Spatiotemporally chaotic dynamics of a Kuramoto-Sivashinsky system is described by means of an infinite hierarchy of its unstable spatiotemporally periodic solutions. An intrinsic parametrization of the corresponding invariant set serves as accurate guide to the high-dimensional dynamics, and the periodic orbit theory yields several global averages characterizing the chaotic dynamics.

We investigate existence and exact multiplicity of phase-locked solutions of an integro-differential equation derived from a Kuramoto system of coupled oscillators. Under general assumptions on the form of frequency distribution, we derive new, easily verified criteria which guarantee that either (i) exactly one solution exists, or (ii) exactly two solutions coexist over an entire interval of v...

In this work, the fractional KdV–Burgers–Kuramoto equation is studied. He's variational iteration method (VIM) and Adomian's decomposition method (ADM) are applied to obtain its solution. Comparison with HAM is made to highlight the significant features of the employed methods and their capability of handling completely integrable equations.

A simple model for dendritic growth is given by S2d'" + 9' — cos(9). For S ss 1 we prove that there is no bounded, monotonic solution which satisfies d(-oo) = -7t/2 and Q(oo) = n/2. We also investigate the existence of bounded, monotonic solutions of an equation derived from the Kuramoto-Sivashinsky equation, namely y" + y = 1 y1 /2. We prove that there is no monotonic solution which satisfies ...

We present a method of self-consistent a-priori bounds, which allows to study rigorously dynamics of dissipative PDEs. As an application present a computer assisted proof of an existence of a periodic orbit for the Kuramoto-Sivashinsky equation ut = (u )x− uxx− νuxxxx, u(t, x) = u(t, x + 2π), u(t, x) = −u(t,−x),

This paper presents a control problem for a one-dimensional nonlinear parabolic system, which consists of a Kuramoto–Sivashinsky–Korteweg de Vries equation coupled to a heat equation. We address the problem of controllability by means of a control supported in an interior open subset of the domain and acting on one equation only. The local null-controllability of the system is proved. The proof...

We study the Kuramoto-Sivashinsky equation on the infinite line with initial conditions having arbitrarily large limits ±Y at x = ±∞. We show that the solutions have the same limits for all positive times. This implies that an attractor for this equation cannot be defined in L∞. To prove this, we consider profiles with limits at x = ±∞, and show that initial conditions L-close to such profiles ...

In this article, a nonic B-spline collocation approach is applied to solve the approximate solution of Kuramoto-Sivashinsky equation (KSE). Here nonlinear term KSE linearizing using Taylo...