Asymptotics of a Slow Manifold

Approximately invariant elliptic slow manifolds are constructed for the Lorenz-Krishnamurthy model of fast-slow interactions in the atmosphere. As is the case for many other two-time-scale systems, the various asymptotic procedures that may be used for this construction diverge, and there are no exactly invariant slow manifolds. Valuable information can however be gained by cap- turing the details of the divergence: this makes it possible to define exponentially accurate slow manifolds, identify one of these as optimal, and predict the amplitude and phase of the fast oscilla- tions that appear for trajectories started on it. We demonstrate this for the Lorenz-Krishnamurthy model by studying the slow manifolds obtained using a power-series expansion procedure. We de- velop two distinct methods to derive the leading-order asymptotics of the late coefficients in this expansion. Borel summation is then used to define a unique slow manifold, regarded as optimal, which is piecewise analytic in the slow variables. This slow manifold is not analytic on a Stokes surface: when slow solutions cross this surface, they switch on exponentially small fast oscillations through a Stokes phenomenon. We show that the form of these oscillations can be recovered from the Borel summation. The approach that we develop for the Lorenz-Krishnamurthy model has a general applicability; we sketch how it generalizes to a broad class of two-time-scale systems.

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