Stability boundary characterization of nonlinear autonomous dynamical systems in the presence of a type-zero saddle-node equilibrium point

F.M. Amaral, L. F. C. Alberto, N. G. Bretas

Abstract


Abstract. Under the assumption that all equilibrium points are hyperbolic, the stability boundary of nonlinear autonomous dynamical systems is characterized as the union of the stable manifolds of equilibrium points on the stability boundary. The existing characterization of the stability boundary is extended in this paper to consider the existence of non-hyperbolic equilibrium points on the stability boundary. In particular, a complete characterization of the stability boundary is presented when the system possesses a type-zero saddle-node equilibrium point on the stability boundary. It is shown that the stability boundary consists of the stable manifolds of all hyperbolic equilibrium points on the stability boundary and of the stable manifold of the type-zero saddle-node equilibrium point.

References


[1] L.F.C. Alberto, H. D. Chiang, Uniform Approach for Stability Analysis of Fast Subsystem of Two-Time Scale Nonlinear Systems, Int J Bifurcat Chaos Appl Sci Eng, 17, (2007), 4195–4203.

[2] H.D. Chiang, M. W. Hirsch, F.F. Wu, Stability region of nonlinear autonomous dynamical systems, IEEE Trans. on Automatic Control, 33, No. 1 (1988), 16–27.

[3] H.D. Chiang, F.Wu, P.P. Varaiya, Foundations of the potential energy boundary surface method for power transient stability analysis, IEEE Trans. on Circuits and Systems, 35, No. 6 (1988), 712–728.

[4] H.D. Chiang, L. Fekih-Ahmed, Quasi-stability regions of nonlinear dynamical systems: optimal estimations, IEEE Trans. on Circuits and Systems, 43 , No. 8 (1996), 636–643.

[5] J. Guckenheimer, P. Holmes, “Nonlinear Oscilations,Dynamical Systems and Bifurcations of Vector Fields”, Springer -Verlag, New York, 1983.

[6] V. Guillemin, A. Pollack, “Differential Topology”, Englewood Cliffs, NJ: Prentice-Hall, 1974.

[7] M. W. Hirsch, C.C. Pugh, M. Shub, Invariant manifolds, Bull. Amer. Math. Soc., 76, No. 5 (1970), 1015–1019.

[8] W. Hurewicz, H. Wallman, “Dimension Theory”, Princeton, NJ: Princeton University




DOI: https://doi.org/10.5540/tema.2010.011.02.0111

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Trends in Computational and Applied Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)

 

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