Classification of boundary equilibrium bifurcations in planar Filippov systems

Area of Science:

• Dynamical Systems Theory
• Bifurcation Analysis
• Differential Equations

Background:

• Piecewise smooth systems are crucial for modeling phenomena with abrupt changes.
• Boundary equilibrium bifurcations occur when system equilibria cross a switching surface.
• Existing classifications of these bifurcations have known omissions.

Purpose of the Study:

• To derive a normal form for boundary equilibrium bifurcations in planar systems.
• To provide a transparent method for classifying these bifurcations.
• To reconcile differing results in the literature regarding bifurcation cases.

Main Methods:

• Derivation of leading order terms for the normal form of boundary equilibrium bifurcations.
• Analysis of stationary points intersecting the switching surface.
• Comparison with established results from Filippov and Hogan et al.

Main Results:

• A complete classification of boundary equilibrium bifurcations for planar systems is achieved.
• The derivation confirms Filippov's classic results using novel methods.
• The 'missing' cases identified by Hogan et al. are precisely those omitted in recent analyses.

Conclusions:

• The derived normal form offers a clear framework for understanding boundary equilibrium bifurcations.
• This work provides a unified and transparent approach to classifying these critical events in dynamical systems.
• The study resolves discrepancies in the literature concerning the completeness of bifurcation classifications.