54
3 Area-Preserving Maps
Fig. 3.3 If we magnify each
elliptic fixed point, we find a
mixture of elliptic and
hyperbolic fixed points
surrounding it. This structure
repeats itself for every elliptic
fixed point
neighborhood of these fixed points, indicates that half of the fixed points must be
elliptic and half must be hyperbolic (see Fig. 3.2c). If we move the origin of our
mapping to any elliptic point, this picture will repeat itself (see Fig. 3.3).
3.2.4 The Tangent Map
If we know the location of a fixed point X (0) =
ρ (0)
φ (0)
, where X (0) = T M
X (0) ,
we can determine its character by linearizing the mapping, T M
, about the fixed
point. The linearized mapping, ∇T M
, is called the tangent map. Its eigenvalues are
sometimes called the “multipliers” of the fixed point.
To linearize the point X n =
ρ n
φ n
about the fixed point X (0) , let X n = X (0) +
δX n , where δX n is small. Then δX n+1 = ∇T M
δX n , where
∇T
M
=
∂ρ n+1
∂ρ n
∂ρ n+1
∂φ n
∂φ n+1
∂ρ n
∂φ n+1
∂φ n
X (0)
.
(3.13)
The eigenvalues, λ, of ∇T M
are given by det[λ ¯
1 − ∇T M
] = 0 or
λ
2
− λTr(∇T
M
) + det(∇T
M
) = 0.
(3.14)
For area-preserving maps, det(∇T M
) = 1, so the eigenvalues are given by
λ ± =
t
2
±
t 2
4
− 1,
(3.15)
where t = Tr[∇T M
]. Thus, the eigenvalues come in reciprocal pairs, λ + = λ
−1
− .