The Logistic Difference Equation

ρ

1.5

initial value (

u

0

)

0.8

number of terms

10

graph

solution plot

line

u

n+1

= ρ

u

n

(1-

u

n

)

equilibrium:

u

n

= 0.333333

The logistic difference equation (or logistic map)

u

n+1

=ρ

u

n

(1-

u

n

)

, a nonlinear first-order recurrence relation, is a time-discrete analogue of the logistic differential equation,

dx

dt

=ρx(1-x)

. Like its continuous counterpart, it can be used to model the growth or decay of a process, population, or financial instrument.

Depending on the value of the constant

ρ

, the solution of the difference equation can approach an equilibrium, move periodically through some cycle of values, or behave in a chaotic, unpredictable way.

A visualization of solutions to the logistic difference equation can be obtained using what can be called a "stairstep diagram." A green line intersects back and forth between the graphs of

y=x

and

y=ρx(1-x)

, beginning at the point

(

u

0

,0)

. Every intersection of the green line and the red parabola represents a value of

u

n

. It is easy to see if the solution converges to a single point, oscillates in "square-like" fashion, or is completely unpredictable.