Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale “mechanism.” Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.

/pdf/mixed-mode-oscillations-with-multiple-time-scales-2gobomm5x9.pdf

Mixed-Mode Oscillations with Multiple Time Scales

Detailed modeling of complex reaction systems is becoming increasingly important in the development, analysis, design, and control of chemical reaction processes. For industrial processes, complete incorporation of the chemistry into process models facilitates the minimization of byproduct and pollutant formation, increased efficiency, and improved product quality. Processes that involve complex reaction networks include a variety of noncatalytic and homogeneous or heterogeneous catalytic processes (such as fluid catalytic cracking, combustion, chemical vapor deposition, and alkylation). For some systems, large sets of relevant reactions have been identified for use in simulations.1-3 For others, the availability of advanced computing environments has enabled the automated generation of reaction networks and their models, based on computational descriptions of the reaction types occurring in the system.4-6 The use of such complex models is hindered by two obstacles. First, because of their sheer size and the presence of multiple time scales, these models are difficult to solve. Second, the models contain large numbers of uncertain (and sometimes unknown) kinetic parameters; regression to determine the parameters of complex nonlinear models is both difficult and unreliable, and the sensitivity of simulations to parameter uncertainties cannot be easily ascertained. Furthermore, for the purpose of gaining insights into the reaction system’s behavior, it is usually preferable to obtain simpler models that bring out the key features and components of the system. For these reasons, model simplification and order reduction are becoming central problems in the study of complex reaction systems. The simulation, monitoring, and control of a complex chemical process benefit from the derivation of accurate and reliable reduced models tailored to particular process modeling tasks. Model simplification is directly linked to identification of key reactions and sets of species that give valuable insights into the behavior of the network and how it may be influenced. Advanced control schemes such as model predictive control7 or multiple model adaptive control8 must be based on selecting appropriate reduced models and tracking key sets of species. Ideally, a model order reduction algorithm should have broad applicability, enable analysis at several levels of detail, and provide an assessment of the modeling error.

http://depa.fquim.unam.mx/amyd//archivero/mODELOSDEcINETICA_3417.pdf

Simplification of mathematical models of chemical reaction systems

Introduction.- The Source of Examples.- Invariance Equation in the Differential Form.- Film Extension of the Dynamics: Slowness as Stability.- Entropy, Quasi-Equilibrium and Projector Field.- Newton Method with Incomplete Linearization.- Quasi-chemical Representation.- Hydrodynamics from Grad's Equations: Exact Solutions.- Relaxation Methods.- Method of Invariant Grids.- Method of Natural Projector.- Geometry of Irreversibility: The Film of Nonequilibrium States.- Slow Invariant Manifolds for Open Systems.- Estimation of Dimension of Attractors.- Accuracy Estimation and Post-Processing.- Conclusion.

http://www.math.le.ac.uk/people/ag153/homepage/Beginning.pdf

Invariant Manifolds for Physical and Chemical Kinetics

A selective classified bibliography of symbolic computation in some areas of chemistry is provided together with some examples of computer algebra algorithms and techniques to facilitate future joint work of chemists and computer scientists. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Symbolic calculation in chemistry: Selected examples

http://www.math.le.ac.uk/people/ag153/homepage/ChemEngSci2003.pdf

Method of invariant manifold for chemical kinetics

We present a geometrical picture of the steady state (SSA) and equilibrium (EA) approximations for simple chemical reactions Geometrically, SSA and EA correspond to surfaces in the phase space of species concentrations In this space, chemical reaction is represented by trajectory motion governed by first order differential equations Initially, during transient decay, trajectories move quickly into the (narrow) region R between the equilibrium (E) and steady state (S) surfaces, puncturing E or S orthogonal to a variable axis This is rigorously what the differential equations say Once in R, trajectories approach another surface, the slow manifold M, sandwiched between E and S SSA or EA is the restriction of the system motion to S or E as a ‘‘shadow’’ of its true, nearby motion along M We illustrate these ideas in relation to the Lindemann mechanism, showing how M is related to S and E, and how it may be obtained by iteration, yielding higher order SSA and EA formulas This procedure is asymptotic wit

