An Efficient Symbolic Algorithm for Computing Lyapunov Constants in Planar Switching Systems

RESUMO

Lyapunov constants are a crucial tool for analyzing bifurcation behavior in switching systems. This paper proposes an efficient algebraic symbolic algorithm for computing Lyapunov constants in planar switching systems. By mapping the vector fields into the complex domain, we construct a normal form algorithm based on Laurent poly?nomials. The classical Poincaré map method requires repeated symbolic integration, which may lead to substantial computational difficulties at high orders. Our ap?proach improves computational efficiency by replacing continuous integration with direct algebraic operations. We apply this approach to investigate two switching models. For a generalized switching quartic Liénard system, the algebraic method extracts the center conditions and proves the existence of 10 small-amplitude limit cycles, establishing a new lower bound for its cyclicity. Furthermore, a physically motivated circuit-level modification of the Alpazur oscillator is investigated. We show that at most one small-amplitude limit cycle can bifurcate from the center and that this upper bound is attainable, illustrating the applicability of the proposed algorithm to a practical switching circuit. This application shows that our method provides a systematic framework for analyzing and interpreting complex dynamical phenomena arising in practical piecewise smooth systems.