Stochastic Properties of Simple Differential — Delay Equations

TL;DRAbstract

The logistic difference equation (1) $$ {P_k}f\int\limits_{{{\left| m \right|}^2} = k} {{f^ \wedge }} (m){e^{imx}}. $$ 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 (2) $$ {P_k}f\int\limits_{{{\left| m \right|}^2} = k} {{f^ \w

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