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

The peak-and-end rule and differential equations with maxima: a view on the unpredictability of happiness