Lyapunov Exponents of the Finite-Difference Approximation of the Generalized Kuramoto–Sivashinsky Equation
https://doi.org/10.1134/S2304487X20060073
Abstract
The generalized Kuramoto–Sivashinsky equation is used to describe a number of nonlinear physical processes.
The aim of this work is to study the influence of the dispersion term of the generalized Kuramoto–Sivashinsky equation on the nature of the dynamics of the system at three different degrees of nonlinearity.
The system dynamical regime can be qualitatively determined by calculating the largest Lyapunov exponent. If the largest Lyapunov exponent is positive, then the nearby trajectories diverge exponentially and the behavior of the system is chaotic. To find the largest Lyapunov exponent, one needs to repeatedly perturb the trajectory of the dynamical system and find the distance between the perturbed and unperturbed trajectories. To do this for a partial differential equation, it is necessary to transform it into the finite-dimensional system of ordinary differential equations. In this work, for the Kuramoto–Sivashinsky equation, this transition is carried out using the approximation of spatial derivatives on a discrete grid. The boundary conditions are chosen to be periodic. For the resulting system of ordinary differential equations, the largest Lyapunov exponent has been calculated as a function of the dispersion parameter for three degrees of nonlinearity of the equation. It has been found that an increase in the bifurcation parameter leads to a decrease in the largest Lyapunov exponent, and, consequently, to the suppression of chaotic motion. Using the method of lines, the dependences of the solution of the generalized Kuramoto–Sivashinsky equation on the spatial and time variables for several values of the dispersion parameter are constructed. The resulting plots illustrate the transition from a chaotic to a regular regime with an increase in the bifurcation parameter.
Keywords
Supporting agencies: The work was supported by the Ministry science and higher education of the Russian Federation (draft state assignment No. 0723-2020-0036) and 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. Lyapunov Exponents of the Finite-Difference Approximation of the Generalized Kuramoto–Sivashinsky Equation. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2020;9(6):529-537. (In Russ.) https://doi.org/10.1134/S2304487X20060073
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