The document discusses fixed points in the context of maps on R^m, defining them as sinks, sources, or saddle points based on the behavior of points in their ε-neighborhoods. It explains linear maps and their stability at the origin using eigenvalues, and introduces the Jacobian matrix, which is used to analyze the stability of fixed points and periodic orbits. Additionally, the document provides conditions for determining whether fixed points are attractive, repelling, or saddle points based on eigenvalue magnitudes.

1. - Kamran Ansari
CBS 6th Semester09 – May - 2018

2.  Sinks, Sources and Saddles
 Linear Maps
 Jacobian Matrix
Contents

3.  Let f be a map on ℝ 𝑚
and let p in ℝ 𝑚
such that f(p)=p then
p is called fixed point.
 If there is 𝜖 > 0 such that for all v in the 𝜖-neighborhood
𝑁𝜖(p), lim
𝒌→∞
𝐟 𝒌
𝐯 = 𝐩, then p is a sink or attracting fixed
point.
 If the 𝜖-neighborhood 𝑁𝜖(p) is such that each v in 𝑁𝜖(p)
except for p itself eventually maps outside of 𝑁𝜖(p) then p is
a source or repelling fixed point.
 If v in 𝑁𝜖(p) is such that it has at least one attracting
direction and at least one repelling direction then p is a
saddle point.
Sinks, Sources and Saddles

4. Fig. :Local dynamics near a fixed point.

5. Example of source fixed point with
Mathematica

7. Example of sink fixed point with
Mathematica

9. Example of saddle fixed point with
Mathematica

11. Linear Maps
 A map A(v) from ℝ 𝑚
to ℝ 𝑚
is linear if for each a, b ∈ ℝ,
and v, w ∈ ℝ 𝑚
, A(av+bw) = aA(v) + bA(w). Equivalently, a
linear map A(v) can be represented as multiplication by an
m×m matrix.
 Every linear map has a fixed point at the origin.

12.  If A(v) be a linear map on ℝ 𝑚
, which is represented by the
matrix A then,
1. The origin is a sink if all eigenvalues of A are smaller than
one in absolute value.
2. The origin is a source if all eigenvalues of A are larger than
one in absolute value.
3. The origin is called a saddle, if a hyperbolic map A has at
least one eigenvalue of absolute value greater than one and
at least one eigenvalue of absolute value smaller than one.

13. Jacobian Matrix
 Let f = (𝑓1, 𝑓2,…., 𝑓𝑚) be a map on ℝ 𝑚
, and let p ∈ ℝ 𝑚
than
the Jacobian matrix of f at p, denoted Df(p), is the matrix
Df 𝐩 =
𝜕𝑓1
𝜕𝑥1
(𝐩) … . .
𝜕𝑓1
𝜕𝑥 𝑚
(𝐩)
: … . . :
𝜕𝑓𝑚
𝜕𝑥1
(𝐩) … . .
𝜕𝑓𝑚
𝜕𝑥 𝑚
(𝐩)

14. f(p + h) − f(p) ~ Df(p).h
 Let f be a map on m, and assume f(p) = p.
1. If the magnitude of each eigenvalue of Df(p) is less than 1,
then p is a sink.
2. If the magnitude of each eigenvalue of Df(p) is greater than
1, then p is a source.
3. If p is hyperbolic and if at least one eigenvalue of Df(p) has
magnitude greater than 1 and at least one eigenvalue has
magnitude less than 1, then p is called a saddle.

15.  If there is a periodic orbit {𝐩1, 𝐩2,….., 𝐩 𝑘} of period k for a
map on ℝ 𝑚
. Using the chain rule,
𝐃𝐟 𝒌
𝐩1 = 𝐃𝐟 𝐩1 . 𝐃𝐟 𝐩2 …. 𝐃𝐟 𝐩 𝑘
The eigenvalues of m×m Jacobian matrix evaluated at 𝐩1,
𝐃𝐟 𝒌
𝐩1 will determine the stability of the period – k orbit.

16. Thank You