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On the problem of total stability for periodic differential systems. (English) Zbl 1237.34103

The authors consider in \({\mathbb R}^{n} \times {\mathbb R}^{n}\) a smooth \(\omega\)-periodic system of ordinary differential equations. Under explicitly stated conditions, it is shown that the total stability of an equilibrium of a periodic system is equivalent to the existence of a fundamental family of asymptotically stable compact neighborhoods of the equilibrium. This extends a known theorem of P. Seibert [Acta Mex. Cienc. Tecnol. 2, 154–165 (1968; Zbl 0215.15104)] on autonomous systems to periodic systems. Furthermore, by using this extension and in addition to some previously known results of the authors, it is further shown that if the equilibrium is totally stable on a smooth invariant manifold and some conditions are satisfied, then the equilibrium is unconditionally totally stable.

MSC:

34D20 Stability of solutions to ordinary differential equations
70K20 Stability for nonlinear problems in mechanics
37C60 Nonautonomous smooth dynamical systems

Citations:

Zbl 0215.15104
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