The Kuramoto–Sivashinsky equation revisited: Low-dimensional corresponding systems

• The Kuramoto–Sivashinsky equation revisited.
• Low-dimensional corresponding systems is obtained and analyzed.
• Periodic and chaotic behaviors of the Kuramoto–Sivashinsky equation and are studied.

The main aim of this paper is to study the behaviors of the spatially periodic initial value problem for the Kuramoto–Sivashinsky (K–S) equation with the viscosity parameter. This is done by using spatially truncated Fourier decomposition with Fourier coefficients a system of ordinary differential equations in time variable. As a low-dimensional dynamical system we start with a system of four ordinary differential equations which has by itself interesting behaviors, specially a new behavior is found for that system. Then these results are applied to the K–S equation where some behaviors are in good agreement with some previous numerical experiments. Finally the order of truncation is increased with the resultant: chaotic behavior of the K–S equation for a value of the parameter is shown by calculation of the Lyapunov exponents.