Basins of
Attraction
Dr. David Chan
NCSSM
TCM Conference
February 1, 2002
Outline
• Definitions and a Simple Example
• Newton’s Method in the Real Plane
• Newton’s Method in the Complex Plane
• The Biology of a Species
Dynamical Systems
A Dynamical System is a set of equations
which model some changing phenomena.
They often take the form of
• Difference Equation(s)
• Ordinary Differential Equation(s)
• Partial Differential Equation(s)
Examples:
•Precalculus
- Population growth
Pn  rPn1
- Drug Dosage
Dn  Dn1  kDn1  q
- Loans
Ln  Ln1  iLn1  p
More Examples:
•Calculus
-Function Iteration
-Fixed points
-Bifurcations
-Periodic Orbits
-Newton’s Method
Attractors
An attractor is a point or a collection
of points on which the system can
limit. These often take the form of
-Fixed Points
-Periodic Orbits
-Strange Attractors
Basins of Attraction
The Basin of Attraction for an
attractor is the set of points which
limit on the attractor.
Example: Function iteration
2
F ( x)  x
Two fixed points
x=0
Has a basin of attraction of (-1,1).
x=1
Has a basin of attraction of {-1,1}.
Everything else goes to infinity!
Calculus—Newton’s Method
• Used to find roots of a function by using
tangent lines.
• Formula:
f ( xn1 )
xn  xn1 
f ( xn1 )
Location of a horizontal tangent line.
4  4  4(3)(8)
x
2(3)
2
4  112
x
 2.4305,1.0971
6
Consider: F ( x)  sin( x)
Questions:
What is the basin of attraction
for 0?
Are there other attractors other
than the roots?
In what way(s) can Newton’s
Method fail?
Question:
What is the basin of attraction
for 0?
Answer:
There is a part of each ‘hump’ of
sine which will give 0 as a root.
Question:
Are there other attractors other
than the roots?
Answer:
There are periodic points.
sin( x)
x
 2  x
cos( x)
Question:
In what way(s) can Newton’s
Method fail?
Answer:
Move to the next hump at the
same location.
sin( x)
x
 x  2
cos( x)
Newton’s Method in the
Complex Plane
•Same method but involves using
complex arithmetic.
•This is 2-dimensional.
n
• x  1 has n different solutions.
• And…
2
Z -
1
3
Z -
1
4
Z -
1
5
Z -
1
2
Z -
1
2.001
Z
-1
2.005
Z
-1
2.01
Z
-1
2.02
Z
-1
2.03
Z
-1
2.04
Z
-1
2.06
Z
-1
2.1
Z
-1
2.2
Z
-1
2.3
Z
-1
2.4
Z
-1
2.5
Z
-1
2.6
Z
-1
2.7
Z
-1
2.8
Z
-1
2.9
Z
-1
2.95
Z
-1
3
Z -
1
1x1
0.1 x 0.1
0.01 x 0.01
0.000001 x 0.000001
Newton’s Method:
3
z 1
zn  zn 1 
2
3z
•Method fails at z=0.
•Method fails at lots of points which
map to zero (eventually).
•All these points have points of all
three colors near them.
Precalculus: Biology
One species-Adult & Children
Cn  An1e
( r  aCn1 bAn1 )
An  kCn1
Simplified Equations
Cn  An 1e
An  Cn 1
( r  aCn1  An1 )
Questions:
1. What happens if r>0 and a>0.
Cn  An 1e
( r  aCn1  An1 )
An  Cn 1
This models competition.
Questions:
2. What happens if r<0?
Cn  An 1e
( r  aCn1  An1 )
An  Cn 1
Everything dies out!
Questions:
3. What happens if r>0 and a<0?
Cn  An 1e
( r  aCn1  An1 )
An  Cn 1
This models cannibalism.