### Synthesizing a return map

Since the mean field curve does not always follow the empirical data, provision has been made to generate an empirical curve, by prescribing the coefficients of a Bernstein polynomial. Cursor arrows or mouse movements are used to select the coefficient or to establish its value.

• MOD - moves the bernstein pointer down along right margin of panel C.
• MOU - moves the bernstein pointer up along right margin of panel C.
• MOR - increases corresponding Bernstein coefficient.
• MOL - decreases corresponding Bernstein coefficient.
• MOA - reduces Bernstein increment used by MOL , MOR .
• MOB - increases Bernstein increment used by MOL , MOR .
• MOX - resets all Bernstein parameters.
• * - lists the Bernstein coefficients in panel M.

Provision has been made for a modifier, inspired by the idea of a probabilistic automaton, which can multiply each Bernstein monomial. The default values are all 1.0.

• X - set Bernstein modifier bx[bi] (in mils).
• I - sets the index (i.e., bi) of the Bernstein monomial.

Another service which has been provided is to fit a parabola to a portion of the return map, the intention being to see how closely the dynamics of the classical logistic function can be matched. These options are very specific to the Chaté-Manneville automata, and must be used cautiously; otherwise floating point overflow is likely to occur. Some control over the points may be obtained by generating an initial field of known probability.

• L - fit a parabola to 50 points of empirical return map.
• H - interpolate a parabola to 3 points of empirical return map.
• # - # a b c f(a) f(b) f(c) interpolates a parabola to 3 points.

The option # requires that the coordinates of the points be stated explicitly.

Harold V. McIntosh
E-mail:mcintosh@servidor.unam.mx