Stochastic Properties of Simple Differential — Delay Equations

Abstract

The logistic difference equation

$$ {P_k}f\int\limits_{{{\left| m \right|}^2} = k} {{f^ \wedge }} (m){e^{imx}}. $$

((1))

where α is a constant parameter satisfying 0 ≤ α ≤ 4, has been vastly and intensively studied in the last decade because of the immense variety of its different solution types and the infinity of bifurcations occuring when α varies from 0 to 4. The solutions, corresponding to initial conditions x₀∈ [0,1], may be periodic, but nevertheless very complicated, or aperiodic, in some sense “chaotic”. The solutions are most erratic in the case of α = 4. This can be seen most directly by observing that the function

$$ {P_k}f\int\limits_{{{\left| m \right|}^2} = k} {{f^ \wedge }} (m){e^{imx}}. $$

is topologically conjugate to the “roof map” given by

$$ {P_k}f\int\limits_{{{\left| m \right|}^2} = k} {{f^ \wedge }} (m){e^{imx}}. $$

$$ {P_k}f\int\limits_{{{\left| m \right|}^2} = k} {{f^ \wedge }} (m){e^{imx}}. $$

namely

$$ {P_k}f\int\limits_{{{\left| m \right|}^2} = k} {{f^ \wedge }} (m){e^{imx}}. $$

((2))

(as noticed by Ulam and von Neumann, see (Stein, Ulam 1964)).