The logistic map: stability one iteration at a time

The dimensionless update $x_{n+1}=r x_n(1-x_n)$ maps the interval from zero to one into itself when the parameter lies between zero and four. The index counts discrete generations or updates, not an infinitesimal time. The parabola's maximum is r/4 at x equal to one half, which proves the interval bound. Outside those assumptions, iterates may leave the interval; a logistic population interpretation must not be extended automatically to negative values or arbitrary growth parameters.

A fixed point solves $f(x_*)=x_*$, giving zero and, when r is nonzero, $1-1/r$. Linearizing one iteration gives an error multiplier $f'(x_*)$. Attraction requires its modulus below one. A negative multiplier makes the error alternate sides while shrinking; it does not mean instability until its magnitude reaches one. This differs from a continuous-time scalar flow, where a negative derivative produces decay without alternating discrete overshoots. Always identify whether the model is a map or a flow before applying a stability test.

At r equal to three the nonzero fixed point's multiplier reaches minus one. The next local stable object can repeat only after two applications. Solve $f(f(x))=x$ and remove the factors corresponding to the old fixed points. The remaining roots form a two-cycle only when distinct and real. They are exchanged by f. Their stability is determined by the product of the two local derivatives, since an error must return to the same phase of the two-cycle before its growth is assessed.

The logistic two-cycle exists above three and is stable until $r=1+\sqrt6$, where its two-step multiplier reaches minus one and another period doubling occurs. Continuing the cascade produces an accumulation near r equal to 3.56995, but this is not a boundary beyond which every parameter is chaotic. Periodic windows interrupt chaotic parameter regions. Superstable parameter choices can even have a derivative product of zero. A bifurcation diagram should discard transients and show several initial conditions when coexistence or exceptional orbits matter.

A digital calculator has finitely many representable states, so a deterministic finite-precision iteration must eventually repeat or hit an exceptional state. That eventual machine period is not a theorem that the mathematical real-number orbit is periodic. Conversely, a long nonrepeating machine trace does not prove a positive asymptotic Lyapunov exponent. Compare precision levels and theoretical invariants, and report the parameter, starting state, transient length, and observation length whenever presenting a numerical route to chaos.

The map is an intentionally simple laboratory for feedback and saturation, not a complete ecological model. Its noiseless single-state update omits overlapping generations, spatial structure, and uncertain parameters. Those omissions do not weaken its mathematical value; they define the question it answers. The meaningful skill is connecting a multiplier calculation to a visible alternating error, then checking which predictions survive changes in initial state and numerical precision. That habit transfers directly to return maps constructed from physical oscillators.