The Logistic Difference Equation

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The logistic difference equation (or logistic map) , a nonlinear first-order recurrence relation, is a time-discrete analogue of the logistic differential equation,
. Like its continuous counterpart, it can be used to model the growth or decay of a process, population, or financial instrument.
Contributed by: Victor Hakim (April 2013)
Open content licensed under CC BY-NC-SA
Snapshots
Details
The equilibrium values for determine how or whether the long-term activity of a solution is predictable. If
and
, then
, and the equilibrium solutions are
or
. Further investigation can be done to show that if
, then
is an asymptotically stable value. For
, solutions converge instead to
. For
, solutions do not converge to a fixed point, except when
exactly for some
, in which case
for all
.
Snapshot 1: the solution converges to a single value
Snapshot 3: where , the solution oscillates with period 2 (a "two-cycle")
For larger values of , the long-term activity is highly chaotic, though there may be certain values of
with oscillations of period 4, 8, 16, 32, … . In this chaotic region (
), there is a high sensitivity to the value of
. Even varying
a small amount changes most terms drastically; the solution becomes unpredictable.
Snapshot 5: a solution that is chaotic and ultimately unpredictable; it can, however, be modeled as a simpler, three-cycle approximation
Snapshots 2, 4, and 6: the stairstep diagrams of snapshots 1, 3, and 5, respectively
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