Basins of Attraction

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Basins of Attraction

• Dr. David Chan
• NCSSM
• TCM Conference
• February 1, 2002

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Outline

• Definitions and a Simple Example
• Newtons Method in the Real Plane
• Newtons Method in the Complex Plane
• The Biology of a Species

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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)

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• Examples
• Precalculus

- Population growth
- Drug Dosage
- Loans
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• More Examples
• Calculus

-Function Iteration
-Fixed points -Bifurcations -Periodic Orbits
-Newtons Method
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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
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Basins of Attraction The Basin of
Attraction for an attractor is the set of points
which limit on the attractor.
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Example Function iteration
Two fixed points
x0 Has a basin of attraction of
(-1,1).
x1 Has a basin of attraction of
-1,1.
Everything else goes to infinity!
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CalculusNewtons Method

• Used to find roots of a function by using tangent
  lines.
• Formula

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Location of a horizontal tangent line.
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Consider
Questions

• What is the basin of attraction for 0?

• Are there other attractors other than the roots?

• In what way(s) can Newtons Method fail?

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• 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.

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Question

• Are there other attractors other than the roots?

• Answer
• There are periodic points.

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Question

• In what way(s) can Newtons Method fail?

• Answer
• Move to the next hump at the
• same location.

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Newtons Method in the Complex Plane

• Same method but involves using
• complex arithmetic.

• This is 2-dimensional.

• has n different solutions.

• And

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Z2 - 1
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Z3 - 1
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Z4 - 1
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Z5 - 1
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Z2 - 1
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Z2.001 - 1
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Z2.005 - 1
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Z2.01 - 1
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Z2.02 - 1
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Z2.03 - 1
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Z2.04 - 1
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Z2.06 - 1
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Z2.1 - 1
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Z2.2 - 1
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Z2.3 - 1
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Z2.4 - 1
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Z2.5 - 1
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Z2.6 - 1
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Z2.7 - 1
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Z2.8 - 1
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Z2.9 - 1
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Z2.95 - 1
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Z3 - 1
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1 x 1
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0.1 x 0.1
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0.01 x 0.01
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0.000001 x 0.000001
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Newtons Method

• Method fails at z0.

• Method fails at lots of points which
• map to zero (eventually).

• All these points have points of all
• three colors near them.

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Precalculus Biology
One species-Adult Children
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Simplified Equations
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Questions 1. What happens if rgt0 and agt0.
This models competition.
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Questions
2. What happens if rlt0?
Everything dies out!
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Questions

1. What happens if rgt0 and alt0?

This models cannibalism.
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