Nonlinear Dynamical Processes Described by the Generalized Duffing Equation with an External Force
https://doi.org/10.56304/S2304487X20050107
Abstract
The generalized nonlinear Duffing equation, which is obtained from the equation for description of the pulse propagation in optical fibers using traveling wave variables, is considered. An ordinary second-order differential equation is written in the form of a dynamical system. For the generalized Duffing equation without external force, stationary points are found and their stability is investigated. The results of the study are presented in the table, where the type of stability is indicated for each of the three equilibrium points depending on the equation parameter. Using the presented Hamiltonian of the considered system of equations, phase portraits of the generalized Duffing equation are constructed excluding perturbation. The considered dynamical system in the presence of a periodic external force is numerically analyzed. For different amplitudes of the driving force, Poincaré sections of the dynamical system are constructed for two different values of the parameter. It is shown that with an increase in the amplitude of the perturbing force, the periodic trajectories of the solution of the system are destroyed and the area of the region of chaotic dynamics of the equations increases. According to Benettin’s algorithm, the senior Lyapunov exponent of the dynamical system is calculated as a function of the amplitude of the driving force. It has been found that with an increase in the amplitude, the senior Lyapunov exponent increases, and as a consequence, the exponential divergence of the trajectories increases, which agrees with the previously obtained Poincaré mappings.
Supporting agencies: The work was supported by the Ministry of Science and Higher Education of the Russian Federation (draft state assignment No. 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. Nonlinear Dynamical Processes Described by the Generalized Duffing Equation with an External Force. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2020;9(5):442-448. (In Russ.) https://doi.org/10.56304/S2304487X20050107
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