Stability Analysis of Nonlinear Ordinary Differential Equations: Derivations, Comparative Criteria, and Geometric Illustrations; Article Review

The local and global stability problems of the nonlinear ordinary differential equation are discussed in a common mathematical perspective with a critical consideration. These linearization test, direct method of Lyapunov, estimates of exponential decays and the invariance principle of LaSalle are developed and contrasted under the corresponding conditions. Scalar and planar examples are presented that show the difference between stability, attractivity, asymptotic stability and exponential convergence. The theoretical results are complemented with complementary geometric interpretations in the form of analytical phase portraits. Special emphasis is placed on the insufficient stability tests, the interpretation of negative semidefinite Lyapunov derivative, and the extra conditions needed to impose a global stability. The review shows that sometimes a failure of a sufficient condition is not necessarily a sign of instability and convergence cannot be deduced from a non-increasing Lyapunov function. Last, the advantages and disadvantages of the methods investigated are summarized and future research directions involving the use of computational stability certificates, uncertain systems and systematic literature assessment are introduced to further mathematical research and methodological development.