Grazing and border-collision in piecewise-smooth systems: a unified analytical framework

Area of Science:

• Dynamical Systems and Control Theory
• Nonlinear Dynamics
• Mathematical Physics

Background:

• Piecewise smooth systems are crucial for modeling real-world processes with abrupt changes.
• Grazing bifurcations, where trajectories repeatedly approach a discontinuity boundary, are common in such systems.
• Understanding these bifurcations is key to predicting system behavior and stability.

Purpose of the Study:

• To provide a comprehensive derivation of normal form maps for grazing bifurcations.
• To establish a clear link between grazing phenomena and border-collision bifurcations in nonsmooth maps.
• To characterize the types of singularities that arise in these maps.

Main Methods:

• Derivation of normal form maps for grazing bifurcations.
• Analysis of piecewise smooth models in dynamical systems.
• Classification of singularities in nonsmooth maps, including square-root and (3/2)-type.

Main Results:

• A unified framework linking grazing bifurcations and border-collision bifurcations is presented.
• It is demonstrated that piecewise linear maps only produce nonsmooth discontinuity boundaries.
• All other piecewise smooth maps associated with grazing bifurcations exhibit either square-root or (3/2)-type singularities.

Conclusions:

• The derived normal form maps offer a powerful tool for analyzing complex behaviors in piecewise smooth systems.
• This work clarifies the nature of singularities in grazing bifurcations, correcting previous assumptions in the literature.
• The findings have implications for the design and control of physical systems exhibiting such dynamics.