. To control it, a damping term has been added to the system. Constraints are found on the parameters of the new system at which homoclinic chaos is realized in it. Basins of attraction have been plotted. It has been found that they have a fractal structure when the damping parameter are less than the critical ones obtained by the Melnikov criterion. The results of the numerical analysis are in agreement with those obtained theoretically by the Melnikov method.
Supporting agencies: The work was carried out with financial support Russian Foundation for Basic Research (project No. 18-29-10025) and the Ministry of Science and Higher Education of the Russian Federation (draft state task 0723-2020-0036)
About the Authors
S. F. Lavrova
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
N. A. Kudryashov
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
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Review
For citations:
Lavrova S.F., Kudryashov N.A. Application of the Melnikov Method to the Triki–Biswas Equation. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2021;10(4):308-317. (In Russ.) https://doi.org/10.1134/S2304487X2104009X
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