Normal form method of center-focus problem in piecewise-smooth systems and algorithm design
Abstract
We investigate the normal form method in piecewise-smooth monodromic systems to distinguish center from focus and determine the order of focus as well as degenerate Hopf bifurcations. For the normal form of piecewise-smooth monodromic systems given in previous publications in the sense of topologically equivalence, we prove the algebraic equivalence between the Lyapunov constants of the original piecewise-smooth system and the coefficient series of its normal form. With this algebraic equivalence, we prefer to compute the normal form coefficient series to avoid cumbersome integrals of trigonometric functions in the computation of Lyapunov constants, and find the relations of foci of the piecewise-smooth system and its subsystems: among the orders and among the stabilities. Moreover, we design algorithms for computing the normal form coefficient series and combine them with the obtained stability relation to construct stabilizing switching signals for state-dependent switched nonlinear systems, one basic problem in the control theory.
Related
- A sequential regularized piecewise affine algorithm for nonconvex nonsmooth multicomposite optimization in RNN training
- Global Linear Convergence of the Proximal Bundle Method under Unknown Piecewise Smoothness and Quadratic Growth
- Super-approximation and optimal L2 convergence of the stabilized BGN method with piecewise linear parametric finite elements for curve-shortening flow
- Stabilized SQP Method for Convex Piecewise Linear-Quadratic Composite Optimization Problems
- A stable rank-adaptive step-and-truncate finite volume method for Vlasov transport on domains with piecewise linear boundaries
BibTeX
@misc{chen2026normal,
title = {{Normal form method of center-focus problem in piecewise-smooth systems and algorithm design}},
author = {Xingwu Chen and Jiahao Li and Tao Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.17167},
eprint = {2607.17167},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.17167},
}
RIS
TY - UNPB
TI - Normal form method of center-focus problem in piecewise-smooth systems and algorithm design
AU - Xingwu Chen
AU - Jiahao Li
AU - Tao Li
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2607.17167
UR - https://arxiv.org/abs/2607.17167
AB - We investigate the normal form method in piecewise-smooth monodromic systems to distinguish center from focus and determine the order of focus as well as degenerate Hopf bifurcations. For the normal form of piecewise-smooth monodromic systems given in previous publications in the sense of topologically equivalence, we prove the algebraic equivalence between the Lyapunov constants of the original piecewise-smooth system and the coefficient series of its normal form. With this algebraic equivalence, we prefer to compute the normal form coefficient series to avoid cumbersome integrals of trigonometric functions in the computation of Lyapunov constants, and find the relations of foci of the piecewise-smooth system and its subsystems: among the orders and among the stabilities. Moreover, we design algorithms for computing the normal form coefficient series and combine them with the obtained stability relation to construct stabilizing switching signals for state-dependent switched nonlinear systems, one basic problem in the control theory.
ER -