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

1n nP rP

1 1n n nD D kD q

1 1n n nL L iL p - Loans

- Drug Dosage

- Population growth

More Examples:•Calculus

-Newton’s Method

-Fixed points-Bifurcations-Periodic Orbits

-Function Iteration

Attractors

-Fixed Points-Periodic Orbits-Strange Attractors

An attractor is a point or a collection of points on which the system can limit. These often take the form of

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 pointsx=0

Has a basin of attraction of (-1,1). x=1Has 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:

11

1

( )

( )n

n nn

f xx x

f x

24 4 4(3)( 8)

2(3)

4 1122.4305,1.0971

6

x

x

Location of a horizontal tangent line.

Questions:

In what way(s) can Newton’s Method fail?

Are there other attractors other than the roots?

What is the basin of attraction for 0?

Consider: ( ) sin( )F x x

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( )2

cos( )

xx x

x

Question:

In what way(s) can Newton’s Method fail?Answer:Move to the next hump at the same location.

sin( )2

cos( )

xx x

x

Newton’s Method in the Complex Plane

•Same method but involves usingcomplex arithmetic.

•This is 2-dimensional.• has n different solutions.1nx • And…

Z2 - 1

Z3 - 1

Z4 - 1

Z5 - 1

Z2 - 1

Z2.001 - 1

Z2.005 - 1

Z2.01 - 1

Z2.02 - 1

Z2.03 - 1

Z2.04 - 1

Z2.06 - 1

Z2.1 - 1

Z2.2 - 1

Z2.3 - 1

Z2.4 - 1

Z2.5 - 1

Z2.6 - 1

Z2.7 - 1

Z2.8 - 1

Z2.9 - 1

Z2.95 - 1

Z3 - 1

1 x 1

0.1 x 0.1

0.01 x 0.01

0.000001 x 0.000001

Newton’s Method:3

1 2

1

3n n

zz z

z

•Method fails at z=0.

•Method fails at lots of points whichmap to zero (eventually).•All these points have points of allthree colors near them.

Precalculus: Biology

One species-Adult & Children

1 1( )1

1

n nr aC bAn n

n n

C A e

A kC

1 1( )1

1

n nr aC An n

n n

C A e

A C

Simplified Equations

Questions:1. What happens if r>0 and a>0.

This models competition.

1 1( )1

1

n nr aC An n

n n

C A e

A C

Questions:

2. What happens if r<0?

1 1( )1

1

n nr aC An n

n n

C A e

A C

Everything dies out!

Questions:

3. What happens if r>0 and a<0?

This models cannibalism.

1 1( )1

1

n nr aC An n

n n

C A e

A C