Normal forms of piecewise-smooth systems with a monodromic singular point

TL;DR

This paper develops a comprehensive normal form theory for piecewise-smooth systems with various equilibrium types, providing explicit formulas, generalizations to higher orders, and new methods for analyzing bifurcations and stability.

Contribution

It introduces a novel approach to derive explicit, any-order normal forms for piecewise-smooth systems, extending previous results and enabling new bifurcation and stability analyses.

Findings

Explicit normal forms for all orders and types of equilibria.

A new method for computing Lyapunov constants.

Applications to center problems and Hopf bifurcations.

Abstract

Normal form theory is developed deeply for planar smooth systems but has few results for piecewise-smooth systems because difficulties arise from continuity of the near-identity transformation, which is constructed piecewise. In this paper, we overcome the difficulties to study normal forms for piecewise-smooth systems with FF, FP, or PP equilibrium and obtain explicit any-order normal forms by finding piecewise-analytic homeomorphisms and deriving a new normal form for analytic systems. Our theorems of normal forms not only generalize previous results from second-order to any-order, from FF type to all FF, FP, PP types, but also provide a new method to compute Lyapunov constants, which are applied to solve the center problem and any-order Hopf bifurcations of piecewise-smooth systems.