Recurrence Table Attractors

x

n+1

function variables

a

0

b

0.7

c

1

d

0

e

0

f

0

y

n+1

function variables

g

-0.8

h

0

i

0

j

1

k

0

l

0

initial conditions

x

0.14

y

0.33

modulus

m

3

number of points

n

2000

randomize

x

n+1

0.7

x

n

+

y

n

y

n+1



2

x

n

-0.8

A recurrence equation such as

x

n+2

=

x

n+1

+

x

n

generates a sequence of terms based on an initial condition. For example, with

x

0

=

x

1

=1

, the Fibonacci sequence 1, 1, 2, 3, 5, 8, … is generated. A recurrence table can generate a multidimensional list of values. In this Demonstration, a range of two-variable recurrence equations is explored. When plotted, a variety of behaviors can be seen. The points may scatter randomly, or quickly converge to a set of values. About one time out of 50, a strange attractor is generated.