Characteristics of Stability Boundary of Nonlinear Continuous Dynamical Systems
Main Article Content
Abstract
The theory of differential equations has been widely known and developed in recent years. One of the issues that many authors give their undivided attention to is the stability boundary of nonlinear dynamical systems. In this work, we first review several properties of equilibrium points on the stability boundary. We next extend the characteristics of the stability boundary for a fairly large class of nonlinear dynamical systems. These characteristics are the key to completely determine the stability boundary of nonlinear dynamical systems.
Keywords:
Dynamical systems, equilibrium point, stability boundary, stability region.
References
[1] F. M. Amaral, L. F. C. Alberto, Stability Boundary Characterization of Nonlinear Autonomous Dynamical Systems in the Presence of Saddle-Node Equilibrium Points, TEMA (São Carlos), Vol. 13, No. 2, 2012, pp. 143-154.
[2] H. D. Chiang, Direct Methods for Stability Analysis of Electric Power Systems: Theoretical Foundation, BCU Methodologies, and Applications, John Wiley & Sons, Hoboken, 2011.
[3] C. W. Liu, S. Thorp, A Novel Method to Compute the Closest Unstable Equilibrium Point for Transient Stability Region Estimate in Power Systems, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Vol. 44, No. 7, 1997, pp. 630-635.
[4] L. Luyckx, M. Loccufier, E. Noldus, Computational Methods in Nonlinear Stability Analysis: Stability Boundary Calculations, Journal of Computational and Applied Mathematics, Vol. 168, No. 1-2, 2004, pp. 289-297.
[5] L. Ya. Adrianova, Introduction to Linear Systems of Differential Equations, American Mathematical Society, Providence, 1995.
[6] J. Lee, Introduction to Topological Manifolds, Springer, Berlin, 2010.
[7] M. W. Hirsch, Differential topology, Springer Science & Business Media, Berlin, 2012.
[8] H. D. Chiang, L. F. C Alberto, Stability Regions of Nonlinear Dynamical Systems: Theory, Estimation, and Applications, Cambridge University Press, Cambridge, 2015.
[9] H. D. Chiang, M. W. Hirsch, F. F. Wu, Stability Regions of Nonlinear Autonomous Dynamical Systems, IEEE Transactions on Automatic Control, Vol. 33, No. 1, 1988, pp. 16-27.
[10] J. Milnor, D. W. Weaver, Topology from the Differentiable Viewpoint, Princeton University Press, Princeton, 1997.
[11] T. J. Koo, H. Su, A Computational Approach for Estimating Stability Regions, 2006 IEEE International Symposium on Intelligent Control, 2006, pp. 62-68.
[12] A. N. Milchel, N. R. Sarabudla, R. K. Miller, Stability Analysis of Complex Dynamical Systems: Some Computational Methods, Circuits, Systems and Signal Processing, Vol 1, No. 2, 1982, pp. 171-202.
[13] H. D. Chiang, J. S. Thorp, Stability Regions of Nonlinear Dynamical Systems: A Constructive Methodology, IEEE Transactions on Automatic Control, Vol. 34, No. 12, 1989, pp. 1229-1241.
[14] S. Osher, R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer, New York, 2005.
[2] H. D. Chiang, Direct Methods for Stability Analysis of Electric Power Systems: Theoretical Foundation, BCU Methodologies, and Applications, John Wiley & Sons, Hoboken, 2011.
[3] C. W. Liu, S. Thorp, A Novel Method to Compute the Closest Unstable Equilibrium Point for Transient Stability Region Estimate in Power Systems, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Vol. 44, No. 7, 1997, pp. 630-635.
[4] L. Luyckx, M. Loccufier, E. Noldus, Computational Methods in Nonlinear Stability Analysis: Stability Boundary Calculations, Journal of Computational and Applied Mathematics, Vol. 168, No. 1-2, 2004, pp. 289-297.
[5] L. Ya. Adrianova, Introduction to Linear Systems of Differential Equations, American Mathematical Society, Providence, 1995.
[6] J. Lee, Introduction to Topological Manifolds, Springer, Berlin, 2010.
[7] M. W. Hirsch, Differential topology, Springer Science & Business Media, Berlin, 2012.
[8] H. D. Chiang, L. F. C Alberto, Stability Regions of Nonlinear Dynamical Systems: Theory, Estimation, and Applications, Cambridge University Press, Cambridge, 2015.
[9] H. D. Chiang, M. W. Hirsch, F. F. Wu, Stability Regions of Nonlinear Autonomous Dynamical Systems, IEEE Transactions on Automatic Control, Vol. 33, No. 1, 1988, pp. 16-27.
[10] J. Milnor, D. W. Weaver, Topology from the Differentiable Viewpoint, Princeton University Press, Princeton, 1997.
[11] T. J. Koo, H. Su, A Computational Approach for Estimating Stability Regions, 2006 IEEE International Symposium on Intelligent Control, 2006, pp. 62-68.
[12] A. N. Milchel, N. R. Sarabudla, R. K. Miller, Stability Analysis of Complex Dynamical Systems: Some Computational Methods, Circuits, Systems and Signal Processing, Vol 1, No. 2, 1982, pp. 171-202.
[13] H. D. Chiang, J. S. Thorp, Stability Regions of Nonlinear Dynamical Systems: A Constructive Methodology, IEEE Transactions on Automatic Control, Vol. 34, No. 12, 1989, pp. 1229-1241.
[14] S. Osher, R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer, New York, 2005.