Stability criteria for second order linear ordinary differential equations

TL;DR

This paper establishes new stability criteria for second order linear ODEs using Riccati equation properties, highlighting differences from classical methods like WKB, Lyapunov, and others.

Contribution

It introduces novel stability conditions based on Riccati solutions that are not implied by traditional WKB applicability criteria.

Findings

Derived new boundedness and stability criteria for second order linear ODEs.

Compared these criteria with classical methods like Lyapunov, Bogdanov, and Wazevski.

Showed that the new conditions are independent of WKB-based assumptions.

Abstract

We use some properties of solutions of Riccati equation for establishing boundedness and stability criteria for solutions of second order linear ordinary differential equations. We show that the conditions on coefficients of the equations, appearing in the proven criteria, do not follow from the conditions, which ensure the application of the WKB approximation to the second order linear equations. On these examples we compare the obtained results wit the results obtained by the Liapunov and Bogdanov methods, by a method involving estimates of solutions in the Lozinski's logarithmic norms, and by the freezing method. We compare these results with the Wazevski's theorem as well.