DISCRETE-TIME DYNAMICAL SYSTEMS GENERATED BY A QUADRATIC OPERATOR

S.K. Shoyimardonov, U.A. Rozikov

correcthigh confidence

Category: math.DS
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper defines the quadratic map H as in (1.1) and proves that every nonzero fixed point E has an eigenvalue 2 for the Jacobian J(E) (Theorem 8), yielding the corollary that no fixed point except the origin can be attracting. This is explicitly stated in the abstract and in Corollary 2. The candidate solution establishes the same conclusions via a short and general argument using Euler’s theorem for homogeneous functions (J(x) x = 2 H(x)), which directly implies J(E) E = 2 E and rules out attraction at E since |2|>1 for a discrete-time C^1 map. The paper’s proof is correct but more elaborate, while the model’s proof is concise and conceptually cleaner. Definition of H: . General Jacobian formula (2.16): . Theorem 8 (eigenvalue 2 at nonzero fixed point): . Corollary 2 (no attracting fixed point except the origin): .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript is mathematically correct and addresses a well-defined class of quadratic discrete-time operators. It identifies all fixed points, classifies their types in various dimensions, proves that the Jacobian at any nonzero fixed point has eigenvalue 2, and deduces the absence of attractive fixed points outside the origin. The presentation can be improved by explicitly stating the stability criterion for discrete-time systems and by including a short, general proof of the eigenvalue-2 fact via the homogeneity identity, thereby streamlining the exposition.