The peak-end rule and its dynamic realization through differential equations with maxima ^*

In the 1990s, after a series of experiments, behavioral psychologist and economist Daniel Kahneman his colleagues formulated following Peak-End evaluation rule: "The remembered utility pleasant or unpleasant episodes is accurately predicted by averaging Peak (most intense value) instant (or disutility) recorded during an episode near end experience", (D. et al., 1997, QJE, p. 381). Hence, simplest mathematical model for time evolution experienced function $u=u(t)$ can be given scalar differential equation $u'(t)=a u(t) + b \max \{u(s) : s\in [t-h,t]\}+f(t) \ (*),$ where $f$ represents exogenous stimuli, $h$ maximal duration experience, $a, \in \mathbb R$ are some weights. this work, we study $(*)$ show that, range parameters b, h$ periodic sine-like term $f$, dynamics completely described in terms associated one-dimensional dynamical system generated piece-wise continuous map from finite interval into itself. We illustrate our approach with two examples. particular, that hedonic $u(t)$ (`happiness') exhibit chaotic behavior.