Darboux first integrals of Kolmogorov systems with invariant $n$-sphere

TL;DR

The paper addresses the problem of characterizing polynomial Kolmogorov vector fields that leave the standard sphere $\mathbb{S}^n$ invariant, and develops a canonical form for such fields of degree $m$ with explicit Darboux-type integrability results. It proves a complete algebraic structure: any degree $m$ Kolmogorov vector field on $\mathbb{S}^n$ has $P_i = x_i\big((1-\sum_{k=1}^{n+1} x_k^2)\tilde f_i + \sum_{j=1}^{n+1} \tilde A_{ij} x_j^2\big)$ with a skew-symmetric $\tilde A$, and establishes the existence of completely integrable examples for all $m\ge 3$ while showing a nonexistence result for cubic Hamiltonian cases on odd spheres. In the cubic case, the authors derive Darboux first integral criteria, relate invariant hypersurfaces to a rank condition on a matrix $B$, and prove that under rank$(B)\le 2$ there are $n$ independent first integrals, yielding complete integrability on suitable domains. These results clarify the structure of Kolmogorov systems on spheres and connect invariant geometry with integrability properties, providing explicit constructive examples.

Abstract

In this paper, we characterize all polynomial Kolmogorov vector fields for which the standard $n$-sphere is invariant. We exhibit completely integrable Kolmogorov vector fields of degree $m$ on $\mathbb{S}^n$ for any $m >2$. Then, we show that there is no cubic Hamiltonian Kolmogorov vector field that makes an odd-dimensional sphere invariant. We examine the conditions under which a cubic Kolmogorov vector field has a Darboux first integral. In many cases, we determine whether they constitute necessary and sufficient conditions. Moreover, we study the complete integrability of cubic Kolmogorov vector fields having an invariant $n$-sphere.

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Proposition 2.2
• Definition 2.3
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• Definition 2.5
• Lemma 2.6: jana2024dynamics