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

Abstract 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.

[1]  G. Sivashinsky,et al.  Nonlinear saturation of Rayleigh–Taylor instability in thin films , 1983 .

[2]  F. Kaiser,et al.  An intermittency route to chaos via attractor merging in the Laser-Kuramoto-Sivashinsky equation , 1996 .

[3]  J. Eckmann,et al.  A global attracting set for the Kuramoto-Sivashinsky equation , 1993 .

[4]  C. H. Lee,et al.  Reduced-order-based feedback control of the Kuramoto-Sivashinsky equation , 2005 .

[5]  Milija Zupanski,et al.  Comparison of sequential data assimilation methods for the Kuramoto–Sivashinsky equation , 2009 .

[6]  Y. Kuramoto,et al.  Persistent Propagation of Concentration Waves in Dissipative Media Far from Thermal Equilibrium , 1976 .

[7]  Roger Temam,et al.  Estimates on the lowest dimension of inertial manifolds for the Kuramoto-Sivashinsky equation in the general case , 1994, Differential and Integral Equations.

[8]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[9]  Roger Temam,et al.  Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attr , 1985 .

[10]  Nastaran Vasegh,et al.  Suboptimal control of linear systems with delays in state and input by orthonormal basis , 2011, Int. J. Comput. Math..

[11]  Roger Temam,et al.  Some Global Dynamical Properties of a Class of Pattern Formation Equations , 1989 .

[12]  R. Temam,et al.  Nonlinear Galerkin methods , 1989 .

[13]  I. Kevrekidis,et al.  Back in the saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation , 1990 .

[14]  James C. Robinson.  Inertial manifolds for the Kuramoto-Sivashinsky equation , 1994 .

[15]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[16]  James C. Robinson.  The asymptotic completeness of inertial manifolds , 1996 .

[17]  Ioannis G. Kevrekidis,et al.  The Oseberg Transition: Visualization of Global bifurcations for the Kuramoto-Sivashinsky equation , 2001, Int. J. Bifurc. Chaos.

[18]  G. Sivashinsky,et al.  On Irregular Wavy Flow of a Liquid Film Down a Vertical Plane , 1980 .

[19]  A. Chian,et al.  High-dimensional chaotic saddles in the Kuramoto-Sivashinsky equation , 2003 .

[20]  William Tang,et al.  Nonlinear Saturation of the Trapped-Ion Mode , 1974 .

[21]  G. Sivashinsky.  Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations , 1977 .

[22]  E. Macau,et al.  Analysis of chaotic saddles in high-dimensional dynamical systems: the Kuramoto-Sivashinsky equation. , 2004, Chaos.

[23]  R. Wittenberg.  Dissipativity, analyticity and viscous shocks in the (de)stabilized Kuramoto-Sivashinsky equation , 2002 .

[24]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[25]  J. Hyman,et al.  The Kuramoto-Sivashinsky equation: a bridge between PDE's and dynamical systems , 1986 .