Finding a stable solution of a system of nonlinear equations arising from dynamic systems

Carl T. Kelley, Liqun Qi, Xiaojiao Tong, Hongxia Yin

Research output: Journal article publicationJournal articleAcademic researchpeer-review

2 Citations (Scopus)


In this paper, we present a new approach for finding a stable solution of a system of nonlinear equations arising from dynamical systems. We introduce the concept of stability functions and use this idea to construct stability solution models of several typical small signal stability problems in dynamical systems. Each model consists of a system of constrained semismooth equations. The advantage of the new models is twofold. Firstly, the stability requirement of dynamical systems is controlled by nonlinear inequalities. Secondly, the semismoothness property of the stability functions makes the models solvable by effcient numerical methods. We introduce smoothing functions for the stability functions and present a smoothing Newton method for solving the problems. Global and local quadratic convergence of the algorithm is established. Numerical examples from dynamical systems are also given to illustrate the effciency of the new approach.
Original languageEnglish
Pages (from-to)497-521
Number of pages25
JournalJournal of Industrial and Management Optimization
Issue number2
Publication statusPublished - 1 May 2011


  • Hopf bifurcation
  • Saddle-node bifurcation
  • Smoothing Newton method
  • Stability functions
  • Stable solutions
  • System of nonlinear equations

ASJC Scopus subject areas

  • Business and International Management
  • Strategy and Management
  • Control and Optimization
  • Applied Mathematics

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