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

The paper constructs three closed intervals I1 = [p0, α], I2 = [α, κ], I3 = [p1, R(0)] with pairwise disjoint interiors, verifies the covering relations I2 ∪ I3 ⊂ R(I1), I1 ⊂ R(I2), and I2 ∪ I3 ⊂ R(I3), and then invokes standard symbolic-dynamics results to obtain a closed invariant set J and a continuous surjection h: J → ΩA+ with h ∘ R = σ ∘ h; this is Theorem 29 and is accompanied by the explicit adjacency matrix A = [[0,1,1],[1,0,0],[0,1,1]] and its Perron eigenvalue (1+√5)/2 . Theorem 29 also states that each periodic orbit in the shift has a preimage periodic point for R, and provides the entropy lower bound log((√5+1)/2), yielding infinitely many periodic solutions of the Magomedov equation (19) via the return-map correspondence . The candidate solution reproduces the same Markov-interval coding argument: choose subintervals inside the Ii that map into the interiors of the target intervals, construct an invariant Cantor-like set J as nested pullbacks, define the itinerary map to ΩA+, deduce periodic points via closed paths in the Markov graph, derive the entropy lower bound, and pass these periodic points back to periodic solutions of (19). This matches the paper’s approach in substance. Minor issues: (i) the candidate’s explicit construction of the subinterval family {Cjk} omits the standard Markov covering property needed to guarantee nonempty cylinders for all finite admissible words; this can be repaired by a standard refinement or by citing the general theorem the paper relies on. (ii) Counting: both the paper and the candidate conflate “fixed points of Rn” (which are ≥ Tr A^n) with “n-periodic orbits”; strictly speaking, Tr A^n counts points fixed by σ^n (period dividing n), so the claim about “n-periodic orbits” should be qualified. Aside from this terminological/counting quibble, both arguments are essentially the same and correct on the main claims.