Stability by linear approximation for time scale dynamical systems
The paper focuses on the matter of stability for linear systems and small nonlinear perturbations of such systems on time scales. Firstly, the stability of a solution is obtained by applying a time scale version of the Gronwall-Bellman inequality, by first approximation. It is proved afterwards that a linear time scale system with bounded, uniformly regressive matrix of coefficients, can be embedded into a linear system of ODEs with a bounded matrix. Finally, an analog of Chetaev's theorem on instability by first approximation is provided for time scale systems. The authors have thus proven that when studying stability/instability, methods from the theory of non-autonomous linear ODEs can be used in the framework of time scale dynamical systems.
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