Symbolic methods for invariant manifolds in chemical kinetics
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Abstract
Abstract Chemical reactions show a separation of time scales in transient decay due to the stiffness of the ordinary differential equations (ODEs) that describe their evolution. This evolution can be represented as motion in the phase space spanned by the concentration variables of the chemical reaction. Transient decay corresponds to a collapse of the “compressible fluid” representing the continuum of possible dynamical states of the system. Collapse occurs sequentially through a hierarchy of nested, attracting, slow invariant manifolds (SIMs), i.e., sets that map into themselves under the action of the phase flow, eventually reaching the asymptotic attractor of the system. Using a symbolic manipulative language, explicit formulas for the SIMs can be found by iterating functional equations obtained from the system's ODEs. Iteration converges geometrically fast to a SIM at large concentrations and, if necessary, can be stabilized at small concentrations. Three different chemical models are examined in order to show how finding the SIM for a model depends on its underlying dynamics. For every model the iterative method provides a global SIM formula; however, formal series expansions for the SIM diverge in some models. Repelling SIMs can be also found by iterative methods because of the invariance of trajectory geometry under time reversal. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006
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