The Asymptotic Behavior of the Stable Initial Manifolds of a System of Nonlinear Differential Equations

Content Provider	Semantic Scholar
Author	Levin, Janna J.
Copyright Year	2010
Abstract	Here we shall show that those solutions of (1.1) whose initial y vector lies on a certain "stable" initial manifold, which depends on e and the initial x vector, are very well approximated for small t and e by the corresponding solutions of a boundary layer equation, in which the initial x vector enters as a parameter, associated with (1.1). .Moreover, it will also be shown that as e—>0+ the stable initial manifolds associated with (1.1) tend to the stable initial manifold associated with the boundary layer equation. Problems of this nature, i.e., problems in which there is a system of differential equations possessing the property that the setting of a parameter equal to zero reduces the order of the system, have been treated under various hypotheses by several authors, e.g., [2; 3; 4; 5; 6 ] and [7 ]. The same hypotheses as in [5] will be assumed here. We now state these hypotheses as well as the principal theorem of [5]; this will then serve to introduce the problem and the results of the present paper. Let fx=fz(t, x, y, e) be the matrix with dfi/dxj in the ith. row andjth column and let/x(<) be the matrix fx{t, p(t), q(t), 0). The matrices /„, gx, gy as well as /„(/), gx{t), g„(t) are similarly defined. HI:/, g, fx,fv, gx, gv are continuous in (t, x, y, e) for Og/gT if | jc — p (/) | + |y — q(t)\ +« is sufficiently small. H2: There exists a real nonsingular matrix P(/)GC'(0^;gr) such that
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Alternate Webpage(s)	http://www.ams.org/journals/tran/1957-085-02/S0002-9947-1957-0088613-X/S0002-9947-1957-0088613-X.pdf
Language	English
Access Restriction	Open
Subject Keyword	Approximation algorithm Arabic numeral 0 Assumed Emoticon H2 Database Engine Nonlinear programming Population Parameter SMO wt Allele Solutions The Matrix manifold
Content Type	Text
Resource Type	Article