CHAPTE R 1
Introduction
Let S, A be real analytic manifolds. An analytic family of vector fields (or,
shortly, an analytic family) Xx is a collection of analytic vector fields on S (called
the phase space) which depends analytically on A £ A (called the parameter space).
Let us suppose that S is homeomorphic to an open subset of the two-sphere S 2 .
Then, for each fixed parameter A, the global structure of the c^-limit set Q\ C S of
the vector field Xx is easily described by the Poincare-Bendixson Theorem.
In fact, each connected subset r C Qx is either a point or homeomorphic to the
circle S
1
with a finite number of its points identified. Moreover, it can be classified
as follows
regular uo-limit sets: T is a periodic orbit;
singular UJ-limit sets: T contains at least one singular point of X\.
If we let the parameter A change, the evolution of such c^-limit sets can be highly
non-trivial (for instance, periodic orbits can collapse into singular points or graph-
ics). The study of such behavior is one of the main topics of Bifurcation Theory
(see e.g. [R2]).
A first interesting issue is how the solution curves behave near the set of singu-
larities Z(X\) (i.e. the set of points p = (xp, Xp) £ S x A such that Xxp(xp) 0).
A singular point p £ Z{X\) can be either isolated in its fiber or not. Th e
former case means that there exists a neighborhood U C S of x such that
Z(XX) n(Ux{\ = Ap}) = {xp}.
This situation is the only possibility in the so-called finite-codimension phenomena.
However, some analytic families which appear in applications can also have
singularities which are non-isolated in the fibers. For instance, a general singular
perturbation problem
X£a = \ X = f(X;y'a'£\, with x,yeR, ( a , £ ) £ R n + 1
is a typical situation where the singularities in the fiber {a = £ = 0},
Z(Xea) n{a = s = 0} = {f(x, y, 0, 0) = 0}
can define an analytic set of positive dimension.
E X A M P L E 1.1. On suitable coordinates, the (singularly perturbed) Van Der
Pol's analytic family ex +
(x2
+ x)x + x a = 0 can be written as
( L 1 ) X^a =
\y = e (a - x) + y {-x -
x2)
where (x,y) £
M2
and (e,a) £ (R
2
,0). For the parameter value (e,a) (0,0),
this family presents a line of singular points at {y 0}. All these singularities are
l
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