Generalization to differential–algebraic equations of Lyapunov–Schmidt type reduction at Hopf bifurcations

• Published in: Communications in nonlinear science & numerical simulation 2024-04, Vol.131, p.107833, Article 107833
• Main Author: Ehrenstein, Uwe
• Format: Article
• Language: English
• Subjects: Differential–algebraic equations; Hopf bifurcation; Lyapunov–Schmidt reduction; Nonlinear Sciences
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2024.107833

The Lyapunov–Schmidt procedure, a well-known and powerful tool for the local reduction of nonlinear systems at bifurcation points or for ordinary differential equations (ODEs) at Hopf bifurcations, is extended to the context of strangeness-free differential–algebraic equations (DAEs), by generalizing the comprehensive presentation of the method for ODEs provided in the classical textbook by Golubitsky and Schaeffer (1985). The appropriate setting in the context of DAEs at Hopf bifurcations is first detailed, introducing suitable operators and addressing the question of appropriate numerical algorithms for their construction as well. The different steps of the reduction procedure are carefully reinterpreted in the light of the DAE context and detailed formulas are provided for systematic and rational construction of the bifurcating local periodic solution, whose stability is shown, likely to the ODE context, to be predicted by the reduced equations. As an illustrative example, a classical DAE model for an electric power system is considered, exhibiting both supercritical and subcritical Hopf bifurcations, demonstrating the prediction capability of the reduced system with regard to the global dynamics. •Explicit formulas for Lyapunov–Schmidt reduction of strangeness-free DAEs.•Stability prediction for periodic bifurcating solutions of nonlinear DAEs.•Application to Hopf bifurcations for a real power-system DAE model.