Abstract
This chapter deals with stability analysis of the origin, both for continuous-time and discrete-time systems. After a brief review of some well known results, a detailed study of the scalar case is carried out. Then, for planar systems, first, the connection between the center manifold theory and semi-invariants is pointed out; secondly, the critical cases for which it is possible to derive stability conditions that are easy to check are studied. It turns out that only some critical cases remain difficult to solve in general, in particular those with a zero linear part and a subclass of those with nilpotent but non-zero linear part (confirming the difficulty of distinguishing a center from a focus). Finally, the last sections deal with the use of semi-invariants as elementary bricks for the construction of Lyapunov functions; such methods are developed both for continuous-time and discrete-time systems of arbitrary order.
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Menini, L., Tornambè, A. (2011). Stability Analysis. In: Symmetries and Semi-invariants in the Analysis of Nonlinear Systems. Springer, London. https://doi.org/10.1007/978-0-85729-612-2_8
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DOI: https://doi.org/10.1007/978-0-85729-612-2_8
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