The steady state and equilibrium approximations: A geometrical picture

The time evolution of two model enzyme reactions is represented in phase space Γ. The phase flow is attracted to a unique trajectory, the slow manifold M, before it reaches the point equilibrium of the system. Locating M describes the slow time evolution precisely, and allows all rate constants to be obtained from steady‐state data. The line set M is found by solution of a functional equation derived from the flow differential equations. For planar systems, the steady‐state (SSA) and equilibrium (EA) approximations bound a trapping region containing M, and direct iteration and perturbation theory are formally equivalent solutions of the functional equation. The iteration’s convergence is examined by eigenvalue methods. In many dimensions, the nullcline surfaces of the flow in Γ form a prism‐shaped region containing M, but this prism is not a simple trap for the flow. Two of its edges are EA and SSA. Perturbation expansion and direct iteration are now no longer equivalent procedures; they are compared in a...

Geometry of the steady-state approximation: Perturbation and accelerated convergence methods

After the decay of transients, the behavior of a set of differential equations modeling a chemical or biochemical system generally rests on a low-dimensional surface which is an invariant manifold of the flow. If an equation for such a manifold can be obtained, the model has effectively been reduced to a smaller system of differential equations. Using perturbation methods, we show that the distinction between rapidly decaying and long-lived (slow) modes has a rigorous basis. We show how equations for attracting invariant (slow) manifolds can be constructed by a geometric approach based on functional equations derived directly from the differential equations. We apply these methods to two simple metabolic models.

Invariant manifold methods for metabolic model reduction.

Coupled chemical reactions are often described by (stiff) systems of ordinary differential equations (ODEs) with widely separated relaxation times. In the phase space Γ of species concentration variables, relaxation can be represented as a cascade through a nested hierarchy of smooth hypersurfaces (inertial manifolds) {Σ}: If d is the number of independent concentration variables, then Γ≡Γd⊇Σd−1⊇Σd−2⋅⋅⋅. The last three sets in this hierarchy have special chemical importance: Σ0 is the stagnation point of the ODEs, i.e., chemical equilibrium; M(≡Σ1) is the linelike slow manifold describing the dynamical steady state in closed systems; Σ(≡Σ2) is the two‐dimensional surface containing the slowest transient flow that reaches M. Thus M and Σ are the structures underlying most steady‐state and transient kinetics experiments. The ODEs describe the velocity field in Γ, which may be used to define functional equations for M, Σ, and other members in the hierarchy {Σ}. These functional equations can be solved to give explicit formulas for M, Σ, etc. In a model three‐step mechanism, M is described by a vector functional equation involving ordinary derivatives, whereas Σ is described by a scalar functional equation involving partial derivatives. We show how Σ may be found by iterative solution of this functional equation if decay is monotonic, and comment on the complications introduced by oscillatory transients. The functional equation approach may be generally applied in higher dimensional systems.

On the geometry of transient relaxation

A generalization of the steady‐state and equilibrium approximations for solving the coupled differential equations of a chemical reaction is presented. This method determines the (steady‐state) reaction velocity in closed form. Decay from rapidly equilibrating networks is considered, since many enzyme mechanisms belong to this category. In such systems, after transients have died away, the phase‐space flow lies close to a unique trajectory, the slow manifold M. Locating M reduces the description of the system’s progress to a one‐dimensional integration. This manifold is a solution of a functional equation, derived from the differential equations for the reaction. As an example, M for Michaelis–Menten–Henri mechanism is found by direct iteration. The solution is very accurate and the appropriate boundary conditions are obeyed automatically. In an arbitrary mechanism, at vanishing decay rate, the slow manifold becomes a line of equilibrium states, which solves the functional equation exactly; it is thus a good initial approximation at finite decay rate. These ideas are applied to a mechanism with both enzyme–substrate and enzyme–product complexes.

Simon J. Fraser

Papers

The steady state and equilibrium approximations: A geometrical picture

Geometry of the steady-state approximation: Perturbation and accelerated convergence methods

Invariant manifold methods for metabolic model reduction.

On the geometry of transient relaxation

Geometrical picture of reaction in enzyme kinetics