Melnikov Method for the Generalized Duffing Equation
https://doi.org/10.1134/S2304487X21020073
Abstract
The generalized Duffing equation obtained using traveling wave variables in the equation describing the propagation of pulses in an optical fiber has been considered. An ordinary second-order differential equation is represented as a dynamical system. For the generalized Duffing equation without the external force, equilibrium points are found and their stability is investigated. The results of the study are presented in the table, where the type of stability for each of the five equilibrium points is indicated depending on the parameters of the equation. The parameters at which the system has homoclinic and heteroclinic orbits are found. Using the Melnikov method, it has been found that homoclinic chaos is always implemented in the studied system for some values of the parameters. Damping is used to control chaotic dynamics in the system. Constraints on the parameters at which chaos is implemented are obtained for the damped system. Bifurcation diagrams of the system are plotted in the absence and presence of the controller. According to the Bennetin algorithm, the largest Lyapunov exponent of a dynamical system is calculated as a function of the amplitude of the driving force. The results obtained by numerical analysis are in agreement with those obtained theoretically by the Melnikov method.
Supporting agencies: The work was supported by the Russian Fund for Basic Research (RFBR) (project 18-29-10025)
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. Melnikov Method for the Generalized Duffing Equation. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2021;10(2):135-142. (In Russ.) https://doi.org/10.1134/S2304487X21020073
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