8,753 research outputs found
An iterative method for the approximation of fibers in slow-fast systems
In this paper we extend a method for iteratively improving slow manifolds so
that it also can be used to approximate the fiber directions. The extended
method is applied to general finite dimensional real analytic systems where we
obtain exponential estimates of the tangent spaces to the fibers. The method is
demonstrated on the Michaelis-Menten-Henri model and the Lindemann mechanism.
The latter example also serves to demonstrate the method on a slow-fast system
in non-standard slow-fast form. Finally, we extend the method further so that
it also approximates the curvature of the fibers.Comment: To appear in SIAD
Computation of saddle type slow manifolds using iterative methods
This paper presents an alternative approach for the computation of trajectory
segments on slow manifolds of saddle type. This approach is based on iterative
methods rather than collocation-type methods. Compared to collocation methods,
that require mesh refinements to ensure uniform convergence with respect to
, appropriate estimates are directly attainable using the method of
this paper. The method is applied to several examples including: A model for a
pair of neurons coupled by reciprocal inhibition with two slow and two fast
variables and to the computation of homoclinic connections in the
FitzHugh-Nagumo system.Comment: To appear in SIAM Journal of Applied Dynamical System
Analysis of the accuracy and convergence of equation-free projection to a slow manifold
In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis, and A. Zagaris, Projecting to a
Slow Manifold: Singularly Perturbed Systems and Legacy Codes, SIAM J. Appl.
Dyn. Syst. 4 (2005) 711-732], we developed a class of iterative algorithms
within the context of equation-free methods to approximate low-dimensional,
attracting, slow manifolds in systems of differential equations with multiple
time scales. For user-specified values of a finite number of the observables,
the m-th member of the class of algorithms (m = 0, 1, ...) finds iteratively an
approximation of the appropriate zero of the (m+1)-st time derivative of the
remaining variables and uses this root to approximate the location of the point
on the slow manifold corresponding to these values of the observables. This
article is the first of two articles in which the accuracy and convergence of
the iterative algorithms are analyzed. Here, we work directly with explicit
fast--slow systems, in which there is an explicit small parameter, epsilon,
measuring the separation of time scales. We show that, for each m = 0, 1, ...,
the fixed point of the iterative algorithm approximates the slow manifold up to
and including terms of O(epsilon^m). Moreover, for each m, we identify
explicitly the conditions under which the m-th iterative algorithm converges to
this fixed point. Finally, we show that when the iteration is unstable (or
converges slowly) it may be stabilized (or its convergence may be accelerated)
by application of the Recursive Projection Method. Alternatively, the
Newton-Krylov Generalized Minimal Residual Method may be used. In the
subsequent article, we will consider the accuracy and convergence of the
iterative algorithms for a broader class of systems-in which there need not be
an explicit small parameter-to which the algorithms also apply
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