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The Kuramoto-Sivashinsky equation: Spatio-temporal chaos and intermittencies for a dynamical system

Description: We survey some recent results on the finite-dimensional behavior of the Kuramoto-Sivashinsky equation. We outline how it is rigorously equivalent to a finite dimensional dynamical system on a finite ''inertial'' manifold; a geometric approach to the construction of such a manifold is given. We give some examples of computational simulations supporting the evidence for a low-dimensional vector field which rules the bifurcations of the inertial manifold. 45 refs., 21 figs.
Date: January 1, 1986
Creator: Nicolaenko, B.
Partner: UNT Libraries Government Documents Department

General class of nonlinear bifurcation problems from a point in the essential spectrum: application to shock wave solutions of kinetic equations

Description: An abstract class of bifurcation problems is investigated from the essential spectrum of the associated Frechet derivative. This class is a very general framework for the theory of one-dimensional, steady-profile traveling- shock-wave solutions to a wide family of kinetic integro-differential equations from nonequilibrium statistical mechanics. Such integro-differential equations usually admit the Navier--Stokes system of compressible gas dynamics or the MHD systems in plasma dynamics as a singular limit, and exhibit similar viscous shock layer solutions. The mathematical methods associated with systems of partial differential equations must, however, be replaced by the considerably more complex Bifurcation Theory setting. A hierarchy of bifurcation problems is considered, starting with a simple bifurcation problem from a simple eigenvalue. Sections are entitled as follows: introduction and background from mechanics; the mathematical problem: principal results; a generalized operational calculus, and the derivation of the generalized Lyapunov--Schmidt equations; and methods of solution for the Lyapunov--Schmidt and the functional differential equations. 1 figure. (RWR)
Date: January 1, 1976
Creator: Nicolaenko, B.
Partner: UNT Libraries Government Documents Department

Remarks on the Kuramoto-Sivashinsky equation

Description: We report here a joint work in progress on the Kuramoto-Sivashinsky equation. The question we address is the analytical study of a fourth order nonlinear evolution equation. This equation has been obtained by Sivashinsky in the context of combustion and independently by Kuramoto in the context of reaction diffusion-systems. Both were motivated by (nonlinear) stability of travelling waves. Numerical calculations have been done on this equation. All the results seem to indicate a chaotic behavior of the solution. Therefore, the analytical study is of interest in analogy with the Burger's and Navier-Stokes equations. Here we give some existence and uniqueness results for the equation in space dimension one, and we also study a fractional step method of numerical resolution. In a forthcoming joint paper with R. Temam, we will study the asymptotic behavior, as t approaches infinity, of the solution of (0.1) and give an estimate on the number of determining modes.
Date: January 1, 1983
Creator: Nicolaenko, B. & Scheurer, B.
Partner: UNT Libraries Government Documents Department

Singular perturbation approach to flame theory with chain and competing reactions

Description: We investigate the structure of laminar flames with two chemical reactions in the limit of high activation energy asymptotics. Depending on the specific reaction network and the other given chemical data, a wide variety of flame configurations are possible. Here we fully explore these possibilities in the case of sequential and competing reaction pairs. Our methods are general enough to extend to most reaction networks with two or three exothermic reactions with high activation energy.
Date: January 1, 1982
Creator: Fife, P.C. & Nicolaenko, B.
Partner: UNT Libraries Government Documents Department

Asymptotic flame theory with complex chemistry

Description: We investigate the structure of laminar flames with general complex chemistry networks in the limit of high activation energy asymptotics. Depending on the specific reaction network and other given thermomechanical data, a wide variety of flame configurations are possible. Here we present a first version of a systematic asymptotic reduction of complex chemistry networks and give practical criteria to determine the dominant reactions when transport and chemistry are coupled.
Date: January 1, 1982
Creator: Fife, P. C. & Nicolaenko, B.
Partner: UNT Libraries Government Documents Department

Traveling-wave solutions to reaction-diffusion systems modeling combustion

Description: We consider the deflagration-wave problem for a compressible reacting gas, with species involved in a single-step chemical reaction. In the limit of small Mach numbers, the one-dimensional traveling-wave problem reduces to a system of reaction-diffusion equations. Thermomechanical coefficients are temperature-dependent. Existence is proved by first considering the problem in a bounded domain, and taking an infinite-domain limit. In the singular limit of high activation energy in the Arrhenius exponential term, we prove strong convergence to a limiting free-boundary problem (discontinuity of the derivatives on the free boundary).
Date: January 1, 1982
Creator: Berestycki, H.; Nicolaenko, B. & Scheurer, B.
Partner: UNT Libraries Government Documents Department

Finite dimensionality in the complex Ginzburg-Landau equation

Description: Finite dimensionality is shown to exist in the complex Ginzburg-Landau equation periodic on the interval (0,1). A cone condition is derived and explained which gives upper bounds on the number of Fourier modes required to span the universal attractor and hence upper bounds on the attractor dimension itself. In terms of the parameter R these bounds are not large. For instance, when vertical bar ..mu.. vertical bar less than or equal to ..sqrt..3, the Fourier spanning dimension is 0(R/sup 3/2/). Lower bounds are estimated from the number of unstable side-bands using ideas from work on the Eckhaus instability. Upper bounds on the dimension of the attractor itself are obtained by bounding (or, for vertical bar ..mu.. vertical bar less than or equal to ..sqrt..3, computing exactly) the Lyapunov dimension and invoking a recent theorem of Constantin and Foias, which asserts that the Lyapunov dimension, defined by the Kaplan-Yorke formula, is an upper bound on the Hausdorff dimension. 39 refs., 7 figs.
Date: January 1, 1987
Creator: Doering, C.R.; Gibbon, J.D.; Holm, D.D. & Nicolaenko, B.
Partner: UNT Libraries Government Documents Department