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.

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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.

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.

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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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