Delay Differential Equations Recent Advances and New Directions
Delay Differential Equations: Recent Advances and New Directions cohesively presents contributions from leading experts on the theory and applications of functional and delay differential equations (DDEs). Researchers will benefit from a unique focus on t
- PDF / 365,974 Bytes
- 24 Pages / 439.37 x 666.142 pts Page_size
- 76 Downloads / 215 Views
Calculating Center Manifolds for Delay Differential Equations Using Maple TM
Sue Ann Campbell
Abstract In this chapter, we demonstrate how a symbolic algebra package may be used to compute the center manifold of a nonhyperbolic equilibrium point of a delay differential equation. We focus on a specific equation, a model for high speed drilling due to Stone and Askari, but the generalization to other equations should be clear. We begin by reviewing the literature on center manifolds and delay differential equations. After outlining the theoretical setting for calculating center manifolds, we show how the computations may be implemented in the symbolic algebra package MapleTM . We conclude by discussing extensions of the center manifold approach as well as alternate approaches. Keywords: Delay differential equation · Center manifold · Symbolic computation
8.1 Introduction To begin, we briefly review some results and terminology from the theory of ordinary differential equations (ODEs). Consider an autonomous ODE x = f(x),
(8.1)
which admits an equilibrium point, x∗ . The linearization of (8.1) about x∗ is given by x = Ax,
(8.2)
where A = Df(x∗ ). Recall that x∗ is called nonhyperbolic if at least one of the eigenvalues of A has zero real part. Given a complete set of generalized eigenvectors for the eigenvalues of A with zero real part, one can construct a basis for the subspace solutions of (8.2) corresponding to these eigenvalues. This subspace is called the center eigenspace of (8.2). Nonhyperbolic equilibrium points are important as they often occur at bifurcation points of a differential equation. The center manifold B. Balachandran et al. (eds.), Delay Differential Equations: Recent Advances and New Directions, DOI 10.1007/978-0-387-85595-0 8, c Springer Science+Business Media LLC 2009
221
222
S.A. Campbell
is a powerful tool for studying the behavior of solutions (and hence the nature of the bifurcation) of (8.1) in a neighborhood of a nonhyperbolic equilibrium point. It is a nonlinear manifold, which is tangent to the center eigenspace at x∗ . For a more detailed review of the theory and construction of center manifolds for ODES see [17, Sect. 3.2], [49, Sect. 2.1] or [33, Sect. 2.12]. In this eighth chapter, we study the center manifolds for nonhyperbolic equilibrium points of delay differential equations (DDEs). In general, one cannot find the center manifold exactly, thus one must construct an approximation. Some authors have performed the construction by hand, e.g., [11, 12, 14, 18–20, 23–25, 28, 30–32, 38,46,47,51,53]. However, the construction generally involves a lot of computation and is most easily accomplished either numerically or with the aid of a symbolic algebra package. Here we focus on the symbolic algebra approach, which has been used by several authors, e.g., [3, 7, 34–36, 45, 48, 54–56]. Unfortunately, there is rarely space in journal articles to give details of the implementation of such computations. Thus, the purpose here is to give these details, for a particular example DD
Data Loading...
