On the polynomial integrability of the critical systems for optimal eigenvalue gaps

Abstract

This exploration consists of two parts. First, we will deduce a family of critical systems consisting of nonlinear ordinary differential equations, indexed by the exponent p ∈ (1, ∞) of the Lebesgue spaces concerned. These systems can be used to obtain the optimal lower or upper bounds for eigenvalue gaps of Sturm–Liouville operators and are equivalent to non-convex Hamiltonian systems of two degrees of freedom. Second, with appropriate choices of exponents p, the critical systems are polynomial systems in four dimensions. These systems will be investigated from two aspects. The first one is that by applying the canonical transformation and the Darboux polynomial, we obtain the necessary and sufficient conditions for polynomial integrability of these polynomial critical systems. As a special example, we conclude that the system with p = 2 is polynomial completely integrable in the sense of Liouville. The second is that the linear stability of isolated singular points is characterized. By performing the Poincaré cross section technique, we observe that the systems have very rich dynamical behaviors, including periodic trajectories, quasi-periodic trajectories, and chaos.