Workshop on
A two-step nonlinear sequential mechanism is studied as a dimensionless model of kinetic control. The model is: A -> B; A + B -> C with rate constant epsilon. For epsilon << 1, little C is formed, whereas for epsilon >> 1, the concentration of B approaches a steady-state and C is the main product. Because of the coupling between the two reaction steps the steady-state approximation (SSA) is both a nullcline and a solution of the ordinary differential equations (ODEs) for the model. All concentrations can be expressed in terms of a timelike, real phase variable phi, whose evolution is described by a first-order, nonlinear ODE. This simple system does not have a slow manifold in the usual sense, although a separation of time scales occurs for large epsilon. The perturbation solution of this ODE for large and small epsilon is described.
This talk is about the numerical continuation of normally hyperbolic invariant manifolds of dynamical systems. Examples of such manifolds include limit sets (attractors), stable and unstable manifolds, co-dimension 1 manifolds separating basins of attraction (separatrices), manifolds in phase space arising from bifurcations, and manifolds in phase plus parameter space on which bifurcations occur. The algorithm allows the computation and visualization of dynamical structures which standard numerical methods can't compute. Examples of computations of tori and closed curves will be given, both attracting and saddle-type, with and without non-uniform adaptive refinement for both maps and differential equations.
Topics will cover:
I will describe work that our research group is doing in chemistry reduction. Using a low-dimensional manifold approach in conjunction with a wavelet spatial approximation we are able to model realistic reactive flow problems. Examples will be used to demonstrate the effectiveness of the approach. In particular I will discuss issues related to the reaction manifold idea and discuss strategies for generalizing the approach to infinite dimensional systems.
Models which are not of the classical mass-action form are common, particularly in biochemistry. The use of non-mass-action equations is normally itself the product of a reduction using the steady-state approximation. However, these models are often far from minimal. Their reduction sometimes proceeds smoothly, but not always. I will speak both about what we can do and about some of the difficulties.
The repro-modelling approach in chemical kinetics includes carrying out several thousand simulations using a detailed reaction mechanism and fitting the results with simple algebraic functions, like polynomials. If the fitted function has less variables than that of the original mechanism, a reduced dimension model is produced. The repro-modelling approach has been successfully used in atmospheric chemistry and in several areas of combustion chemistry, like in the simulation of hydrocarbon ignition, shock waves, laminar and turbulent flames.
In this talk, I shall present a method to identify unstable modes and to study bifurcations to both Steady States and Periodic Orbits based on the continuous fractions. The method is presented in the Navier-Stokes equations setting with the hope that it can be applied to dimension reduction problems in Chemical Kinetics.
The bifurcation of the Navier-Stokes equations is a fundamental problem in turbulence theory, and its study is still in its early stage due essentially to two main difficulties. First, it is extremely subtle to estimate the crucial information on the spectrum of the linearized non self-adjoint Navier-Stokes operator around the basic flow. Second, the eigenvalues of the linearized operator always have even multiplicity, and special care is needed for bifurcation analysis where oddness of the eigenvalues is crucial.
I shall present recent progress made by Chen and myself, which goes beyond the early results in the sixties by Velte, Yudovich, Rabinowitz and others. For the 3D Navier-Stokes equations with Kolmogorov forcing, there is a basic steady-state flow for all Reynolds numbers. We show that there exist a number of critical Reynolds numbers such that four or more different steady states branch off the basic flow when the Reynolds number increases across each of such numbers. More precisely, there exist two or more different flow invariant subspaces with respect to each of such numbers so that the basic flow undergoes supercritical pitchfork bifurcations within each of such subspaces. The proof of these results are based on a) a very careful analysis of the spectrum of the linearized Navier-Stokes operator via continuous fraction method, b) the Krasnolselskii-Rabinowitz topological theory, c) asymptotic analysis, and d) some basic PDE theory associated with the Navier-Stokes equations. Issues related to Hopf bifurcations will also be explored.