We explore methods to locate subcritical branches of spatially periodic solutions in pattern forming systems with a nonlinear finite-wavelength instability. We do so by means of a direct expansion in the amplitude of the linearly least stable mode about the appropriate reference state which one considers. This is motivated by the observation that for some equations fully nonlinear chaotic dynam...

A numerical method is proposed for computing time-periodic and relative time-periodic solutions in dissipative wave systems. In such solutions, the temporal period, and possibly other additional internal parameters such as the propagation constant, are unknown priori and need to be determined along with the solution itself. The main idea of the method is to first express those unknown parameter...

subject to appropriate initial and boundary conditions has been introduced in [15],[16] and in [23],[24] in studying phase turbulence and the flame front propagation in combustion theory. In the absence of any a priori estimates for the solutions of the scalar equation (1), most authors find it more convenient, for the mathematical study, to consider the differential form of the equation for u ...

A new theorem is applied to the Kuramoto-Sivashinsky equation with L-periodic boundary conditions, proving the existence of an asymptotically complete inertial manifold attracting all initial data. Combining this result with a new estimate of the size of the globally absorbing set yields an improved estimate of the dimension, N ∼ L.

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 an accurate guide to the highdimensional dynamics, and the periodic orbit theory yields several global averages characterizing the chaotic dynamics. PACS nu...

A model consisting of a mixed Kuramoto-Sivashinsky-Kortewegde Vries equation, linearly coupled to an extra linear dissipative equation has been proposed in [1] in order to describe the surface waves on multi-layered liquid films and stability criteria are discussed using wave mode analysis. In this paper, we study the linear stability of solutions to the model from the viewpoint of energy estim...