# The Logistic Difference Equation

The Logistic Difference Equation

The logistic difference equation (or logistic map) =ρ(1-), a nonlinear first-order recurrence relation, is a time-discrete analogue of the logistic differential equation, =ρx(1-x). Like its continuous counterpart, it can be used to model the growth or decay of a process, population, or financial instrument.

u

n+1

u

n

u

n

dx

dt

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

ρ

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 and , beginning at the point . Every intersection of the green line and the red parabola represents a value of . It is easy to see if the solution converges to a single point, oscillates in "square-like" fashion, or is completely unpredictable.

y=x

y=ρx(1-x)

(,0)

u

0

u

n