12.2 Transversality 165
to put it more cynically, one needs to know the answer before asking the
question. It might therefore seem that these ideas are of little use to the
applied scientist; however, this is not exactly true, since the theorems describing
structural stability and generic properties do give one a good idea
of what to expect, although they cannot tell what is precisely happening in
a specific system. Also, the reader should always ask him or herself whether
or not the dynamics are stable and/or typical in some sense. Probably the
best way of mathematically quantifying these two notions for the applied
scientist has yet to be determined.
12.2 Transversality
Before leaving this section let us introduce the idea of transversality, which
will play a central role in many of our geometrical arguments.
Transversality is a geometric notion which deals with the intersection of
surfaces or manifolds. Let M and N be differentiable (at least C1
) manifolds
in Rn
.
Definition 12.2.1 (Transversality) Let p be a point in Rn
; then M and
N are said to be transversal at p if p ∈ M ∩ N; or, if p ∈ M ∩ N, then
TpM + TpN = Rn
, where TpM and TpN denote the tangent spaces of M
and N, respectively, at the point p. M and N are said to be transversal if
they are transversal at every point p ∈ Rn
; see Figure 12.2.1.
Whether or not the intersection is transversal can be determined by
knowing the dimension of the intersection of M and N. This can be seen
as follows. Using the formula for the dimension of the intersection of two
FIGURE 12.2.1. M and N transversal at p.
166 12. Structural Stability, Genericity, and Transversality
vector subspaces we have
dim(TpM + TpN) = dim TpM + dim TpN − dim(TpM ∩ TpN). (12.2.1)
From Definition 12.2.1, if M and N intersect transversely at p, then we
have
n = dim TpM + dim TpN − dim(TpM ∩ TpN). (12.2.2)
Since the dimensions of M and N are known, then knowing the dimension
of their intersection allows us to determine whether or not the intersection
is transversal.
Note that transversality of two manifolds at a point requires more than
just the two manifolds geometrically piercing each other at the point. Consider
the following example.
Example 12.2.1. Let M be the x axis in R2
, and let N be the graph of the
function f(x) = x3
; see Figure 12.2.2. Then M and N intersect at the origin in
R2
, but they are not transversal at the origin, since the tangent space of M is
just the x axis and the tangent space of N is the span of the vector (1, 0); thus,
T(0,0)N = T(0,0)M and, therefore, T(0,0)N + T(0,0)M = R2
.
End of Example 12.2.1
FIGURE 12.2.2. Nontransversal manifolds.
The most important characteristic of transversality is that it persists
under sufficiently small perturbations. This fact will play a useful role in
many of our geometric arguments; we remark that a term often used synonymously
for transversal is general position, i.e., two or more manifolds
which are transversal are said to be in general position.
Let us end this section by giving a few “dynamical” examples of transver-
sality.
Example 12.2.2. Consider a hyperbolic fixed point of a Cr
, r ≥ 1, vector
field on Rn
. Suppose the matrix associated with the linearization of the vector