Slow attractive canonical invariant manifolds for reactive systems

We analyze the efficacy of a standard manifold-based reduction method used to simplify reaction dynamics and find conditions under which the reduction can succeed and fail. In the standard reduction, a heteroclinic trajectory linking saddle and sink equilibria is taken as a candidate reduced manifold which we call a Canonical Invariant Manifold (CIM). We develop and exercise analytic tools for studying the local behavior of trajectories near the CIM. In so doing, we find conditions under which nearby trajectories are attracted to the CIM (ACIM) as well as conditions for which the dynamics on theACIM are slow (SACIM). Themethod is demonstrated on a (1) simple model problem, (2) Zel’dovich mechanism for nitric oxide production, and (3) hydrogen–air system. For systems that evolve in a three-dimensional composition space, we find that normal stretching away from the CIM in a volume-shrinking vector field is admitted and that depending on the magnitude of the local rotation rate, may or may not render the CIM to be attractive. The success and failure of the candidate CIM as a SACIM is displayed for the model system. Results for the Zel’dovich mechanism The authors recognize the partial support of the National Science Foundation (NSF) under Grant No. CBET-0650843. J. M. Powers (B) · S. Paolucci Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA e-mail: [email protected] J. D. Mengers Office of Energy Efficiency and Renewable Energy, U.S. Department of Energy, Washington, DC, USA A. N. Al-Khateeb Department of Aerospace Engineering, Khalifa University, Abu Dhabi, UAE