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